SOLSTICE:  AN ELECTRONIC JOURNAL OF GEOGRAPHY AND MATHEMATICS.
Volume IV, Number 1.  Summer, 1993.
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\def\righthead{\sl\hfil SOLSTICE }
\def\lefthead{\sl Summer, 1993 \hfil}
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\centerline{\big SOLSTICE:}
\vskip.5cm
\centerline{\bf AN ELECTRONIC JOURNAL OF GEOGRAPHY AND MATHEMATICS}
\vskip5cm
\centerline{\bf SUMMER, 1993}
\vskip12cm
\centerline{\bf Volume IV, Number 1}
\smallskip
\centerline{\bf Institute of Mathematical Geography}
\vskip.1cm
\centerline{\bf Ann Arbor, Michigan}
\vfill\eject
\hrule
\smallskip
\centerline{\bf SOLSTICE}
\line{Founding Editor--in--Chief:
{\bf Sandra Lach Arlinghaus}. \hfil}
\smallskip
\centerline{\bf EDITORIAL BOARD}
\smallskip
\line{{\bf Geography} \hfil}
\line{{\bf Michael Goodchild},
University of California, Santa Barbara. \hfil}
\line{{\bf Daniel A. Griffith},
Syracuse University. \hfil}
\line{{\bf Jonathan D. Mayer},
University of Washington;
joint appointment in School of Medicine.\hfil}
\line{{\bf John D. Nystuen},
University of Michigan
(College of Architecture and Urban Planning).\hfil}
\smallskip
\line{{\bf Mathematics} \hfil}
\line{{\bf William C. Arlinghaus},
Lawrence Technological University. \hfil}
\line{{\bf Neal Brand},
University of North Texas. \hfil}
\line{{\bf Kenneth H. Rosen},
A. T. \& T. Bell Laboratories. \hfil}
\smallskip
\line{{\bf Engineering Applications} \hfil}
\line{{\bf William D. Drake},
University of Michigan, \hfil}
\smallskip
\line{{\bf Education} \hfil}
\line{{\bf Frederick L. Goodman},
University of Michigan, \hfil}
\smallskip
\line{{\bf Business} \hfil}
\line{{\bf Robert F. Austin, Ph.D.} \hfil}
\line{President, Austin Communications Education Services \hfil}
\smallskip
\hrule
\smallskip

The purpose of {\sl Solstice\/} is to promote  interaction
between geography and mathematics.   Articles in which  elements
of   one  discipline  are used to shed light on  the  other  are
particularly sought.   Also welcome,  are original contributions
that are purely geographical or purely mathematical.   These may
be  prefaced  (by editor or author) with  commentary  suggesting
directions  that  might  lead toward  the  desired  interaction.
Individuals  wishing to submit articles,  either short or full--
length,  as well as contributions for regular  features,  should
send  them,  in triplicate,  directly to the  Editor--in--Chief.
Contributed  articles  will  be refereed by  geographers  and/or
mathematicians.   Invited articles will be screened by  suitable
members of the editorial board.  IMaGe is open to having authors
suggest, and furnish material for, new regular features.

The opinions expressed are those of the authors, alone, and the
authors alone are responsible for the accuracy of the facts in
the articles.
\smallskip
\noindent {\bf Send all correspondence to:
Institute of Mathematical Geography, 2790 Briarcliff,
Ann Arbor, MI 48105-1429, (313) 761-1231, IMaGe@UMICHUM,
Solstice@UMICHUM}
\smallskip
Suggested form for citation.   If  standard  referencing  to the
hardcopy in the  IMaGe Monograph Series is not used (although we
suggest that reference  to that  hardcopy be included along with
reference  to  the  e-mailed  copy  from which  the hard copy is
produced), then we suggest the following  format for citation of
the electronic copy.  Article,  author, publisher (IMaGe) -- all
the usual--plus a notation as to the time marked electronically,
by the process of transmission,  at the  top  of the  recipients
copy.   Note  when  it was sent from Ann Arbor (date and time to
the  second)  and  when  you  received  it (date and time to the
second)  and  the  field characters covered by the article  (for
example FC=21345 to FC=37462).
\vfill\eject

This  document is produced using the typesetting  program,
{\TeX},  of Donald Knuth and the American Mathematical  Society.
Notation  in  the electronic file is in accordance with that  of
Knuth's   {\sl The {\TeX}book}.   The program is downloaded  for
hard copy for on The University of Michigan's Xerox 9700 laser--
printing  Xerox  machine,  using IMaGe's commercial account with
that University.

Unless otherwise noted, all regular features"  are  written by
the Editor--in--Chief.
\smallskip
{\nn  Upon final acceptance,  authors will work with IMaGe
to    get  manuscripts   into  a  format  well--suited  to   the
requirements   of {\sl Solstice\/}.  Typically,  this would mean
that  authors    would  submit    a  clean  ASCII  file  of  the
manuscript,  as well as   hard copy,  figures,  and so forth (in
camera--ready form).     Depending on the nature of the document
and   on   the  changing    technology  used  to  produce   {\sl
Solstice\/},   there  may  be  other    requirements  as   well.
Currently,  the  text  is typeset using   {\TeX};  in that  way,
mathematical formul{\ae} can be transmitted   as ASCII files and
downloaded   faithfully   and   printed   out.    The     reader
inexperienced  in the use of {\TeX} should note that  this    is
not  a what--you--see--is--what--you--get"  display;  however,
we  hope  that  such readers find {\TeX} easier to  learn  after
exposure to {\sl Solstice\/}'s e-files written using {\TeX}!}

{\nn  Copyright  will  be taken out in  the  name  of  the
Institute of Mathematical Geography, and authors are required to
transfer  copyright  to  IMaGe as a  condition  of  publication.
There are no page charges; authors will be given  permission  to
make reprints from the electronic file,  or to have IMaGe make a
single master reprint for a nominal fee dependent on  manuscript
length.   Hard  copy of {\sl Solstice\/} is  available at a cost
of \$15.95 per year (plus shipping and handling; hard copy is issued once yearly, in the Monograph series of the Institute of Mathematical Geography. Order directly from IMaGe. It is the desire of IMaGe to offer electronic copies to interested parties for free. Whether or not it will be feasible to continue distributing complimentary electronic files remains to be seen. Presently {\sl Solstice\/} is funded by IMaGe and by a generous donation of computer time from a member of the Editorial Board. Thank you for participating in this project focusing on environmentally-sensitive publishing.} \vskip.5cm \copyright Copyright, June, 1993 by the Institute of Mathematical Geography. All rights reserved. \vskip1cm {\bf ISBN: 1-877751-55-3} {\bf ISSN: 1059-5325} \vfill\eject \centerline{\bf TABLE OF CONTENT} \smallskip \noindent{\bf 1. WELCOME TO NEW READERS} \smallskip \noindent{\bf 2. PRESS CLIPPINGS---SUMMARY} \smallskip \noindent{\bf 3. GOINGS ON ABOUT ANN ARBOR--ESRI AND IMaGe GIFT} \smallskip \noindent{\bf 4. ARTICLES} \smallskip\noindent \noindent{\bf Electronic Journals: Observations Based on Actual Trials, 1987-Present} \noindent {\bf Sandra L. Arlinghaus and Richard H. Zander}. \noindent Abstract; Content issues; Production issues; Archival issues; References. \smallskip \noindent {\bf Wilderness As Place} \noindent {\bf John D. Nystuen} \noindent Visual paradoxes; Wilderness defined; Conflict or synthesis; Wilderness as place; Suggested readings; Sources; Visual illusion authors. \smallskip \noindent {\bf The Earth Isn't Flat. And It Isn't Round Either: Some Significant and Little Known Effects of the Earth's Ellipsoidal Shape} \noindent {\bf Frank E. Barmore} \noindent reprinted from {\sl The Wisconsin Geographer\/}. \noindent Abstract; Introduction; The Qibla problem; The geographic center; The center of population; Appendix; References \smallskip \noindent {\bf Microcell Hex-nets?} \noindent {\bf Sandra Lach Arlinghaus} \noindent Introduction; Lattices; Microcell hex-nets; References. \smallskip \noindent {\bf Sum Graphs and Geographic Information} \noindent {\bf Sandra L. Arlinghaus, William C. Arlinghaus, Frank Harary} \noindent Abstract; Sum graphs; Sum graph unification: construction; Cartographic application of sum graph unification; Sum graph unification: theory; Logarithmic sum graphs; Reversed sum graphs; Augmented reversed logarithmic sum graphs; Cartographic application of ARL sum graphs; Summary \vskip.5cm \smallskip\noindent {\bf 5. DOWNLOADING OF SOLSTICE} \smallskip \noindent{\bf 6. INDEX to Volumes I (1990), II (1991), and III (1992) of {\sl Solstice}.} \smallskip \noindent{\bf 7. OTHER PUBLICATIONS OF IMaGe } \vfill\eject \centerline{\bf 1. WELCOME TO NEW READERS} Welcome to new subscribers! We hope you enjoy participating in this means of journal distribution. Instructions for downloading the typesetting have been repeated in this issue, near the end. They are specific to the {\TeX} installation at The University of Michigan, but apparently they have been helpful in suggesting to others the sorts of commands that might be used on their own particular mainframe installation of {\TeX}. New subscribers might wish to note that the electronic files are typeset files---the mathematical notation will print out as typeset notation. For example, $$\Sigma_{i=1}^n$$ when properly downloaded, will print out a typeset summation as i goes from one to n, as a centered display on the page. Complex notation is no barrier to this form of journal production. Many thanks to the members of the Editorial Board of {\sl Solstice\/}. Some of them have refereed articles and offered suggestions, as have others. Thanks to all. \vskip.5cm \centerline{\bf 2. PRESS CLIPPINGS---SUMMARY} \noindent Brief write-ups about {\sl Solstice\/} have appeared in the following publications: \noindent 1. {\bf Science}, Online Journals" Briefings. [by Joseph Palca] 29 November 1991. Vol. 254. \smallskip \smallskip \noindent 2. {\bf Science News}, Math for all seasons" by Ivars Peterson, January 25, 1992, Vol. 141, No. 4. \smallskip \smallskip \noindent 3. {\bf Newsletter of the Association of American Geographers}, June, 1992. \smallskip \noindent 4. {\bf American Mathematical Monthly}, Telegraphic Reviews" --- mentioned as one of the World's first electronic journals using {\TeX}," September, 1992. \smallskip \noindent 5. {\bf Harvard Technology Window}, 1993. \smallskip \noindent 6. {\bf Graduating Engineering Magazine}, 1993. If you have read about {\sl Solstice\/} elsewhere, please let us know the correct citations (and add to those above). Thanks. \vfill\eject \centerline{\bf 3. GOINGS ON ABOUT ANN ARBOR} 1. ESRI, negotiating with IMaGe, has agreed to give a University Lab Kit to the University of Michigan, to be housed in the School of Education. All here are very happy and thank ESRI for their generosity. We look forward to pursuing the research projects that we explained to ESRI. Bob Austin, Sandy Arlinghaus, John Nystuen, Fred Goodman, and Bill Drake were all involved in various aspects of developing research and educational projects. 2. In the Fall of 1992, Bill Drake taught a course in Transition Theory" (and invited Sandy Arlinghaus to co-teach it) in the School of Natural Resources and the Environment. It was quite popular, and this course that was experimental in nature in 1992-93 has just become part of the permanent graduate curriculum. A monograph written primarily by the students, and published by SNR and E, came from that course. 3. Book co-edited and co-authored by Bill Drake. {\sl Population --- Environment Dynamics\/}, edited by Gayl D. Ness, William D. Drake, and Steven R. Brechin, Ann Arbor: The University of Michigan Press, 1993. This book has 15 chapters organized into four sections plus a final section Summary, conclusions, and next steps" by the editors. It also has a Reference listing, information about the contributing authors, and an index. The book is 456 pages and costs \$45.
The titles of the four dominant sections are:

Global Perspectives:
History, Ideas, Sectoral Changes, and Theories.

The State as Actor:
Population --- Environment Dynamics in Large Collectivities.

The State as Environment:
Population --- Environment Dynamics in Small Communities.

Emergent Ideas:
Theory and Method.

4. Fred Goodman of the School of Education has been very helpful
in  finding  space  and  resources  so  that  IMaGe can give the
software  it's  trying  to  line  up  to  UM.   Fred  has   been
instrumental  in  providing constructive, diplomatic liason with
other  units  within  UM.   We  also  welcome  Fred  to the {\sl
Solstice\/} Board with this issue.
\vfill\eject
\centerline{\bf 4.  ARTICLES}
\smallskip
\centerline{\bf ELECTRONIC JOURNALS:}
\centerline{\bf OBSERVATIONS BASED ON ACTUAL TRIALS, 1987-PRESENT}
\vskip.5cm
\centerline{\bf Sandra L. Arlinghaus and Richard H. Zander.$^*$}
\vskip.5cm

\noindent{\bf ABSTRACT}

Electronic journals offer a 21st-century forum for the interchange
of scholarly ideas.  They are  inexpensive,  fast,  easy to store,
easy  to  search,  and  they  have  long-term archivability; these
advantages easily justify the time  spent  learning  to  deal with
the new technology.  The authors, both editors of nationally-noted
electronic  journals,  share  with  others their interdisciplinary
experiences in dealing with this new medium  for producing online,
refereed journals.
\vskip1cm

During  the  past  six  years  each of us has created and edited a
successful electronic journal (E-journal) in our respective fields
of geography ({\sl Solstice:   An  Electronic Journal of Geography
and  Mathematics\/}  first  appeared in June of 1990) (Palca 1991;
Peterson 1992) and botany  ({\sl Flora Online\/} first appeared in
January of 1987)  (Palca 1991).   Both journals are peer-reviewed;
both  are  available,  free, over standard computer networks; and,
both  have  editors  who served as authors in early issues--to get
the journal off the ground.  E-journals provide an  opportunity to
share  computerized  information  with  others  in  an orderly and
responsible  fashion,  within  the  context of current technology.
They offer:

\item{1.}
An inexpensive way to share information,  quickly,  with  a  large
number of individuals;
\item\item{a.}
As direct, online,  transmissions  from  editor  to individual; in
this case,  the  transmission  should be free of charge,  in  much
the   way   that  a  library   card   is   free   of  charge.  The
editor/publisher   bears   the   cost   of  journal  creation  and
manufacture;  the  reader  bears the cost of maintaining on online
mail box;
\item\item{b.}
As  direct  transmissions  to  libraries -- libraries  should  pay
for diskettes, hard copy, online transmission,  or  whatever  they
desire.  The  cost to the library  is  generally  greatly  reduced
from  that  of  conventional  journals,  thereby  freeing  library
funds for other useful projects.  Funds generated from this source
may make the E-journal(s) self-sustaining;
\item\item{c.}
As posted messages" on an electronic bulletin board or files  on
an anonymous FTP" server. The reader bears the cost of accessing
the board or server and downloading the article.

\item{2.}
When E-journals are highly specialized, they can serve  as  a more
formal alternative to large (archived) data banks  in the  natural
sciences  and elsewhere.  Indeed, when the E-journal is downloaded
into  a  wordprocessor  or  a  data  manager,  the  content can be
manipulated and edited carefully to  fit the research needs of the
individual user.

There  are  many  systematic   electronic  communications  already
available and there  are  apparently  more in the planning stages.
The  first  edition  of   Michael  Strangelove's   Directory  of
Electronic Journals and Newsletters"  (1991)  catalogues  about 30
journals  and  over  60  newsletters.   Major  academic societies,
notably  the  American  Mathematical  Society   and  the  American
Association  for  the  Advancement  of  Science,   have  announced
far-reaching plans to produce other electronic journals;   (Janusz
1991; Palca, 1991:  1480).   A  glance  at  a flyer for the Annual
Meeting  for  the  Society  for  Scholarly  Publishing (July 1992)
suggests  that  more  than  half of  the four-day meetings will be
devoted to issues related to electronic publication.

There are:
\item{1.}
Genuine" electronic journals.
\item{2.}
Mere computerized versions of hardcopy titles.
\item{3.}
Non-archived electronic databases that are not really citable in a
scientific paper since the data used may have been  changed or may
no  longer  be  available,  even  though  these  databases  may be
copyrighted.

What makes a systematic electronic communication a journal" is a
difficult issue (Ni\-chol\-son 1992); concern for  rigid,   {\sl a
priori\/}, definition might better  be  replaced  with open regard
for  all  entries  and  suitable  concern  for the broad issues of
journal  production.   For,  an  E-journal is first and foremost a
journal" that has simply been  {\bf modified}  as electronic,"
both  linguistically  and  technologically,  by  the method of its
transmission and production.

Thus,  we offer  a  generalized summary  of observations that have
come from six years of actual trials with {\sl Flora Online\/} and
three years of actual trials with {\sl Solstice\/}.   It is useful
to separate these results into three broad categories:
content issues, production issues, and archival issues.
\vskip.5cm
\noindent{\bf Content issues.}

The most important concern is to obtain good manuscripts.  And, to
be  acceptable  as an outlet for scholarly publication, E-journals
should  approximate  standard  formats  for professional journals,
have high standards of scholarship, and  be refereed.  It does not
matter  how sophisticated the technological production becomes; if
the journal  does not have interesting and useful material of high
quality, it will fail.   This point should be obvious; however, it
can  become  obscured,  particularly  in  light  of  the  exciting
capability of the computerized format.

Thus, author perceptions of E-journals are critical; the most
serious problem involves citation.  Will others see the work? Will
the work be taken seriously?   The following strategies help:

\item{1.}
the editor should see to it that the E-journal (and when necessary
hard  copy  derived  from  it)  is listed,  housed,  or  otherwise
recognized in

\item\item{a.}
standard  reviews  that  are  specific  to the  discipline  of the
journal;
\item\item{b.}
the usual  indexing services (publications are often judged by the
bibliographic and citation services that mention them --- services
that  accept electronic files are particularly easy to deal with);
\item\item{c.}
news media, including field-specific  conferences  and meetings as
well as mass media;
\item\item{d.}
standard  book/journal  registers of documents  using conventional
book/journal codes (such as ISBN and ISSN); and,
\item\item{e.}
library  archives.   Libraries  apparently  dislike  the  idea  of
downloading  journals;  they  appreciate diskettes mailed to them.
Archiving  is  important  for  E-journals  so  that  data  can  be
retrieved long after publication.

\item{2.}
The  editor  should  consider  the  unusual  to  boost  regard and
readership for this mode of journal transmission, such as:

\item\item{a.}
the  use  of  reprints (with appropriate copyright permission)  of
hard-to-find  works  of  field-leaders  (prospective  authors---of
lesser  fame---usually  perceive some  benefit-by-association  and
field leaders often are interested in participating in a different
venture);
\item\item{b.}
the  use  of  interactive  review of material --- post-publication
review followed by online alteration of the original document as a
later version (coded appropriately--original  is  version  1.0 and
updates carry larger numbers according to the extent of change);
\item\item{c.}
the  use  of  taxonomic,  bibliographic,   and  other   data  sets
consisting  of long lists of records that can easily be downloaded
and sorted according to user need.  Several agencies are preparing
monolithic  data  banks from  which scientists can  extract  items
of   information   using  specialized  data  management  programs.
Unfortunately,  such data banks usually employ in  each  different
management  system,   complex  and  difficult for the scientist to
learn, and the data banks give second-hand data (digested by those
who  run  the  data  bank and who are not necessarily scientists).
With the advent, however, of electronic publishing, information in
the sciences developed by individual  scientists can now be easily
and directly shared;
\item\item{d.}
the use of novel typesetting or other electronic capabilities that
display the power of the vehicle of transmission (Horstmann 1991);
and,
\item\item{e.}
the sharing of experiences in E-journal editorship with others ---
through  professional  associations directly promoting  electronic
journal  editorship  (such  as an E-journal editor's  association)
and with other organizations indirectly promoting it  (such as the
{\TeX\/}  Users' Group; {\TeX\/}" is a trademark of The American
Mathematical Society).

Readers  who  are initial skeptics can become  more receptive when
they see actual output; hence, the early need for editor to become
author.  To increase E-journal availability, and to convert a wide
variety of skeptics, E-journals should be distributed in more than
one manner (e.g.   diskette, File Transfer Protocol (FTP), Bitnet,
on a listserv, U.S. mail, hardcopy).

When editor becomes author, then a mechanism for review is all
the more important.  Pre-publication  peer-review by an  editorial
board  or  by  other  colleagues  is effective and easy to achieve
electronically;  post-publication  feedback  in  an open or closed
forum is also simple electronically.  In addition, it is important
that the editor continue to publish in various  other outlets held
in high regard.

There  are  also a number of other reasonable, but less important,
concerns that authors might have.  These include:

\item{1.}
Manuscript  security;  because E-journals can be forwarded easily,
alteration of original  manuscripts can occur.  There are a number
of ways to deal with this problem:

\item\item{a.}
Copyright  a  hard  copy  of  the  original  transmission (thereby
placing it in the Library of Congress);
\item\item{b.}
Advertise that the original computerized version or a hard copy of
the  original  transmission  is  available  (on-demand)  to  those
wishing it--including libraries;
\item\item{c.}
Store single hard copies in selected libraries (including  that of
the author's institution);
\item\item{d.}
Transmit  E-copies  directly  from  editor  to  individuals,  over
standard  electronic  networks,  using  an electronic distribution
list automatically marked with the sender's name and time;
\item\item{e.}
Download  from  an  electronic bulletin board.  A persistent worry
here  is  that  a  file  made  available  for  downloading  is not
published"  in  the  sense  of  being  distributed.   This worry
underscores, again, the need for  adequate  reviewing and indexing
of the document.  However, the prospective author should note that
a  file  made  available  for  downloading  is in  fact  published
because
\item\item{\quad i.}
this  is  the  same  way hardcopy books are published --- they are
simply advertised as available for purchase, and
\item\item{\quad ii.}
in  bibliographic  research,  the  date of publication is the date
advertised as available,  since  it  is  impossible  to track down
the date of first purchase or first mailing of the book.
\item\item{f.}
Copy-protect  diskettes  (using  some  sort  of seal unique to the
journal) to prevent unthinking abuse.

\item{2.}
Virus and other crank programming prevention.   Downloaded  FTP or
regular  phone  modem  files  from  other   computers  can  spread
electronic viruses if they are executable," and only if they are
actually   run   as   programs.   Downloaded  text  files   cannot
spread viruses: downloaded executable files (.EXE, .COM in MS-DOS)
can be examined by commercial programs for viruses before they are
run.   When the E-journal is  made  available  through  a  network
server,  the  E-journal's  health is simply transferred elsewhere;
the network  supervisor  has  considerable  responsibility in this
regard. Of course, good backup habits and a procedure in place for
dealing  with viruses if they  happen are a must in all workplaces
that use programs obtained from outside the workplace.
\vskip.5cm
\noindent{\bf Production issues.}

Production  issues  generally  appear  to  fall  into  one  of two
categories:   Document manufacture  and editing, and transmission.
Warehousing is not  an issue of  any significance, nor is the sort
of    marketing    that    requires   a   network  of  publishers'
representatives to sell hardcopy documents.

\centerline{\sl Document manufacture and editing.}

The manufacture  involves creating,  or being supplied, electronic
files.  Editing at this stage in journal computerization generally
requires in-house  manufacture and distribution of files and their
media.   It  is  useful  to aim for the lowest common denominator:
currently,  that means ASCII text  and .GIF or .PCX graphics files
if needed --- such  files  are  easily  read on a IBM PC clone,  a
Mackintosh,  or  Unix  machine  (Xwindows  or  whatever),  by  any
wordprocessor  and most graphic file viewers.  It would be nice if
the files could be set up with the format of one of  the  new  GUI
wordprocessors  (e.g. WordPerfect, MSWord) but  it  seems  prudent
to  wait  until  a  multiplatform  wordprocessor that creates text
files incorporating graphics images becomes commonly  used.   Most
prospective  authors  can  provide  manufacture-ready"  copy  in
the  form  of  an  ASCII  file sent over the e-mail or provided on
diskette.  Indeed,  for MS-DOS  environments DCA or RTF  (Document
Content  Architecture  or Rich Text Format) are also standard file
formats  retaining  formatting  commands;  these  may  be  used to
transfer a formatted text to any of  most  commercially  available
major  word  processors.  It  is  thus an easy matter  to ship the
E-file to referees and to provide  authors  with E-proof  to check
prior to final production.
\vfill\eject

There  are  a  number  of  issues,  found  also   in  conventional
publishing,  that remain difficult.  For this reason (also), it is
useful for editors  to  be  experienced as authors of conventional
articles;  it is  additionally  desirable  for  them  to  have had
editorial experience in dealing with a conventional publisher.

\item{1.}
When  the  ASCII  file  is  typeset  using  {\TeX\/}, mathematical
notation,  tables,  and  figures that are rectilinear in shape are
easy  to  handle;  otherwise,  complex  mathematical  notation  is
difficult even to approximate in ASCII.  The typeset {\TeX\/} file
is itself an ASCII file with ASCII formatting commands, and so can
be transmitted easily.

\item\item{a.}
The computerized typeset {\TeX\/} file is not strictly what-you-
see-is-what-you-get"; however, the file is of  traditional quality
typesetting, and the file of electronic text  and notation can now
be downloaded and cheaply typeset or  printed in  hard copy by the
journal  receiver at his or her expense.  To typeset the file, the
receiver  must  first  convert  the transmitted {\TeX\/} file to a
.DVI  file  and  then print it on any available downloading device
(such as a Xerox 9700 series machine or an APS phototypesetter).
\item\item{b.}
The  receiver  can  view  the  transmitted {\TeX\/} file on screen
(with the {\TeX\/} commands visible). The editor can right-justify
the {\TeX\/} file in a word processor (prior to transmission), and
bitstrip it to retain it as an ASCII file, in order  to  produce a
journal-like  electronic  page  in  the transmitted E-file without
interfering with (or influencing) the typesetting of the hardcopy.
Right-justified  electronic copy tends to reduce the visual impact
of the unnatural  looking  typesetting commands that appear in the
{\TeX\/} file as it is viewed online.
\item\item{c.}
{\TeX}  produces  device   independent  files;   however,  because
different installations of {\TeX} support different features it is
good, at present at least, to keep the typesetting simple. To this
end, the editor should consider supplying a set of  {\TeX}  macros
to authors wishing to do their own typesetting using {\TeX}; these
can be supplied over the electronic mail in much the way that  the
American  Mathematical  Society   encourages  the  submission   of
abstracts for its meetings.
\item\item{d.}
Not  all  individuals  have  access  to {\TeX\/} even though their
university  has  it;   individuals   in   mathematics  departments
generally do have access to it and know how to use it.
\item\item{e.}
Figures, charts, and tables  that  can be considered  as a  matrix
(such as a crossword puzzle) can be typeset using {\TeX\/}.   Maps
and non-rectilinear figures generally cannot.
\item\item{f.}
One approach to dealing with figures, that  works  easily,  is  to
scan complicated  maps  and figures and to incorporate the scanned
file  into  any  distributed  hardcopy  by  electronic cutting and
pasting.   The Xerox DocuTech  stores scanned images as electronic
files  on  a  hard  disk  and  permits  such  electronic  editing.
Hardcopy,  complete  with figures, can be produced in an on-demand
fashion  for  sale  to  standing orders and to others who inquire.
Warehousing  is  thus  converted  to  a  just-in-time"  approach
requiring virtually no extra  space or cost.  Hard copies can then
be made available in a variety of bindings.
\item\item{g.}
If the  scanned  electronic files are downloaded as part of a text
file,  then  the  reader's  electronic  cutting  and   pasting  is
unnecessary.  The capability of  future word  processors holds the
answer to the possibility of shipping mathematical notation, maps,
and photos in a single easy-to-read, typeset, transmission.
\item\item{h.}
Graphics  transmission  can  be  executed  immediately  by  making
available for distribution binary files of graphics images  on  an
Internet  server  for downloading via FTP (File Transfer Protocol)
or from a standard bulletin board.
\item\item{i.}
Yet another approach to the graphics issue might  involve  linkage
to  a  Geographic  Information  System  to provide a procedure for
creating  compatible  transmittable  map  files directly from data
managers  into  a  {\TeX\/}-ed  file.  Data files are likely to be
quite large; compressed files should be used with instructions for
decompression and recompression provided online in help files."

\item{2.}
As above, {\TeX\/}  can  be  used  to create an ASCII file that is
typeset,  including  diacritical  marks.  If,  however, the editor
chooses not to use {\TeX\/}, publishers can convert the formatting
codes  of  other  software   such  as   Microsoft   Word,  XyWrite
(Signature), and other robust word processors.  If straight ASCII,
perhaps  employing  the  upper  IBM ASCII set whenever diacritical
marks  are  important,  is  used to transmit the electronic files,
then another set of  issues,  some  similar  to and some different
from using {\TeX\/}, confront the editor.

\item\item{a.}
At present  it is  important never to right-justify straight ASCII
files.   Right-justified  text  introduces  extra  spaces in  word
processors that produce straight ASCII files.  To mend this, users
must  do a  number of search-and-replaces, replacing double spaces
with single spaces.   They need to  do this  to make the text look
like their own text so  they  can  add  items  from a bibliography
to  their  own  bibliographies or add to other downloaded lists of
subjects that are searchable with a word processor or data manager.
\item\item{b.}
Data-intensive text files, either those for which it is  difficult
to find a publisher in hardcopy or,  in particular, those that are
suited to searching and other  computer  text  manipulation  (such
as bibliographies or checklists), are  well-suited   to   journals
employing the straight ASCII format.    Data files take two forms:
article format, similar to paper publications -- searchable with a
standard word processor or with text management" software,  and,
data base  format,  appropriate for importing into a standard data
base manager.  The latter should have data presented with an equal
number  of  lines  per  record  and  information  entered  on  the
appropriate line for each field, or in another delimited" format.
\item\item{c.}
Large text files  should  be divided into  smaller files each less
than  300  kb  in  size.  These  can  be uploaded  as is, or first
converted  into  smaller  compressed (e.g.,  .ARC,  .ZIP, or .LZH)
files.   Split  text  files  can  be  downloaded  and  reconnected
(through DOS copy  command) by  the  user.  Very large files  may,
for now,  be more appropriately distributed on disk.
\item\item{d.}
Foreign  language  characters,  symbols,  and  graphics.   Authors
should expect that downloaders will generally use 8 data  bits and
an error-checking protocol, so binary files and  text  files  with
the IBM  upper ASCII character set (foreign and special characters
and graphics) can be  easily transmitted.  If the text is prepared
in  something  other  than  a  MS-DOS,   pure  ASCII   environment
(non-ASCII  texts  are  created  by many word processors), authors
need  to  remove  all   software-specific  formatting   codes  and
type-style  codes,   before  uploading.   These  can,  however, be
suggested --- underlining codes, for example, might be represented
by symbols  like @ or $|$ so  downloaders can re-underline through
search-and-replace.

Users of operating systems other than MS-DOS generally do not have
access  to  the upper  IBM ASCII set, which has foreign characters
and  symbols  such  as  the  degree  sign ($^{\circ }$) and simple
graphics.   Also,   because   all  users   may  not   have  MS-DOS
microcomputers or compatibles, some authors may wish to substitute
special codes  for the IBM upper ASCII set used in MS-DOS.   It is
recommended  that  instructions  for  translating  (by  search and
replace)  the codes into the actual  character  be  given  at  the
beginning of the  publication.  Any system can be used; however, a
simple  system,  which  can  be  easily  interpreted  even  before
translation  and  may be easily used by non-MS-DOS systems, is the
backspace and  overstrike" method:  many foreign  characters may
be  easily  manufactured  by causing the  printer to backspace and
overstrike a diacritical mark.  Since some  wordprocessors  cannot
deal  with  the ASCII  backspace character (ASCII 8), substituting
an unused lower ASCII character such as @ or $|$ for the backspace
character will allow  search  and replace for  (1)  the  backspace
character itself, (2) for an  acceptable printer  code  substitute
for it, or (3) replacement of the three  characters  with  an  IBM
upper ASCII character.  Examples of backspace substitution:  a$|$
= \a , A$|$o = \AA, u$|$" = \"u; and of direct substitution: deg.
= $^{\circ }$, u = $\mu$ (search  for  space-u-space  and replace
with $\mu$).  Graphics characters have little utility and  cannot
easily  be  coded  for  non-MS-DOS  standard  machines,  so  it is
recommended that these be restricted to special applications.

There are a number of efforts at an enlarged ASCII set for foreign
languages (Hayes 1992). The coming of Unicode or something similar
will  hopefully  provide  a  complete set of multiplatform foreign
characters.
\item\item{e.}
In bibliographies, spell out all duplicate author's names  (do not
use a sequence of hyphens.)  so  that  the  author's  names can be
searched for.  Begin each  entry  flush  left  and  leave an empty
line (two hard rights) between each entry.
\item\item{f.}
Do not  spell any  words with all capital letters (this  may  make
it difficult to search for them; it also looks bad).
\item\item{g.}
If appropriate,  present files in a squeezed" form  as  an  .ARC
file   or  .ZIP  or another archiving' utility file format.  This
allows faster and less costly downloading and keeps diskette files
small.

\item{3.}
File  management  seems  to be relatively easy with an  E-journal.
Keeping track of manuscripts,  and of who  is refereeing them, and
of  their  stage in the production process, is made simpler by the
technology.

\centerline{\sl Transmission.}

E-journals  should  have  standard, and thus easy-to-use, modes of
access. They should be transferable across different systems (e.g.
various micro, mini and mainframe platforms).  Alphabets should be
standard  (ASCII,  ISO  Superalphabet  eventually)  in order to be
available  to a wide number of users.  Transmission can occur in a
number of different ways and have various uses.

\item{1.}
Issues  may  be  obtained  by  anonymous FTP"  or downloaded via
regular telephone lines by modem from an electronic bulletin board.
An  electronic  bulletin  board  system is a computer and software
system that can be accessed from  outside by  a caller, who likely
has a number of options, including perhaps:

\item\item{a.}
Reading  or leaving messages. These are typed while online and may
be public or private (readable only by the addressee).
\item\item{b.}
Depositing or taking  away data or  text files.  These are created
with a word processor or data manager previous  to calling and are
up-" or down-loaded" as a unit.
\item\item{c.}
Extracting information from a large data file. Authors can prepare
compiliative publications  that  they  use  personally and wish to
share.  Then they may,  if they wish,  maintain  the  publications
informally  or  formally  as  a  series of versions in online data
banks.  Users of the bulletin board download online files, and use
the files directly for searching for particular data or by copying
portions to enlarge their own personal files, with due respect, of
course, for copyright privileges of the original author.

\item{2.}
A bulletin  board  can be of interest to scholars in the following
ways:

\item\item{a.}
Messages - For exchange of ideas and information. Speed of contact
is  far  greater  than  with  regular mail.  Special Conference"
sections allow  public  exchanges on single scientific topics that
are equivalent to symposia at national meetings.
\item\item{b.}
Files --- Electronic publications that may be cited in an author's
curriculum  vita.  Such publications should be copyrighted.  These
include: original text  material  and  computer programs;  text or
data  files  of  an  ephemeral or informal nature; and, previously
published  computer  programs   (of reprint" value).   With  the
eventual  realization  of  a network of bulletin boards across the
country, this method of transmission holds considerable promise.

\item{3.}
Ship  the  E-journal  across Bitnet or Internet to a  distribution
list of subscribers who ask to have  the E-journal mailed to them.
Some installations  do  not  have  the  capacity  to send files in
excess of 25,000 characters. In that case, split the journal apart
with  instructions  to  the user to concatenate the files prior to
downloading, printing, or typesetting.
\vskip.5cm
\noindent{\bf Archival issues.}

All journals are useful only for as long as they can be located in
the  holdings  of  some institution.  As technological formats for
producing journals  change,  it will be important to keep not only
the  new,  but also the old --- as back-up with a known life-span.
Some  of the  issues  that  will confront archivists include those
listed below.

\item{1.}
Availability --- the  E-journal should be archived indefinitely in
an  institution  willing  to  provide  copies or the equivalent on
request.

\item{2.}
Durability --- Archives should be maintained so as not  to degrade
with  time,  e.g.  contents  of diskette transferred to hard disk,
then  to  optical  disk,  then  to  solid state or whatever future
technology  provides.   Duplicates  stored   off-site,   and   EMF
protection are also advisable  in  the  long-term. Paper burns and
degrades   with   age,  but  magnetic  images  can  be  maintained
indefinitely if copied periodically  onto new media (diskettes are
said to have a maximum data retention life of 10-15 years).

\item{3.}
Retrievability  and salvageability  -  Standard  operating  system
formats should be changed in a timely fashion: MS-DOS to Unix, etc.
Standard word processing formats should be upgraded so they can be
read decades hence.   Database formats  should be standard or also
available  in  ASCII-delimited  format.   Any  required   programs
(decompression programs, graphics  viewing  programs, special word
processors) should be archived, too, along with necessary hardware
platforms.

We  have  found  that  editorial  and  publishing problems can be
overcome  within  the  limits  of  existing  technology such that
electronic  journals  can  be  successful  in  transmitting   and
presenting   information  to  scholarly  readers.  We  foresee  a
significant  upgrade  in  quality  and  flexibility of electronic
presentations   with   the  advent  of   standard  cross-platform
graphics-capable word processors, standard export-import formats,
and  standard  multi-language  character sets.  The advantages of
electronic publication:  inexpensive,  fast,  easy to store, easy
to search, long-term archivability, easily justify the time spent
learning to deal with the new technology.
\vfill\eject

\noindent{\bf References.}

\ref Hayes, Frank.  1992.  Superalphabet compromise is best of two
worlds.  {\sl UnixWorld\/}, January 1992:  99-100.

\ref Horstmann, Cay S.  1991.  Automatic conversion from a scientific
word processor to {\TeX\/}.  {\sl TUGBoat:  The Communications of
the {\TeX\/} Users Group\/} 12:471-478.

\ref Janusz, Gerald J.  1991.  Reviewing at Mathematical Reviews.
{\sl Notices of the American Mathematical Society\/}, 38:789-791.

\ref Knuth, Donald E.  1984.  {\sl The TeXBook\/}, Reading, MA:
Addison-Wesley and Providence, RI:  The American Mathematical
Society.

\ref Nicholson, Richard S. 1992.  Data make the difference.
{\sl Science News\/}, March 28:195.

\ref Palca, Joseph.  1991.  Briefing.  {\sl Science\/}, November 29:
1291.

\ref Palca, Joseph.  1991.  New journal will publish without paper.
{\sl Science\/}, September 27:1480.

\ref Peterson, Ivars.  1992.  Math for all seasons.  {\sl Science
News\/}, January 25:61.

\ref Strangelove, Michael.  1991.  {\sl Directory of Electronic
Journals and Newsletters\/}.  Washington D.C.:
Association of Research Libraries.
\smallskip
\smallskip
$^*$
Sandra L. Arlinghaus,
Institute of Mathematical Geography,
2790 Briarcliff, Ann Arbor, MI 48105.
Richard H. Zander,
Buffalo Museum of Science,
1020 Humboldt Parkway,
Buffalo, NY 14211.
\vfill\eject
\centerline{\bf WILDERNESS AS PLACE}
\vskip.5cm
\centerline{\bf John D. Nystuen $^*$}
\vskip1cm

Some conflicts are the result of people talking at cross purposes
because  they  interpret  identical  empirical  data   in   quite
different ways.  These  differences  can  arise  from deep seated
differences  in  belief  systems  or  from  the knowledge systems
(theories) applied to understanding a  phenomenon.   The conflict
over the meaning of wilderness is an example.
\vskip.5cm

\noindent{\bf Visual Paradoxes}

The  biologist  Richard  Dawkins  in  his  book {\sl The Extended
Phenotype\/}  uses  the  analogy of the Necker Cube (Louis Albert
Necker, 1832)  to  illustrate  the  fact  that the same empirical
evidence can be  interpreted  in  two  or more perfectly accurate
ways,  each  of  which  is valid but incompatible with the other.
The  Necker Cube  is a visual paradox in which the mind perceives
a flat plane drawing as a three dimensional transparent  cube  in
which the orientation of the cube is arbitrary (Figure 1). At one
moment  it  appears  to be viewed from above but as one stares at
it,  a  reversal  occurs  and  in  the next moment it seems to be
viewed  from  below.    The   visual  paradox  arises  when  full
information is available.  Partial knowledge seems  to  favor one
view or the other.

\midinsert \vskip 3in
\noindent{\bf Figure 1.}  Necker Cube.
A sequence of three cubes shown as line drawings.  The reader
unfamiliar  with  Necker's  Cube  would  be  well-advised  to
reconstruct  this figure.  The left hand cube is one with all
edges showing; the center cube has three edges hidden so that
it appears the reader is looking down at the cube from above;
and, the right cube has three edges hidden so that it appears
that the reader is looking up at the cube from below.
\endinsert

An  additional  set  of  views  is  available --- that  of a  two
dimensional plane  figure  which, of course, is what the drawings
are.  This set of views  may become dominant by rotating the cube
so that the many symmetries of the cube are emphasized (Figure 2).

\topinsert \vskip 3in
\noindent{\bf Figure 2.}  Views Along Axes of Symmetry of a Cube.
This  figure  is also a sequence of three views of cubes
shown  as  line  drawings.   The  left  cube  is a full-
information cube (no hidden edges) seen  head-on, with a
face of the cube closest to the face of the reader.  The
center cube is a cube with all edges showing viewed head
-on  with  an  edge  closest  to  the reader so that the
prominent  edge,  and  the  diametrically  opposed  edge
appear to coincide for part  of their length.  The right
cube is a view of the cube with  one  corner  closest to
the reader so that the plane view of the cube appears as
a hexagon with three diameters.\endinsert

Another  well-known  visual  paradox,  {\sl   face/vase\/},   was
introduced  by  Edgar  Rubin in 1915 (Figure 3).  In this example
additional knowledge seems to resolve the paradox --- as a simple
white, classical vase against a black background, both  vase  and
profiles of faces at either side are evident.  If  baseball  caps
are put on the profiles, the faces dominate; if, instead, flowers
are drawn in the vase, then the vase dominates.

\topinsert \vskip 4in
\noindent{\bf Figure 3.}  Face/vase paradox.
\endinsert

Usually  one has to plan how to seek additional knowledge about a
problem.   If only a certain type of knowledge is pursued because
that is  the  way  the problem is interpreted, then one view will
likely  prevail.   If  only  economic  evidence  is  admitted for
consideration  (for  example),  other  views,  other  values, may
remain invisible.

Past experience may bias one's interpretation  beyond what  seems
reasonable  to  others  with  different  points of view.   Gerald
Fisher's  (1967)  man-girl  paradox  is  a  sequence   of   eight
progressively  modified  drawings --- from man to nymph-like girl
(Figure 4).  The fourth drawing  in  the sequence was  found upon
empirical  testing  to have equal probability of  being seen as a
man's face or a girl's figure.  However,  by viewing the sequence
successively  from  the  top  left  to  the  bottom right one can
maintain a bias towards seeing  the man's face almost to the last
drawing.  There, only a faint, melting ghost of a face remains to
be seen, if seen at all.  The opposite is true if one starts with
the girl's figure and moves in the reverse direction.

\topinsert \vskip 4in
\noindent{\bf Figure 4.}  Man-Girl.
Shows a sequence of eight line drawings--transforming a man's
face to the profile of a girl's body.
\endinsert

\noindent{\bf Wilderness Defined}

The  value  of  wilderness  to  society  resembles  a Necker Cube
paradox.  People  of  goodwill see the same empirical evidence in
very  different  lights.   The  dominant  American  view  of  the
environment is utilitarian and anthropocentric.  The  environment
is for humans to use.  Natural resources are cultural appraisals,
more a matter of society than of nature.  For something  to  be a
resource  we  must  want  to  use it, know how to do so, have the
power to do so, and be entitled to do so.  Nature offers only the
opportunity for use.

A  biocentric  ethic  imbues   nature   with   intrinsic   values
independent of mankind. We are part of nature, not apart from it.
In an anthropocentric  view  we  are distinguished and especially
favored by God.  In  a  biocentric  view all creatures, large and
small,  and  plants  too,  have  a  right  to exist.  Most Native
American cultures held to this belief.  They  apologized to their
fellow life forms when consuming them to meet their own needs.

In Western Society  the  biocentric ethic  is not well understood
perhaps  even  by  many of its advocates.  Preservationists focus
on symbols  of  wilderness  rather than on wilderness in its full
existence.  Tactical  reasons  motivate  this  approach  but then
frequently   wilderness   advocates   are    outmaneuvered.    Do
preservationists  really  care  about  the  snail  darter and the
spotted  owl?   Or  are  these species being used as focal points
to preserve  entire  habitats?   They embody or personify concern
for more  abstract  values.   Do we really want the habitat to be
{\sl preserved\/} unchanged?

I  recall,  when  visiting Disneyland, a frontier scenario of a
settler's log cabin  under  attack and in flames."  The logs were
made of cement and  the  flames came from gas jets --- they  burn
eternally for the tourists, daily during open hours, season after
season.

The  wilderness  worth  saving  is  the  biosphere  process.  The
wilderness  ethic  is  to  let  wild  habitats  exist where human
contact is slight and/or remote (outside--backdrop).  Living wild
habitats change and perhaps spotted owls or  other  species  will
vanish but not as a result of direct human action.  Of  value are
natural processes remote and indifferent to mankind.    John Muir
said, In Wilderness  is the Preservation  of the  Earth."  That
phrase  is  the  motto  of  the Sierra Club which Muir founded in
1892.   Preservation  of the earth as the home of life transcends
societal  concerns.   Beyond  a  species  imperative,  it is life
imperative.
\vskip.5cm

\noindent{\bf Conflict or Synthesis}

M.  C.  Escher,  the  artist  noted  for  his  depictions  of the
complexities  of  time  and space, transcends the choice required
by  the  Necker  Cube.  He  gave the object some attention in his
lithograph  {\sl Belvedere\/}  (see  {\sl The World of Escher\/},
p. 229).   The  man  seated  in  the  foreground  is  holding  an
impossible  cubic  object  while contemplating a drawing of it on
the  ground  in  front  of  him.  In this scene Escher provides a
drawing, a hand-held model,  the embodiment of the concept in the
structure of the castle building.

Escher simultaneously embraces two views of the cube with a model
and a construction process that can only exist in the imagination.
The  paradox is in the images of physical things depicted.  There
are  no  paradoxes in nature.  Nature exists.  Paradoxes observed
in nature mean that our understanding of phenomena is inadequate.
This is  what drives the imagination of physicists.  Theory holds
that  nothing  can  exceed  the  speed  of light --- except human
imagination;  light bends;  space  is warped;  black holes exist;
time  flows  backward; light is both wave and photon.  Deeper and
deeper understanding of nature incorporates these  constructs  of
our imagination.  From the beginning many predictions of  quantum
mechanics were viewed as very strange.  Now after many decades or
resisting refutation, the theory yields new results  that  border
on  the  surreal:   that  quantum phenomena are neither waves nor
particles  but  are intrinsically undefined until the moment they
are  observed  (John  Hortgan,  1992).   Yet  nature exists.  The
problem is our mind set, the position of our understanding.

To understand Escher's impossible cube one must take into account
the  position  of  the observer.  It is like a rainbow; it exists
only for those who  are  in the proper position to appreciate it.
There is no rainbow for the people who are being rained upon.

I  remember  talking to a Gurung woman (the Gurung are a highland
people of Nepal)  who,  under  a government program, had migrated
to a lowland farm on the Nepalese  portion  of the Gangetic Plain
(elevation  600  feet).   I asked her if she missed the mountains
for I had seen the breathtaking panoramas of her homeland in  the
high Himalaya.  She said, What is there to miss?   We have four
bega of good land here and we had only one half bega of very poor
land in the other place."

We do not need to be articulate or  self-conscious  about  things
essential to our being.  For example, food is so  fundamental  to
our  existence  that  we  treat  it  very  emotionally.  Reasoned
discourse is not the only  or  even  dominant  basis for thinking
about food or debating public  policy  about entitlement to food.
A sense  of  place  is  as  deeply  held  and  fundamental to our
existence as food.  We become attached to a place  to the  extent
that  we  fill  the  place  with  meaning.   A  personal and deep
attachment is made to  the  place  called  {\sl home\/}.  Home is
familiar, safe, restoring, and controlled territory.  We fight to
protect it from invasion with deep feeling  and  energy.  We will
die for it.

Wilderness  is  a place that is {\sl not home \/} for humans.  It
becomes  real  and  important  only to the extent that we fill it
with  meaning.  To  give  it  meaning  it  must become foreground
(subject).   Mere  opposites  of  home  values do not capture the
essence.  Is wilderness  strange, dangerous,  stressful, and wild
territory?   Strange  and  wild  are nice but to me stressful and
dangerous are the wrong emphasis, sometimes used by organizations
that  are  trying  to  build  self-confidence  in  adolescents by
thrusting them into  confrontation  with  wilderness.  Recreation
hunters whose  intent  is  to  achieve a kill reveal this sort of
confrontational  approach  to wilderness as well.  I believe that
wilderness should not be taken as hostile, something to overcome,
but rather  one  should enter a wilderness prepared, take prudent
action  and  seek  to  experience  the strange and the wild to be
found there. Admittedly, some views of wilderness are going to be
incompatible.   But  at  least  hunters and preservationists have
visions of the meaning of wilderness, compatible or not.  Certain
vantage  points  must  be  assumed  or  wilderness  will   remain
invisible.  An alliance to build a public edifice is  conceivable
that  might,  like  Belvedere,  provide  positions  for people to
calmly gaze in different directions.

Wilderness is like a rainbow.  Existence depends, in part, on
the position of the viewer.  Do rainbows exist?  Or are they only
latent until observed in some fashion or another?  Are they to be
valued, if so, how is value assigned?  Can you own one?
\vskip.5cm

\noindent{\bf Wilderness As Place}

The {\sl Bureau of Land Management\/} (BLM) is a  federal  agency
that  controls  179  million  acres of land mostly in the western
states  (over  nine  percent  of  the  total  land  area  in  the
coterminous  USA).   The  bureau  was  created  in  1946  through
consolidation  of  two  federal  agencies,  the  {\sl  Land Sales
Office\/} and the {\sl Grazing  Service\/}.  The bureau inherited
from these prior agencies the  mandate to either sell off federal
land to private owners as  quickly and efficiently as possible or
to make federal lands  available  for  use by private individuals
through  issuing  grazing  permits.   In 1976 Congress passed the
{\sl Federal  Land Policy and Management Act\/} which contained a
mandate to the  BLM to inventory, study, and make recommendations
for wilderness  designations  for  BLM  lands.  The bureau was to
report back its actions by 1991.

The  bureau  people  were  somewhat  at  a  loss for words.  What
exactly is wilderness?  Is that a place with no conceivable human
use;  a  place  nobody  wants?   Wouldn't it be what is left over
after  we  do  our  job?  Could we address this mandate simply by
subtraction?  The  answer  was no, that would not do.  Wilderness
did  not  fit into a commodity based, I can own it,' philosophy.
How could humans manage a wilderness?  What would there be to do?

The  bureau  people  were  more  than a little uncomfortable with
their  new  task.   In  the  past two decades a sea of change has
occurred  on  how  to  view  the environment and the BLM has been
caught in its tide.  Today,  environmental groups are a political
force  with  access  to  agency  decisions through new avenues of
public participation.  It is not business as usual.

In the words of C. Ginger (1993):

The philosophical challenge faced by BLM has, at its core,
human perceptions of the value  of land.   These  values are
the same  as those that were at the base of the disagreement
between  John  Muir  and  Gifford  Pinchot at the end of the
nineteenth  century.   Muir and Pinchot debated the ideas of
preservation of land versus conservation of land.  Placed in
the context of the wilderness protection, we might ask if we
are saving  wilderness for wilderness' sake or because it is
a wise use  of  natural resources.   These  two perspectives
(preservation and  conservation) were a challenge to a third
perspective that  dominated the government institutions that
oversaw  public  lands   in   Muir   and   Pinchot's   time:
exploitation of natural resources  in  the  short  run.  All
three points of view are present today  in  our  approach to
land and resources but it is  Pinchot's  view that  provides
the dominant ideal in the form of the multiple-use sustained
-yield  philosophy  established  by Congress for public land
management in the United States.  The debate over wilderness
designations  in  the  West  illustrates  that  the  idea of
preserving  a  chunk  of land is not just an administrative,
legal  or  even  political  issue.   The  sometimes dramatic
conflict reflects an underlying  difference  in  values  and
perceptions of our relationship to the land.  And the values
are  not  simply held by individuals.  They are reflected in
and  perpetuated  by the institutions we have created to act
collectively.   We can find in the Bureau of Land Management
how  the debate over our relationship to the land is defined
and pursued."

Human institutions are not natural phenomena.  They  are  created
by humans  and  some  contain  paradoxes  and ambiguities.  These
ambiguities may be the source of conflict in circumstances  where
identical evidence is interpreted in different ways.

Human  belief  systems  are  mutable  but  they  are  also  quite
resistant to change even in the face  of  accumulating  evidence.
In the  United States race relations and women's roles in society
have changed in the second half of the 20th century to the extent
that certain behaviors and attitudes accepted  as commonplace  in
the  first  half  of  the century are disapproved and are illegal
today.  Equal access to places and roles is now an accepted ideal,
not yet attained in many circumstances, but  with  many instances
of success.  {\sl Justice\/} and {\sl equality\/}  are underlying
moral  imperatives  driving   these   movements   in   particular
directions.

Sustainability and ultimately, {\sl survivability of life\/}  are
the  imperatives  underlying  the  shift  from anthropocentric to
biocentric views.  As  far  as we know, we  alone, among sentient
beings, record history, and thus can be aware of long consequences
of our actions.  As humans gain capacity to control and to destroy
we  must  take  responsibility  to sustain.  We need goals in this
regard.  Sustaining life processes on earth is an  acceptable goal
to be placed on the balance scale along with other values.

Defining  and  managing wilderness by the agencies responsible for
public lands is a skirmish in the paradigm shift over the position
of humans in nature.  Elements of nature must be given standing in
human  value  systems  in  order  that wilderness be recognized in
human affairs.  This is to be done  by  defining  wilderness  as a
place  apart,  imbued  with  boundaries  and  rights, where humans
behave in  prescribed ways as if they were in someone else's home.
For  wilderness  to be a place it must be filled with meaning that
large  segments  of  society  understand and support, otherwise it
will  remain  a  backdrop  in  human  affairs, invisible to policy
makers.
\vfill\eject
\noindent{\bf Suggested Readings}

\ref Fisher, Gerald, (September, 1968)
Ambiguity of form:  Old and new,
{\sl Perception and Psychophysics\/}, v. 4, no. 3:189-192.

\ref {\sl Image, Object, and Illusion, Readings from
Scientific American\/} (1974)  San Francisco:  W. H. Freeman
and Company.

\ref Locher, J. L., Editor (1971)
{\sl The World of M. C. Escher\/},
New York:  Harry N. Abrams, Inc. Publishers.

\ref Hortgan, John (July 1992)
Quantum philosophy,
{\sl Scientific American\/}, v. 267, no. 1:94-101.
San Francisco: W. H. Freeman and Company.

\ref Relph, E. (1976),
{\sl Place and Placelessness\/},
London: Pion Limited.

\ref Oelschlaeger, Max (1991) {\sl The Idea of Wilderness\/},
New Haven:  Yale University Press.
\vskip.5cm

\noindent{\bf Sources}

\ref M. C. Escher, Belvedere" 1958, lithograph.

\ref M. C. Escher, Study for the Lithograph Belvedere'"
1958 pencil.
Plate 228, {\sl World of M. C. Escher\/}

\ref L. S. Penrose and R. Penrose,
Impossible Objects, A Special Type of Visual Illusion,"
{\sl The British Journal of Psychology\/},
February, 1958.  Contains the impossible triangle--basis for
Escher's waterfall."  R. Penrose is, of course, the inventor
(later) of Penrose tilings; he postulated the existence of
five-fold symmetry thought to be impossible in nature by
crystallographers until their recent discovery of
five-fold symmetries in quasi-crystals.

\ref Attneave, Fred, (December 1971)
Multistability in perception,"
{\sl Scientific American\/},
San Francisco:  W. H. Freeman and Company.

\ref  Rabbit-Duck, Joseph Jastrow, 1900.

\ref Young girl --- Old woman, Edwin G. Boring, 1930,
by W. E. Hill, {\sl Puck\/}, 1915
as My Wife and My mother-in-law."

\ref Man-Girl, Gerald Fisher, 1967.

\ref Reversible goblet, Edgar Rubin, 1915.

\ref Necker Cube, Louis Albert Necker, Swiss geologist, 1832.

\ref Slave market with apparition of the invisible bust
of Voltaire, S. Dali, Dali Museum of Cleveland.

\ref  Clare Ginger, doctoral candidate, Urban, Technological
and Environmental Planning Program, the University of Michigan.
She is working on a dissertation about the meaning of
wilderness in the eyes of BLM personnel and spent four summers
collecting taped interviews from BLM employees at federal,
state, and district levels.  She asked them to describe
wilderness and their responses to the wilderness mandate.
Quotation in the text is from an unpublished document, 2/3/93.
\vfill\eject

\noindent{\bf Visual illusion authors}

\ref Marvin Lee Minsky, MIT

\ref Robert Leeper, University of Oregon

\ref Julian Hochberg and Virginia Brooks, Cornell University

\ref Alvin G. Goldstein, University of Missouri

\ref Ernst Mach, Austrian physicist and philosopher,
(Dover Publ., 1959, trans., C. M. Williams).

\ref Murray Eden, MIT

\ref Leonard Cohen, New York University
\smallskip
\smallskip
$^*$ John D. Nystuen,
Professor of Geography and Urban Planning,
College of Architecture and Urban Planning,
The University of Michigan,
Ann Arbor, Michigan, 48109.
\vfill\eject
\centerline{\bf THE EARTH ISN'T FLAT.  AND IT ISN'T ROUND EITHER!}
\centerline{\bf SOME SIGNIFICANT AND LITTLE KNOWN EFFECTS}
\centerline{\bf OF THE EARTH'S ELLIPSOIDAL SHAPE}
\vskip.5cm
\centerline{\bf Frank E. Barmore $^*$}
\vskip.5cm
\centerline{\bf Reprinted, with permission, from}
\centerline{\bf THE WISCONSIN GEOGRAPHER}
\centerline{\bf VOLUME 8, 1992}
\centerline{\bf pp. 1 -- 8}

\noindent {\bf Abstract}

The small difference between the shape of the earth  and a sphere
is usually thought to be negligible except for work of very  high
accuracy such as geodesy.  This is not the case.  There are  some
examples  where  this  small  difference in shape makes an easily
apparent difference in what is observed.  This paper will comment
on three problems and  evaluate  the impact  of the non-spherical
shape of the Earth on the result: 1) the qibla problem of Islamic
geography,  2) the center of area (geographic center) and  3) the
center of population.

\noindent {\bf Introduction}

I have noticed  that some common  considerations in geography are
often  treated  without  due  regard  for the Earth's ellipsoidal
shape.  This is surprising.  The  Earth is not spherical (round).
It is,  rather,  very  nearly  an  ellipsoid  of  revolution with
equatorial radii, $a$ and  $b$, of 6378.2 km.  and  polar radius,
$c$, of 6356.6 km.  ---  a difference of 21.6 km. This difference
is   significantly   larger  than  the  next  largest   pervasive
topographic  feature, the continent --- ocean  basin dichotomy of
5 km.   Also, this  shape, an  ellipsoid  of  revolution, is  not
intrinsic to terrestrial  planets.   Venus  is  nearly spherical,
$a=b=c=$ ca.  6051.5 km. (Head, et al., 1981). Mars is reasonably
well  described  as  a  tri-axial  ellipsoid  of  $a=3399.2$  km,
$b=3394.1$ km. and $c=3376.7$ km. (Mutch, et al., 1976).

This departure  of  the shape of the Earth from a sphere is often
given as the flattening,
$$f=(a-c)/a=0.0034 \quad\hbox{or} \quad 0.34\%,$$
or the eccentricity, $e$, where
$$e^2=(a^2 - c^2)/a^2 = 0.0068.$$
The  departure from a sphere also results in a difference between
geocentric and geographic latitude of (at 45$^{\circ}$ latitude),
$$0.195^{\circ} = 0^{\circ}11.7' = 0^{\circ}11'42''.$$

While  these  are  small  quantities, they are not insignificant.
For  comparison,  consider  the following difference or ratios of
similar magnitude:
\item\item{a.}   one vacation day per year  (which, in  turn,  is
larger  than  the  one  day  calendar adjustment
every fourth or leap'' year),
\item\item{b.}   a watch which  gains  or looses five minutes per
day,
\item\item{c.}   a two inch gap in a 50 foot brick wall,
\item\item{d.}   a 1/6 inch crack in a 48 inch table top,
\item\item{e.}   \$100 per \$30,000 of annual earnings,
\item\item{f.}   an angle  of 1/3 of the apparent diameter of the
sun or moon.

We  routinely  concern  ourselves  with such small differences in
daily life. We expect and receive better accuracy from craftsmen.
Differences in direction of this magnitude are easily seen.

Consistency  would  require  that we be as concerned with equally
small  quantities  in geography as we are in other circumstances.
Therefore,  all  but  the  simplest  considerations  in geography
should routinely take into account the Earth's ellipsoidal shape.
Often  this is not done.   This paper will consider the impact of
the Earth's non-spherical shape on the results in three cases:
1) the qibla problem of Islamic geography,
2) the computation of a geographic center (center of area)
and
3) the computation of a center of population.

\noindent{\bf The Qibla Problem}

As  I have previously  commented (Barmore, 1985), a Koranic line
which may be translated  as $\ldots$ wherever  you  are, turn
your face towards it [the Holy Mosque --- the  Kaaba]"  is often
invoked to establish the correct orientation (the qibla)  during
the  obligatory  prayer  (the salat),  and  hence  the   correct
orientation  for  mosques.   This requirement, in turn, is often
considered  as  satisfied  when  a  mosque  is  aligned with the
direction  of  the  Kaaba  in  Mecca.   There  is,  in   Islamic
scientific literature,  sufficient  discussion  of the direction
of Mecca  to indicate  the usual definition  of direction (King,
1979).   The direction  is that  of the shortest arc  of a great
circle on a spherical Earth between the locality and Mecca. (But
note that medi{\ae}val Islamic religious and legal scholars have
often  argued  otherwise  and,  as  a  result, other orientation
traditions  have  existed  (King,  1972, 1982a, 1982b, and other
work  in  preparation).)   The  direction  is  then specified by
stating the azimuth  of this arc  of a great circle  relative to
the meridian.
Given  the geographic coordinates of a locality and of Mecca the
azimuth  of  Mecca   is   easily   calculated   with   spherical
trigonometry,  {\bf  provided  a  spherical Earth is assumed\/}.
Tables of such information,  both  historical  and contemporary,
exist  in  great  number.   These  tables,  as  well as numerous
individual  calculations  in  the literature discussing the many
facets  of  Islamic  culture,  often  give  their results to the
nearest minute  of arc (or even the nearest second of arc).  The
implication  is  that  the results are correct to the same level
of  accuracy.   But  the  Earth  is not spherical.  The Earth is
ellipsoidal in shape.  If qibla azimuths are calculated assuming
a spherical  Earth,  they do not represent the real case with an
accuracy  approaching  a  minute  of  arc.  In every case I have
examined,  the  calculations  were  done  as if the Earth were a
sphere.  In order  to illustrate  the errors that result, I have
calculated the simple azimuth as well as the geodetic azimuth of
the Kaaba in Mecca for a number of places.   (The simple azimuth
is  calculated  on  a  sphere  while  the  geodetic azimuth more
closely represents the correct case (See Appendix A).) The qibla
error,
$$QE = az(S) - AZ(E),$$
is the amount that must be subtracted from the incorrect but more
easily calculated simple azimuth, $az(S)$, in order to obtain the
more  accurate  geodetic  azimuth,  $AZ(E)$,  calculated  on  the
ellipsoid representing the Earth. The locations of various places
were taken from {\sl The Times Atlas of the World\/} (1990).  The
location of the Kaaba in Mecca was taken from  a large scale  map
of Mecca (1970).  The result,  for Clarke's (1866) Ellipsoid,  is
displayed  in  Table 1  for selected localities and shown for the
world in Figure 1.

\vskip.5cm
\hrule
\vskip.2cm
\hrule
\vskip.2cm
\centerline{Table 1}
The error in the qibla azimuth for various places when calculated
on  a  sphere.  The  results  are given in decimal degrees and in
minutes of arc.  A  tabulated  value  of  the qibla error,  $QE = az(S) - AZ(E)$,  is  the  amount that must be subtracted from the
incorrect but more easily calculated simple azimuth, $az(S)$,  in
order  to  obtain  the  more  accurate geodetic azimuth, $AZ(E)$,
calculated on Clarke's (1866) Ellipsoid representing the earth.
\vskip.2cm
\hrule
\vskip.5cm
\settabs\+\quad&Tombouctou\quad&33.3333\quad&106.7500\qquad\quad
&240.3127\quad&240.4550\qquad\quad
&Qibla Error\quad&min.arc\cr %sample line
\+&&{\bf Place}&
&\qquad{\bf Qibla}&
&\qquad{\bf Qibla Error}&\cr
\vskip.5cm
\hrule
\vskip.5cm
\+&Name&Lat(N+)&Long(E+)
&$az(S)$&$AZ(E)$
°ree&min.arc\cr
\vskip.5cm
\+&Baghdad&\phantom{-}33.3333&\phantom{-}\phantom{1}44.4333
&200.0637&200.1583
&$-.0946$&\phantom{1}$-5.67$\cr
\+&Cairo&\phantom{-}30.0500&\phantom{-}\phantom{1}31.2500
&136.2137&136.0561
&\phantom{$-$}.1576&\phantom{$-$}\phantom{1}9.46\cr
\+&Chicago&\phantom{-}41.8333&$-87.7500$
&\phantom{1}48.5875&\phantom{1}48.5209
&\phantom{$-$}.0666&\phantom{$-$}\phantom{1}3.99\cr
\+&Cordoba&\phantom{-}37.8833&\phantom{0}$-4.7667$
&100.3041&100.1910
&\phantom{$-$}.1131&\phantom{$-$}\phantom{1}6.78\cr
\+&Damascus&\phantom{-}33.5000&\phantom{-}\phantom{1}36.3167
&164.7021&164.6278
&\phantom{$-$}.0743&\phantom{$-$}\phantom{1}4.46\cr
\+&Istanbul&\phantom{-}41.0333&\phantom{-}\phantom{1}28.9500
&151.5875&151.4770
&\phantom{$-$}.1105&\phantom{$-$}\phantom{1}6.63\cr
\+&Jakarta&$-6.1333$&\phantom{-}106.7500
&295.1509&294.9765
&\phantom{$-$}.1744&\phantom{$-$}10.46\cr
\+&Jidda&\phantom{-}21.5000&\phantom{-}\phantom{1}39.1667
&\phantom{1}97.2106&\phantom{1}97.1680
&\phantom{$-$}.0426&\phantom{$-$}\phantom{1}2.55\cr
\+&Kabul&\phantom{-}34.5167&\phantom{-}\phantom{1}69.2000
&250.8290&250.9551
&$-.1261$&\phantom{1}$-7.56$\cr
\+&Khartoum&\phantom{-}15.5500&\phantom{-}\phantom{1}32.5333
&\phantom{1}48.5631&\phantom{1}48.7363
&$-.1732$&$-10.39$\cr
\+&Marrakech&\phantom{-}31.8167&\phantom{0}$-8.0000$
&\phantom{1}91.5913&\phantom{1}91.5139
&\phantom{$-$}.0774&\phantom{$-$}\phantom{1}4.65\cr
\+&Medina&\phantom{-}24.5000&\phantom{-}\phantom{1}39.5833
&175.8084&175.7846
&\phantom{$-$}.0239&\phantom{$-$}\phantom{1}1.43\cr
\+&Mombasa&$-4.0667$&\phantom{-}\phantom{1}39.6667
&\phantom{1}\phantom{1}0.3437&\phantom{1}\phantom{1}0.3460
&$-.0024$&\phantom{1}$-0.14$\cr
\+&Riyadh&\phantom{-}24.6500&\phantom{-}\phantom{1}46.7667
&244.5752&244.7080
&$-.1328$&\phantom{1}$-7.97$\cr
\+&Tashkent&\phantom{-}41.2667&\phantom{-}\phantom{1}69.2167
&240.3127&240.4550
&$-.1423$&\phantom{1}$-8.54$\cr
\+&Tehran&\phantom{-}35.6667&\phantom{-}\phantom{1}51.4333
&218.5599&218.7030
&$-.1431$&\phantom{1}$-8.59$\cr
\+&Tombouctou&\phantom{-}16.8167&\phantom{0}$-2.9833$
&\phantom{1}76.4880&\phantom{1}76.5301
&$-.0421$&\phantom{1}$-2.53$\cr
\+&Trabzon&\phantom{-}41.0000&\phantom{-}\phantom{1}39.7167
&179.6976&179.6962
&\phantom{$-$}.0013&\phantom{$-$}\phantom{1}0.08\cr
\vskip.5cm
\hrule
\vskip.2cm
\hrule
\vskip.5cm

\midinsert \vskip4.0in
\noindent{\bf Figure 1.}  The error in the qibla azimuth for
various places when calculated on a sphere.  The results are
given in  minutes  of  arc.   The plotted value of the qibla
error,  $QE = az(S) - AZ(E)$,  is  the  amount  that must be
subtracted  from  the  incorrect  but more easily calculated
simple  azimuth,  $az(S)$,  in  order  to  obtain  the  more
accurate geodetic azimuth,  $AZ(E)$,  calculated on Clarke's
(1866) Ellipsoid representing the Earth.  The variations are
complex near Mecca, located at 21.4 degrees N., 39.8 degrees
E., and at the antipodes  of  Mecca.   Note  the non-uniform
contour  intervals,  the  incomplete  contours in regions of
high  contour  line  density  and  some intermediate contour
fragments, shown dashed. \endinsert

When  these  results are considered it is clear that qibla errors
on the order of 0.1  degrees (0$^{\circ}$ $06'$) will result when
azimuths are  calculated assuming a spherical earth.  Not only is
this  true  for  qibla azimuths, but it is also true for azimuths
calculated  for  any other purpose.  Clearly, azimuths calculated
assuming a spherical earth will not, in general, be accurate to a
tenth  of  a degree and should not be given in a way that implies
such accuracy.

It  would  not  be  appropriate  to  criticize  historical  works
concerning the qibla problem for lacking such accuracy.  However,
knowledge of the  ellipsoidal  shape  of  the Earth is now widely
known --- clear  descriptions  are  to  be found in many texts on
physical geography.  I wish to raise two questions:
1)   Is  there  an  instance  in  recent  or  contemporary  works
concerning  the  qibla  problem"  where  the  problem  has been
considered  with  due  regard for the ellipsoidal (non-spherical)
shape of the Earth?
2)  Would  Islamic  legal,  religious or geographic scholars have
any  interest  in  this  small  but  noticeable  correction  to a
traditional solution of the qibla problem"?

\noindent{\bf The Geographic Center}

There exists, in north central Wisconsin, less than 3/4 kilometer
to  the north and west of the very small community Poniatowski, a
monument with the following text:
\midinsert
\hrule
\vskip.2cm
\hrule
\vskip.5cm
\centerline{\sl GEOLOGICAL MARKER\/}
\noindent
{\sl This marker in Section 14, in the Town of Rietbrock,
Marathon County  is the exact center of the northern half
of  the  Western  Hemisphere.   It  is here that the 90th
meridian of  longtitude (sic)  bisects the 45 parallel of
latitude, meaning it is exactly halfway between the North
Pole and the Equator,  and is a quarter of the way around
the earth from Greenwich,
England.\/}
\vskip.2cm
\centerline{\sl MARATHON COUNTY PARK COMMISSION}
\vskip.5cm
\hrule
\vskip.2cm
\hrule
\endinsert

The location of Poniatowski near this unique geographic point has
given  it  sufficient fame to be mentioned in newspaper articles,
some tourist  literature  and  even celebrated in song (Berryman,
1989).

If  the  Earth  were  spherical  or much more nearly so, then the
statements  on the marker would be true enough.  But, as a result
of  the  Earth's  ellipsoidal shape:   a) the place marked is not
halfway between the  Equator and the pole, b) the place marked is
well removed from the  center" and c) the halfway point and the
center are well  separated from one another.  (Note, however, the
Earth's ellipsoidal shape  not withstanding,  the  monument  does
mark the place, 90 W longitude, 45 N latitude, well enough.)  The
monument's failure in marking the halfway point and the center is
substantial and each failure will be discussed in turn.

{\sl Halfway Point\/}:

\noindent  Because  of  the  ellipsoidal  shape of the Earth, the
length (measured on the surface) of a degree of geographic  (that
is,  geodetic)  latitude  varies with latitude.  As a result, the
point that is equidistant  from  the  pole and the equator is not
simply the midpoint in latitude.  Using Clarke's (1866) ellipsoid
and the various  relationships  in  the  geometry of an ellipsoid
(Bomford,  1971, Appendix A)  it  is  a  straightforward calculus
problem  to  find the equidistant point.  It is at the geographic
latitude  45.1447  =  45$^{\circ}08'41''$  (see Appendix B).  The
place  with this latitude is about 16 km. from the one marked and
sufficiently far  from  Poniatowski  as to place it well into the
next county to the north, Lincoln County.

{\sl Center\/}:

\noindent  The  concept of the geographic center (center of area)
for a curved surface  is  not as straightforward as when the area
is flat.  What  is usually meant by the center is the average (or
mean)  location.   The  location  coordinates  used (latitude and
longitude)  are  curvilinear rather than rectangular.  Because of
this,  one {\bf may not\/}  average the latitude and longitude of
the  elements of area that make up the whole in order to find the
center  (average location)  of  the whole area.  In order to make
this point  more clear,  consider Figure 2.   Shown shaded is the
northwest quadrant of  the Earth.  On a sphere, this area shows a
great deal of symmetry  about  the point at latitude 45$^{\circ}$
N., longitude 90$^{\circ}$ W.  Surely the center of this quadrant
on the surface of a sphere is at this central point.  But, if one
calculates the average latitude of the various area elements that
make up the northwest  quadrant  on  the surface of a sphere, the
result  is  32.7042  degrees or 32$^{\circ}42'15''$ N. Surely the
center is not there. (Other statistics are no better when applied
to latitude alone --- the median latitude  is 30$^{\circ}$ N. and
the modal latitude is 0$^{\circ}$.)  What must be averaged is the
location,  {\bf not\/} the coordinates of the location.   Phrased
differently, the latitude of the center of area is different from
the average latitude of the same area.

\topinsert \vskip 6in
\noindent{\bf Figure 2.}   The geographic center (center of area)
of  the  northwest  quadrant  of  the  Earth  (or  the upper left
quadrant of a sphere or an ellipsoid) and other statistics.
A)  An oblique view of the Earth showing the northwest quadrant.
B)  The region of the northwest quadrant near the median and mean
latitudes of the quadrant on the 90th meridian.
C)  The  region  of  the  northwest  quadrant near the geographic
center.  The center was  determined  by  the preferred method
(Barmore, 1991);  that is,  calculated  with the computations
and the result restricted  to  the surface.  The ellipsoid is
Clarke's (1866) ellipsoid.  \endinsert

Any  satisfactory  method  of  finding  the center must take into
account  the  curved surface of the Earth in a suitable way.  One
method is to calculate the center by assuming that the quantities
spread  over  the  two  dimensional  surface  of  a   sphere  are
distributed  in  a  three-dimensional  Euclidean space (as indeed
they are). One early geographical use (the earliest I have noted)
of  this  three-dimensional"  method  for  finding  centers  of
population (or area) on the surface of a sphere was derived by I.
D.  Mendeleev  and  used by his father, D. I. Mendeleev (1907 and
earlier)  to  find  the centers of area and population of Russia.
Such a  method  is  easily  extended to calculating the center of
area or population on the surface of an ellipsoid.

I believe  an  alternative method is preferable --- a method that
restricts the  computations  and  the results to the surface of a
sphere.  We are largely confined to the Earth's surface and it is
appropriate  to  adopt this provincial viewpoint when determining
the center of population or geographic center.  This is discussed
elsewhere in some  detail (Barmore, 1991).   Whichever of the two
methods  is  used  (computations  {\bf in\/}  the earth  in three
dimensions  or  computations  {\bf  on\/}  the  surface  in   two
dimensions)  the  geographic  center  (center  of  area)  of  the
northwest  quadrant  of  a  spherical Earth is at 90$^{\circ}$ W.
longitude and 45$^{\circ}$ N. latitude.

But the Earth is  not  spherical.   The  Earth  is ellipsoidal in
shape.   When  these  computations are done for an ellipsoid, one
finds that the geographic center is far removed from 45$^{\circ}$
N. latitude (though it remains on the 90th meridian). I have used
both  methods to calculate the geographic center of the northwest
quadrant  for  Clarke's  (1866)  ellipsoid  using the ellipsoidal
geometry found in Bomford (1977) and find the center is  about 22
km. to the north, well into the next county,  Lincoln County,  at
about  45$^{\circ}12'$  N.  latitude.  In addition  to  being far
above  the  45th  parallel  and  far  removed  from  Poniatowski,
Wisconsin, the center is also far removed from the  point  midway
between  the  equator  and  the  pole (see Figure 2).  Though the
monument marks the intersection of the 45th parallel  of latitude
with the 90th meridian well enough,  it marks {\bf neither\/} the
point  midway  between  the  equator and the pole {\bf nor\/} the
center of the northern half of the western hemisphere. The claims
of the marker that it is at the exact center  of  the  northern
half  of  the  Western  Hemisphere  $\ldots$ " and  $\ldots$ is
exactly  halfway  between  the   North  Pole   and  the  Equator,
$\ldots$ " are simply not true.

\noindent{\bf The Center of Population}

When  calculating  the center of population of the United States,
the Bureau of the  Census explicitly states that it has assumed a
spherical Earth (U.S. Bureau of the Census, 1973).  But the Earth
is  ellipsoidal in shape, not spherical.  The formul{\ae} used by
the  Census Bureau  for the center of population calculation  are
not  particularly  suitable  for the computation of the center of
populations on a sphere, let  alone  an ellipsoid.   As  has been
previously pointed out in considerable  detail  (Barmore,  1991),
the  Census Bureau  formul{\ae}  do  not  take  the  curvature of
Earth's surface into account in an appropriate way.  But, however
the  center  of  population is calculated  for populations on the
surface of a sphere,  the  questions  remains:   What will be the
center  of  population  for  populations  on  the  surface  of an
ellipsoid?   As  indicated in the previous section, there are two
methods  of  computing  centers  on  spherical  surfaces  and the
procedures  can  be  extended  to  the problem of calculating the
center  of  population  of the United States on the surface of an
ellipsoid.

I have  calculated  the center of population of the United States
for  1980  using  Clarke's  (1866)  ellipsoid and the ellipsoidal
geometry  given in  Bomford (1977) two ways:   1) {\bf in\/}  the
Earth in three dimensions  and  2)  {\bf on\/} the surface in two
dimensions as outlined in a  previous paper (Barmore, 1991).  The
same example data set was used in  all  cases.   The  results  of
these computations as well as previously derived  results for the
spherical case are shown in Table 2 and Figure 3.
\topinsert
\vskip.5cm
\hrule
\vskip.2cm
\hrule
\vskip.2cm
\centerline{Table 2.}
The  Center  of  Population  for  1980  for  the   United  States
calculated  by  various methods  for  the  same  example data set
previously used (Barmore, 1991).
\vskip.2cm
\hrule
\vskip.5cm
\centerline{Center of Population}
\settabs\+\qquad&In three dimensions for an ellipsoid\quad
&$COP-E$\quad
&latitude\quad &longitude\quad &\cr %sample line
\vskip.5cm
\+&Method of computation&label&latitude&longitude&depth\cr
\vskip.5cm
\+&Bureau of the Census formul{\ae}
&$BC$
&38.1376&90.5737&---\cr
\+&In three dimensions for a sphere
&$s$
&39.1823&90.3477&165 km\cr
\+&In three dimensions for an ellipsoid
&$e$
&39.1887&90.3469&165 km\cr
\+&On the surface of a sphere
&$COP$
&39.1980&90.4978&\phantom{16}0\cr
\+&On the surface of an ellipsoid
&$COP-E$
&39.2045&90.4969&\phantom{16}0\cr
\vskip.5cm
\hrule
\vskip.2cm
\hrule
\vskip 3.5in
\noindent{\bf Figure 3.}
The  Center  of  Population"  of  the  United States  for  1980
calculated by various methods.  The place shown as an open circle
and labeled $BC$, is the center determined by the U.S. Bureau  of
the Census (1983).  As discussed previously  (Barmore, 1991) this
place should not be called the  center of population.  The places
shown as solid circles and labeled $s$ and $e$,  mark the centers
calculated in three dimensions assuming  the population is on the
surface  of  a  sphere  or  on  the  surface  of  Clarke's (1866)
ellipsoid, respectively.  The calculated centers lie  {\it ca.\/}
165 km below the places marked.  The places shown as an  asterisk
or a plus and labeled $COP$ or $COP-E$ are the centers calculated
using  the  preferred  method  (Barmore,  1991)  and  assumes the
population  is  on  the  surface of a sphere or on the surface of
Clarke's  (1866) ellipsoid,  respectively.   The preferred method
restricts  the  computation and results to the surface (sphere or
ellipsoid) containing the population. \endinsert

When these results are considered it is clear that the difference
between  the  center  obtained  with  the  Bureau  of  the Census
formul{\ae}  and  the  center obtained using the preferred method
(or the other  reasonable  alternative) is substantial.  However,
the  error  in  ignoring  the  ellipsoidal  shape of the earth is
smaller --- less  than a minute of arc difference in the location
of the center of population.

The  Bureau of the Census  gives  the center of population to the
nearest  second of arc  of latitude and longitude.  If one wishes
to pursue the  location of the center of population of the United
States  to  an  accuracy  of  one  second  of  an  arc  then  the
ellipsoidal   shape   of   the   Earth   (and  a  host  of  other
considerations) should be taken into account.

\noindent{\bf Summary}

The Earth is not spherical.   The Earth  is ellipsoidal in shape.
When computations are done without due regard for the ellipsoidal
shape  of  the Earth they may be in error by amounts on the order
of 1/10 degree.  This paper points out:  1)  that errors of  {\it
ca.\/} 1/10 degree result in qibla (and other azimuths calculated
on a sphere, 2) that errors  of  {\it ca.\/} 1/10  degree  result
in  the  location  of  the  geographic center of very large areas
calculated on a sphere, but 3) that the  error in the location of
United States  population  center  when  properly calculated on a
sphere is less than one minute of an arc.
\vfill\eject
\centerline{\bf Appendix A}

Because the Earth is not a sphere  (nor, for that matter, exactly
an  ellipsoid  of  revolution)  a  certain amount of care will be
needed in  using  the terms {\bf azimuth\/} and {\bf distance\/}.
This paper  uses several terms (described below) which correspond
closely  to  those  defined  and  used  by Bomford (1977).  Also,
several other concepts deserve additional comment.

ASTRONOMICAL AZIMUTH:  For places on the physical surface of  the
Earth,  the  astronomical  azimuth  of  one  place  from  another
corresponds to what would be measured with an accurate instrument
located on the {\bf surface of the Earth\/}.

GEODETIC  AZIMUTH:   For  places  on  the surface of an ellipsoid
representing the Earth, the geodetic  azimuth  of  one place from
another  is  what  would  be measured with an accurate instrument
located on the  {\bf  surface of the ellipsoid\/}, the instrument
being  leveled"  relative  to  the  ellipsoid's  normal  at the
instrument's location rather than the gravitational field."

SIMPLE AZIMUTH: For places on the surface of a sphere, the simple
azimuth of one  place  from  another corresponds to what would be
measured with an accurate  instrument located on the {\bf surface
of the sphere\/}, the  instrument  being  leveled'' relative to
the  sphere's normal at the instrument's location rather than the
gravitational field."

DISTANCE:   For places  on the surface of an ellipsoid, distances
between  places  are  often measured along the normal sections"
rather  than  along  geodesics.  For places  on the  surface of a
sphere, distances between places are almost always measured along
geodesics, called great circles.

On  the  sphere  simple  azimuths  and great circle distances are
easily  calculated with spherical trigonometry.  On the ellipsoid
geodetic  azimuths and normal section distances are determined by
more complex calculations. In this paper Cunningham's formula was
used  for  Geodetic Azimuth (Bomford, 1977, Eq. 2.23) and Rudoe's
9-figure"  formula  was  used  for  distances  along the normal
section (Bomford, 1977, p. 136).

LOCATION:   Places are  located  on  a sphere, an ellipsoid or an
accurate  map  according  to their geographic (that is, geodetic)
coordinates.

ACCURACY:   Roughly speaking,  calculations done on a sphere will
represent  distances  and  direction  on  the real surface of the
Earth with an accuracy of one degree or more.   Calculations done
on  a  suitable  ellipsoid will represent distances and direction
on the real surface  of the Earth  with an accuracy of one minute
of  arc  or  more.  For an accuracy of one second of arc or more,
details  such  as  the  choice  of  the ellipsoid parameters, the
Earth's  gravitational  field  and  heights of the various places
must  be  taken  into  account.   For  the purposes of this paper
(accuracy of one minute of arc) geodetic  azimuths  and distances
in  the normal sections  represent the real case well enough.  It
is a  rare case  indeed  that the difference between the geodetic
and the  astronomical quantities  would be so large as one minute
of arc (Bomford, 1977, p. 115, 528).  In  the  main  text  of the
paper  results  are often stated to the nearest second of arc (or
0.0001 degree).  It should be kept in mind that these results are
the  geodetic  results.   This level of accuracy is justified for
comparisons  of  similar  results  but  it  is  not  the absolute
accuracy of quantities on the physical surface of the Earth.

ELLIPSOID:  All  the  calculations  involving  the  ellipsoid and
discussed  in  the  main  part  of  the text used Clarke's (1866)
Ellipsoid, a=6378.2064 km. and e=0.08227185.  The geometry of the
ellipsoid  and  various  series  expansions  for  some   of   the
relationships were those given by Bomford (1977, Appx. A, C).

NOTATION:   All azimuths  are  measured from the North toward the
East and are always positive  (i.e.,  SW = $+225$  degrees, never
$-135$).   Angles  are  given  in  degrees  and  decimal  degrees
(sometimes  without  the  unit  name or symbol) or in degrees and
minutes of arc (and sometimes seconds of arc) and always with the
symbols: dd$^{\circ}$mm$'$ss$''$).

COMPUTATIONS:  All  computations  were  done  on  an  Apple  IIGS
computer using the spreadsheet in AppleWorks 3.0 (Claris Corp.).
\vfill\eject

\centerline{\bf Appendix B}
\centerline{\bf Half-way Point Calculation.}
\vskip.1cm
\centerline{(added to this reprinting at the request of the Editor.)}

\noindent   If the distance from the equator to the pole measured
along a meridian on the surface of the ellipsoid is $s$, then:
$$s=\int_{\hbox{equator}}^{\hbox{pole}} ds .$$
Rewriting  this in  terms of the radius of curvature, $\rho$, and
the  geographic  (geodetic) latitude, $\phi$, the latitude of the
half-way point, $\Phi$, is then given by:
$$\int_0^{\Phi} \rho\,\cdot\,d\phi = {1\over 2}\,\, \int_0^{\pi / 2} \rho\,\cdot\,d\phi = {1\over 2}s.$$
Bomford  (1977, eq. A.53)v gives the radius of curvature in terms
of  the  semi-major  axis  $a$,  the  eccentricity  $e$,  and the
geographic latitude.  Then:
$$\int_0^\Phi {{a(1-e^2)\,d\phi}\over{(1-e^2\hbox{sin}^2\phi)^{3/2}}} = {1\over 2}\,\, \int_0^{\pi /2} {{a(1-e^2)\,d\phi}\over{(1-e^2\hbox{sin}^2\phi)^{3/2}}}$$
Cancelling  common  terms,  using the  binomial expansion ($e$ is
small), and evaluating the resulting series of definite integrals
on the right hand side (RHS) one finds:
$$\hbox{RHS} = {\pi \over 4}\, [1 + {3\over 2} \cdot e^2 \cdot {1 \over 2} + {{3\cdot 5}\over {2\cdot 4}} \cdot e^4 \cdot {{1\cdot 3}\over {2\cdot 4}} + \cdots ].$$
The left hand side (LHS) integrals can be reduced (with a certain
amount of algebraic and trigonometric manipulation) to:
$$\hbox{LHS} = \Phi + {3\over 2} \cdot e^2 \cdot ({\Phi\over 2}-{{\hbox{sin}2\Phi}\over 4}) + {{3 \cdot 5}\over {2 \cdot 4}} \cdot e^4 \cdot ({3\over 8}\,\Phi - {{\hbox{sin}2\Phi}\over 4} + {{\hbox{sin}4\Phi}\over {32}} +\cdots ).$$
Ignoring the smaller terms --- terms containing $e^4$, $e^6$ etc.
(using the eccentricity for Clarke's 1866 ellipsoid) yields:
$$\Phi = 0.787923557 = 45.1447^{\circ} = 45^{\circ}08'41'' .$$
Including terms containing $e^4$ and $e^6$ yields:
$$\Phi = 0.787945019 = 45.145924^{\circ} = 45^{\circ}08'45''.$$
\vfill\eject
\noindent{\bf References}

\ref Barmore, Frank E. 1991.  Where Are We?
Comments on the Concept of the Center of Population'."
{\sl The Wisconsin Geographer\/}, Vol. 7, 40-50.
(Reprinted (with the example data set used and with
several corrections) by the Institute of Mathematical Geography,
Ann Arbor, MI, in their journal, {\sl Solstice:
An Electronic Journal of Geography and Mathematics\/},
Vol. III, No. 2, 22-38, Winter, 1992.)

\ref  Barmore, Frank E.  1985.  Turkish Mosque Orientation
and The Secular Variation of the Magnetic Declination."
{\sl The Journal of Near Eastern Studies\/}.
Vol. 44, No. 2, 81-98.

\ref  Berryman, Peter.  1989.  PONIATOWSKI."
{\sl The New Berryman Berryman Songbook\/},
Madison, WI, Lou and Peter Berryman.

\ref Bomford, G. 1977.  {\sl Geodesy\/}.  Oxford UK,
Clarendon Press.  A reprinting (with corrections)
of the 1971 3rd Ed.

\ref Head, J. W., et al. 1981.  Topography of Venus and Earth:
A Test for the Presence of Plate Tectonics."
{\sl American Scientist\/},
Vol. 69, 614-623.

\ref King, D. A., 1972.  Kibla."  {\sl The Encyclop{\ae}dia of
Islam\/}, 2nd ed., Vol. 5, 83-88.

\ref King, D. A., 1978.  Three Sundials from Islamic Andalusia."
{\sl Journal for the History of Arabic Sciences\/},
Vol. 2, 358-392.

\ref King, D. A., 1982a.  Astronomical Alignments in Medi{\ae}val
Islamic Religious Architecture."
{\sl Annals of the New York Academy of Sciences\/},
Vol. 385, 303-312.

\ref King, D. A., 1982 b.  The World about the Kaaba."
{\sl The Sciences\/}, Vol. 22, 16-20.

\ref Mendeleev, D. I. 1907. {\sl K Poznaniyu Rossii\/},
5th ed. St. Petersburg, A. S. Suvorina, p. 139.

\ref Mutch, T. A., et al.  1976.  {\sl The Geology of Mars\/}.
Princeton NJ, Princeton University Press,
pp. 61-63, 209, and 213.

\ref U. S. Bureau of the Census.  1983.
{\sl 1980 Census of Population, Vol. 1\/},
Chapter A, Part 1 (PC80-1-A1).
Washington DC; U.S. Dep't. of Commerce, Bureau of the Census.
Appendix A, p. A-5 and Table 8, p. 1-43.

\ref U. S. Bureau of the census.  1973.
{\sl 1970 Census of Population and Housing:
Procedural History\/} (PHC(R)-1).
Washington DC; U.S. Dep't. of Commerce, Bureau of the Census.
Appendix B (Computation of the 1970 U.S. center of population),
pp. 3-50.

\ref {\sl The Times Atlas of the World\/}, 8th Comp. Ed.  1990.
New York, Times Books / Random House.

\ref {\sl Mecca al-Mukarrama, 1:15000\/} 1970?
(Riyadh, Kingdom of Saudi Arabia,
Ministry of Pe\-tro\-le\-um and Resources,
Aerial Survey Dep't.)
\smallskip
\smallskip
$^*$
Frank E. Barmore,
Department of Physics, Cowley Hall,
University of Wisconsin, La Crosse,
La Crosse, WI 54601
\vfill\eject
\centerline{\bf MICROCELL HEX-NETS?}
\vskip.5cm
\centerline{\bf Sandra Lach Arlinghaus $^*$}
\vskip1cm

The   ongoing  revolution  in  electronic  communications  offers
exciting  opportunities  to  realize  geographic ideas in perhaps
unimagined  electronic  realms.   It  is  well-known,  throughout
governmental,  business,  and  academic  communities,  that   the
cartographer can make a map from hundreds of electronic layers in
a Geographic Information  System (GIS),  in which the data behind
the  map  work  interactively with the map, so that upgraded data
produces  an  upgraded map.  GIS is certainly one exciting result
of the interaction between traditional science and electronics.

Cordless  telephones  offer  other prospects:  networks of mobile
terminals can be linked together  in networks across city streets
as well as within office skyscrapers.  Chia (1992) notes that the
Research on  Advanced Communications for Europe (RACE) initiative
of  1988,  to  study  techniques  to implement a third generation
Universal Mobile Telecommunications System by the year 2000, is a
significant  step  toward   unifying   communications  and  fixed
networks.

The  concept  of a  mobile  telecommunication  is straightforward
(Chia 1992).  Simply stated, a  set  of  microcell base stations,
each of which can transmit and receive electronic information, is
spread across  a geographic  space as a network of stations, each
with  its  own  tributary  area,   a  microcell,  with  which  it
communicates.  Typically, one might  think  of the microcell base
station as the center of a circular tributary area, with circular
areas packed to cover a larger  circular area.   At the center of
the larger circular area,   a macrocell base station serves as an
umbrella" to relay information  to  the microcell base stations
under it, and from one  network  of  microcells to the next (Chia
1992).  Within this sort of mixed cell architecture," a vehicle
carrying  a  terminal  onboard  passes through the microcells and
receives information on  a continual  basis from the base station
associated with the  microcell  it is currently traversing.  This
sort of hand-off of information in order to traverse a network is
not  new;   indeed,  the Rohrpost ---  an  underground network of
pneumatic tubes  used for  message transmission  in Berlin in the
early 1900s --- was composed of energy regions in which pneumatic
carriers were handed off from one region to the next in order  to
transmit  messages  across  a  fairly   large   geographic   area
(Arlinghaus 1986).  More commonly, a relay foot race involves the
handing off of a baton  from  one  runner  to  a second, once the
first runner has expended much  energy to traverse some specified
geographic space.  There are a host  of  other  illustrations  of
this sort.

There  are  apparently  numerous  engineering concerns associated
with  the  optimal  positioning  of  the  base  stations: antenna
radiation patterns,  natural terrain features,  interference from
tall  buildings,  and  interference and signal attenuation of all
sorts, including difficulties when the mobile unit turns a corner
(Chia, 1992). The geometry of directional paths through Manhattan
space (Krause 1975),  based  on number of  vehicle turns can then
also become of concern (Arlinghaus and Nystuen 1989).

It  is  the  geographic  issues  of  street patterns and building
position  that  are  fundamental  to the  engineering concerns in
implementing these networks  in which moving vehicles interchange
information  with  a  fixed network of base stations (Chia 1992).
Even a brief glance at an atlas shows  the  range of variation in
street  pattern --- from  the  predominantly  rectilinear grid of
Manhattan,  to  the  polar-coordinate  style of rotary and radial
evident in Washington D.C. Thus, many studies involving microcell
networks are done, initially, in an abstract  environment  (Chia,
1992) --- as a benchmark against which to evaluate others in less
than  optimal  environments.  It  is  within  this  spirit that a
microcell  system,  composed  of  layers of microcells of varying
size, is viewed.

\noindent{\bf Lattices}

Viewed  broadly,  microcell  base  stations  are a set of lattice
points.  The  way  in which the lattice is constructed can affect
all  other  considerations  of  the functioning of the consequent
microcell  network.  There  are  an infinite number of general"
environments that one might  use  in which to construct benchmark
networks.   When  the  size  of  the  microcells  is sufficiently
large," the microcell tributary areas might be viewed as curved
surfaces  which  when  pieced  together  form  a  set  of  plates
composing a  broad continental  (for example)  surface.  When the
size  of  the  microcells (or macrocells) is local" rather than
large,"  curvature  may  not be an issue and the cells might be
treated as plane regions.  (What is local," and what is not, is
a  significant  problem  for  pragmatic  implementation;  at  the
abstract  level  it  is  of  importance  to  note  it,  but   not
necessarily to deal with it directly.)  And, if the line-of-sight
geometry is one that excludes parallelism,  or  that permits more
than  one  parallel,  then  it  may be suitable to view microcell
network  architecture/geography  from  the  non-Euclidean vantage
point of elliptic or hyperbolic geometry (Arlinghaus 1990).

Within  a  plane  region, there are two basic ways of creating an
evenly-spread lattice:   one  with  the lattice points lying in a
grid  pattern,  and  the other with the lattice points lying in a
triangular / hexagonal   grid   pattern   (Coxeter  1961).    The
differences  between  the  two  should be clear to anyone who has
played  the  game  of  checkers  on  both a square board and on a
Chinese" board.  What is not evident,  though,  is the sorts of
patterns  that  emerge  when  one  stacks  layers  of  square  or
hexagonal cells of different sizes in varying orientations.  When
a  square  lattice  is  chosen,  one  style  of  space-filling by
tributary  regions emerges;  when an hexagonal lattice is chosen,
another appears.

\noindent{\bf Microcell hex-nets}

Walter  Christaller  (1933, 1966)  grappled  with  the problem of
overlays  of  hexagonal  nets;  he  did  so  in  the German urban
environment.  One  might  question some of the interpretations of
the patterns, but his analysis of the actual patterns of overlays
is correct.  There are  numerous  discussions  of  this  problem,
often  under  the heading of central place theory" --- or,  how
cities might share interstitial  space  (Christaller  1933, 1966;
Dacey 1965).   When  the focus is on the extent to which space is
filled by portions of the hexagonal outlines, as it might be when
signal attenuation and  interference of radio waves are an issue,
then the fractal approach  which  permits the easy measurement of
the extent to which an infinitely  iterated  overlay of nets will
fill space is useful.

One  way to look at the complicated issue of visualizing overlays
of  hexagonal  nets  is  simply  to  think  of  a central hexagon
surrounded  by six hexagons of the same size --- each of these is
centered on a  microcell  base  station.   The central hexagon is
also centered on a macrocell base station which serves the entire
set  of  seven  hexagons  and  has  as  its  own larger tributary
macrocell,  a  hexagon  formed  by  joining  pairs  of   vertices
(separated by two intervening vertices)  of the perimeter of this
snowflake  region.   When  these  microcells  and  macrocells are
iterated across the plane,  a  stack  of  two layers of hexagonal
cells emerges, with the orientation of one relative to the  other
at an angle that insures that each of the macrocells contains the
geometric equivalent of 7 microcells.  If one zooms in or out, to
generate  other  layers  of larger or smaller hexagons, the stack
may be increased; as long as the angle of orientation is fixed by
the  first  two,  the  value  of  7"  will be a constant of the
hierarchy ---no matter which two adjacent layers of the hierarchy
are  considered,  a  large  cell will contain the equivalent of 7
smaller cells.  In the  literature,  this is often referred to as
the $K=7$" hierarchy.

When  one  chooses  different orientations of the nets, different
$K$  values  emerge;  indeed,  there  are  an  infinite number of
possibilities.  When it is also required that vertices of smaller
hexagons coincide with  those of larger hexagons, there are still
an infinite number of  hierarchies  with the $K$ values generated
by the Diophantine equation $x^2+xy+y^2 = K$  (Dacey 1965)  where
$x$ and $y$ are the coordinates of the lattice points arranged in
a triangular lattice (and so relative to a coordinate system with
the y-axis inclined at $60^{\circ}$ to the x-axis).

A  structurally identical process may be employed to make similar
calculations  for layers of squares centered on a square lattice.
Relationships  which  show  the number of small square microcells
within  a  larger  square  macrocell  are  also  constant between
adjacent layers of  a  hierarchy formed from a single orientation
criterion (J" value).

Fractal geometry may be used to generate any of these hierarchies:
hexagonal or square.  All that is needed is to know the number of
sides  in  a  fractal  generator  and the self-similarity pattern
desired (K- or J-value). From these, one can determine completely
and uniquely the entire hierarchy--both  cell size within a layer
and  orientation  of  layer  (Arlinghaus,  1985;  Arlinghaus  and
Arlinghaus 1989).  The  fractal  dimension measures the extent to
which parts of the boundary  of  the  hexagons  or squares remain
under infinite iteration.  When  the  results of the calculations
are  displayed  in  a  table,  it  appears  that  hexagonal  nets
consistently fill less space (Arlinghaus, 1993).

This  Table  suggests  that  individuals   actually  implementing
microcell  systems  might wish to first consider shape, size, and
orientation  of  layers  of  a  mixed  cell architecture prior to
superimposing any of the geographic concerns  of street networks,
or  engineering  concerns  caused  by  interference  and   signal
attenuation.  A mixed cell architecture of low fractal  dimension
might be one that reduces  interference, to some extent, just  by
the relative positions of microcells to macrocells.

\vfill\eject
\vskip.5cm
\hrule
\vskip.2cm
\centerline{Table:  Comparison of fractal dimensions}
\vskip.2cm
\hrule
\vskip.5cm
\settabs\+\indent\quad&Lattice coordinates of\qquad\qquad
&Fractal Dimension\qquad\qquad&\cr %sample line
\+&Lattice coordinates of       &Fractal Dimension    &\cr
\+µcell base station                            &&\cr
\+&adjacent to                  &Squares     &Hexagons \cr
\+µcell base station                            &&\cr
\+&at $(0,0)$.                                       &&\cr
\vskip.5cm
\+&(1,1)                        &2.0         &1.262\cr
\+&(1,2)                        &1.365       &1.129\cr
\+&(0,2)                        &2.0         &1.585\cr
\+&(0,3)                        &1.465       &1.262\cr
\+&(0,4)                        &1.5         &1.161\cr
\+&(0,5)                        &1.365       &1.209\cr
\+&(0,6)                        &1.387       &1.161\cr
\+&(0,7)                        &1.318       &1.129\cr
\+&(0,8)                        &1.333       &1.153\cr
\+&(0,9)                        &1.290       &1.131\cr
\+&(0,10)                       &1.301       &1.114\cr
\vskip.5cm
\hrule
\vskip.2cm
\hrule
\vskip1cm
\vfill\eject

\noindent{\bf References}

\ref Arlinghaus, S. 1993.  Electronic geometry.  {\sl Geographical
Review\/}, to appear, April issue.

\ref Arlinghaus, S. 1990.  Parallels between parallels.
{\sl Solstice\/}, Vol. I, No. 2.

\ref Arlinghaus, S. 1986.  {\sl Down the Mail Tubes:  The
Pressured Postal Era, 1853-1984\/}, Monograph \#2, Ann Arbor:
Institute of Mathematical Geography.

\ref Arlinghaus, S.  1985.  Fractals take a central place.
{\sl Geografiska Annaler\/} 67B, 2, 83-88.

\ref Arlinghaus, S. and Arlinghaus W.  1989.  The fractal theory of
central place geometry:  a Diophantine analysis of fractal
generators for arbitrary Loschian numbers.  {\sl Geographical
Analysis\/}, Vol. 21, No. 2.  pp. 103-121.

\ref Arlinghaus, S. and Nystuen, J. 1989.
Elements of geometric routing theory.  Unpublished.

\ref Chia, Stanley.  December, 1992.  The Universal Mobile
Telecommunication System.  {\sl IEEE Communications\/}, Vol. 30,
No. 12, pp. 54-62.

\ref Christaller, Walter.  1933 (translated into English 1966).
Baskin translation: {\sl The Central Places of Southern Germany\/}.
Englewood Cliffs:  Prentice-Hall.

\ref Coxeter, H. S. M.  1961. {\sl  Introduction to Geometry\/}.
New York:  Wiley.

\ref Dacey, M. F.  1965.  The geometry of central place theory.
{\sl Geografiska Annaler\/} 47, 111-24.

\ref Krause, Eugene. 1975.  {\sl Taxicab Geometry\/}.  Menlo Park:
Addison-Wesley; 1980, Springer-Verlag.
\smallskip
\smallskip
$^*$
Sandra Lach Arlinghaus,
Institute of Mathematical Geography,
2790 Briarcliff, Ann Arbor, MI 48105.
\vfill\eject
\centerline{\bf SUM GRAPHS AND GEOGRAPHIC INFORMATION}
\vskip.5cm
\centerline{\bf Sandra L. Arlinghaus,
William C. Arlinghaus,
Frank Harary$^{*}$}

\noindent{\bf Abstract}

We examine a new graph theoretic concept  called a  sum graph,"
display  a  new  sum  graph construction, and prove a new theorem
about sum graphs  (sum  graph  unification theorem) verifying the
construction.   The  sum graph is then generalized, ultimately as
an  augmented  reversed  logarithmic  sum  graph,  so  that it is
useful in dealing with large sets of geographic information.  The
generalized  form  permits 1) the compression of large data sets,
and  2)  the  simultaneous  consideration of data sets at various
levels of resolution.

The  advantages  of  employing  sum  graph  unification  and  the
augmented reversed logarithmic sum graph to handle data sets  are
illustrated  by  hypothetical  example;  as a data structure, the
various  forms  of  sum  graph  data  management  provide compact
handling  of  data and do so in a manner that permits variability
of  resolution,  at multiple levels (unlike the quadtree), within
a single layer of mathematical manipulation.

Our  interest  in  creating,  and  exploring,  this  sort of data
structure  rests in searching for structures that are translation
invariant.  Data structures resting on geographic direction, such
as  the  quadtree, seem destined not to be translation invariant;
structures  that  are  not  tied  to the ordering of the space in
which  they  are  embedded,  but  only  to an ordering within the
structure itself, have the potential to be translation invariant.
\vskip.2cm

Geography and graph theory have a long history of interaction:
the  Four Color Problem (now Theorem) and the K\"onigsberg Bridge
Problem  of  graph  theory  arose  as geographical questions.  As
geography  has  stimulated  mathematical creation, so too has the
body  of  theory  developed by graph theorists stimulated careful
analysis  of  various  geographical  networks.  It is within this
well-established   spirit   of   interaction,   and   within  the
technological  framework  where electronic processing of data may
be characterized using graph  theory, that we examine a new graph
theoretic  concept,  called  a  sum  graph, as a theoretical data
structure.

In this structure,  the numerical pattern of the labels of the
nodes in the sum graph" will be dictated by the linkage pattern
in  the  underlying data, rather than the other way around, which
is more conventional. Thus, data that are linear" (sequential),
such as  data  streams in a raster mode, will be represented by a
sum  graph  whose  linkage  pattern  is linear, thereby forcing a
certain style of label to be present on the associated nodes.  We
demonstrate the theoretical concepts in this paper using examples
limited  to  the  linear  case  because it is easy to express and
because it has wide applicability.

Thus,  the  first section introduces the reader to elements of
the  abstract  development  of sum graphs, focusing only on those
concepts that will actually be applied.  The second section shows
how to force correct" labelling of sum graphs" to  permit the
simultaneous   consideration   of  data  at  multiple  levels  of
resolution  within  subsets  of  a  data  set  that  is linear in
character.   The  third  section  introduces   the   concept   of
logarithmic sum graph,"  used  to compress large data sets into
subsets within bands of width of one unit --- a critical strategy
as  the  length  of  the linear sequence (data stream) increases.
The  fourth  section  introduces  the reversed sum graph" which
also permits simultaneous consideration  of data at more than one
scale  and  does  so  with  optimal labelling within bands of one
unit.   The  fifth  section  introduces  the augmented reversed
logarithmic  sum graph," a graph combining the desirable elements
of  previously  considered  structures  augmented  by  a  set  of
linkages,  induced  by  the  numerical labelling of subsets, that
permits  inclusion  of  data at variable levels of resolution and
offers  a means to link that data between, in addition to within,
subsets.   Throughout,  we  show  how  to use these concepts in a
small  application  derived  from  a set of data concerning North
American cities.
\vskip.5cm
\centerline{\bf 1.  Sum Graphs}

\noindent{\sl Definition 1\/} (Harary, 1989)

Let  $S$  be a set of $n$ distinct positive integers.  Define
{\sl the sum graph\/} $G^+(S)$ as follows:

\item{1.}  $G^+(S)$ has $n$ nodes, each labelled with a different
element (number) of $S$;
\item{2.}  there  is  an  edge between two nodes labelled $a$ and
$b$ if and only if $a+b\in S$.

\noindent{Example 1}

Figure 1 shows the sum graph of $S_1=\{1,4,5,7,8,9\}$.  $S_1$
is a set of arbitrarily chosen labels for the nodes.  Because the
label  9"  is  an  element  of  $S_1$, it follows that the edge
linking 4 and 5 ($4+5=9$) is present  in  the graph.  Because the
label 6" is {\sl not\/}  an  element  of $S_1$ there is an edge
linking 1 and 5 ($1+5=6$).  A number of  theorems  concerning sum
graphs  appear  in  the  mathematics  literature  (Harary,  1990;
Bergstrand {\it et al.\/}, 1990,  1991).  We  state those results
without proof; others wishing to employ these methods should read
with understanding the proofs in the mathematics literature, lest
the methods be inappropriately applied  in  different situations.
First note that the largest number in $S$  cannot be the label of
a node joined to any other node.

\midinsert
\vskip.5cm
\hrule
\vskip.5cm
$$\matrix{ & &8 & & \cr & &\bullet & & \cr & &\Big\vert & & \cr 4& &1 & &9 \cr \bullet &------------&\bullet & &\bullet \cr \Big\vert & &\Big\vert & & \cr \bullet & &\bullet & & \cr 5& &7 & & \cr }$$
\vskip.5cm
\centerline{\bf Figure 1.}

\centerline{$G^+(S_1)$:  the sum graph of $\{1,4,5,7,8,9\}$.}
\centerline{Reader is to solidify any dashed lines with a pencil}
\vskip.5cm
\hrule
\vskip.5cm
\endinsert

\noindent{\sl Lemma 1\/} (Harary, 1990)

Every sum graph contains at least one isolated node.

\noindent{\sl Example 2\/}:

The sum graph of $S_2=\{2,3,5,6,10\}$ is displayed in Figure 2.

\midinsert
\vskip.5cm
\hrule
\vskip.5cm
$$\matrix{ 2 &\bullet \cr &\Big\vert \cr 3 &\bullet \cr & \cr 5 &\bullet \cr & \cr 6 &\bullet \cr & \cr 10&\bullet \cr }$$
\vskip.5cm
\centerline{\bf Figure 2.}

\centerline{$G^+(S_2)$:  the sum graph of $\{2,3,5,6,10\}$.}
\vskip.5cm
\hrule
\vskip.5cm
\endinsert

Lemma 1 assures that the node labelled 10 is isolated.  Example 2
illustrates that more than one isolated node is possible;  hence,
the phrase at least" in the statement of Lemma 1.

\noindent{\sl Definition 2\/} (Harary, 1969)

Two graphs $G_1$ and $G_2$ are {\sl isomorphic\/} is there  is
a one-to-one  correspondence  $f$  between  their  node sets such
that, for any two nodes $a$ and $b$ in $G_1$,  $(a,b)$ is an edge
in $G_1$ if and only if $(f(a),F(b))$ is an edge  in $G_2$.  Thus
two graphs are isomorphic not only  when  they look the same, but
perhaps have different labellings of the nodes, but they are also
isomorphic when the graphs do not look  alike  but  have the same
connection pattern --- as are views  of  the same digital terrain
model from different vantage points.  Figure 3  illustrates  this
phenomenon   for   the   graph  of  the  octahedron.   Isomorphic
structures are invariant under geometric translation.

\midinsert \vskip2.5in
\centerline{\bf Figure 3.}
\centerline{The octahedron in two different views
(View A on the left; View B on the right)}
\centerline{The reader should draw it}
\endinsert

\noindent{\sl Notation\/}   Given a set $S$ of positive integers,
write $kS = \{kx : x \in S\}$.

\noindent{\sl Theorem 1\/} (Harary, 1990)

If  $G^+(S)$  is  a  sum  graph  and $S'=kS$, $k$ a  positive
integer, then $G^+(S)$ and $G^+(S')$ are isomorphic.

\noindent{\sl Example 3\/}

Consider the sum graph of  Example 1,  $G^+(S_1)$  with  $S_1 = \{1,4,5,7,8,9\}$.   When $k=3$, we have $S_1 = \{3, 12, 15, 21, 24, 27\}$.   The  distributive  law  of  algebra  guarantees that
exactly  the  same  edges  will  appear  in  $G^+(S_1')$  as   in
$G^+(S_1)$.   For  example,  because  $5\in S_1$,  1  and  4  are
adjacent  in  $G^+(S_1)$; because $3\cdot 5 \in S_1'$, $3\cdot 1$
and  $3 \cdot 4$  are  adjacent  in $G^+(S_1')$, since $3 \cdot 1 + 3 \cdot 4 = 3 \cdot (1+4)$.   Thus,  $G^+(S_1)$ and $G^+(S_1')$
have  the  same  edge  structure  (but different node labellings,
hence,  perhaps,  different  geographic  positions),  so they are
isomorphic.

One  interesting  structure  a  sum  graph  might  have  is a
graph-theoretic path (Harary, 1969).

\noindent{\sl Definition 3\/} (Niven and Zuckerman, 1960)

The sequence of Fibonacci numbers $F_n$ is defined as follows:
$F_1 =1$,  $F_2 = 2$,  $F_n = F_{n-2} + F_{n-1}$ when $n \geq 2$.
For example, the first nine elements of this sequence are  1,  2,
3, 5, 8, 13, 21, 34, 55.

\noindent{\sl Theorem 2\/} (Harary, 1990)

If $S = \{F_1,F_2,\ldots , F_p\}$ is the set consisting of the
first  $p$  Fibonacci  numbers, then $G^+(S)$ consists of a path
connecting $F_1$ and $F_{p-1}$ and the isolated node $F_p$.

\noindent{\sl Example 4\/}

Let  $S_3 = \{1,2,3,5,8,13,21,34,55\}$.   Then $G^+(S_3)$ is
the graph of Figure 4.

\midinsert
\vskip.5cm
\hrule
\vskip.5cm
$$\matrix{ 1 &\bullet \cr &\Big\vert \cr 2 &\bullet \cr &\Big\vert \cr 3 &\bullet \cr &\Big\vert \cr 5 &\bullet \cr &\Big\vert \cr 8&\bullet \cr &\Big\vert \cr 13 &\bullet \cr &\Big\vert \cr 21 &\bullet \cr &\Big\vert \cr 34 &\bullet \cr & \cr 55 &\bullet \cr }$$
\vskip.5cm
\centerline{\bf Figure 4.}

\centerline{$G^+(S_3)$:
A Fibonacci sum graph containing a path and an isolate}
\vskip.5cm
\hrule
\vskip.5cm
\endinsert
\vskip.5cm

\centerline{\bf 2.  Sum Graph Unification:  Construction}

One of the  characteristics  that  distinguishes  a  sum graph
from  other  graphs  is  that  the algebraic rule assigning edges
forces the sum graph to have  at  least one isolated node.  Thus,
in aligning this graph-theoretic concept with geographic notions,
one might,  at  the  outset,  be tempted to look for applications
that require isolating" one  geographic  location from a set of
others, as in site-selection for toxic  waste sites, for prisons,
or for other similar societally-obnoxious facilities.

Further reflection suggests,  however,  that  the power behind
this isolation" might  be  best  exploited  by  considering the
isolated node as one with linkages not visible at the graph-scale
shown, much as inset maps generally do not reveal linkages to the
larger-scale  maps  they   modify.    Thus,   this   cartographic
conception of the  isolated  node  as a node with invisible edges
will provide a  systematic  method for shifting scale, or varying
resolution, without disturbing  the associated spatial structure.
The isolated node  acts  as a cataloging" node functioning at a
scale  different  from  the  content  it  catalogues   (the  term
isolated"  will  therefore  be reserved for the graph-theoretic
case; when  viewed  in a geographic context, the isolated" node
will  be  referred  to  as a cataloging" node to emphasize this
role).

Consider three disjoint sets of nodes, $A$, $B$, and $C$, with
a  linear  linkage  pattern  joining each (Figure 5).  The linear
linkage pattern of each path  is based on some serial arrangement
of data, such as data ordered by longitude from east to west.

\midinsert
\vskip.5cm
\hrule
\vskip.5cm
$$\matrix{ &\bullet &&&&& &\bullet &&&&& &\bullet \cr &\Big\vert &&&&& &\Big\vert &&&&& &\Big\vert \cr &\bullet &&&&& &\bullet &&&&& &\bullet \cr &\Big\vert &&&&& &\Big\vert &&&&& &\Big\vert \cr &\bullet &&&&& &\bullet &&&&& &\bullet \cr & &&&&& &\Big\vert &&&&& &\Big\vert \cr & &&&&& &\bullet &&&&& &\bullet \cr & &&&&& & &&&&& &\Big\vert \cr & &&&&& & &&&&& &\bullet \cr }$$
\vskip.5cm
\centerline{\bf Figure 5.}

\centerline{Three graphs, Left, Middle, and Right, representing
serial linkage of data.}
\vskip.5cm
\hrule
\vskip1.5cm
\endinsert

It is not difficult to obtain the paths, $P_3$,  $P_4$,  $P_5$
of  Figure  5  as  three  distinct  sum  graphs  using Theorem 2.
Fibonacci labelling of  the nodes of Figure 5, shown in Figure 6,
generates (as sum graphs)  exactly the path-patterns of Figure 5;
e.g., the edge joining 2 to  3  in $A$ is present because $2+3=5$
is also a node label.   An  additional node, a cataloging one, is
necessarily introduced in  each  sum-graph,  $A$, $B$, and $C$ of
Figure 6.  When the  label  of  a  cataloging  node  is used as a
label for an entire configuration,  this sum graph represents not
only the linear linkage  within  the  path, but also, at the same
time,  represents  information  (as a label) for the entire path.
Information  at  different  cartographic  scales   is   displayed
simultaneously.

\midinsert
\vskip.5cm
\hrule
\vskip.5cm
$$\matrix{ 1 &\bullet &&&&&1 &\bullet &&&&&1 &\bullet \cr &\Big\vert &&&&& &\Big\vert &&&&& &\Big\vert \cr 2 &\bullet &&&&&2 &\bullet &&&&&2 &\bullet \cr &\Big\vert &&&&& &\Big\vert &&&&& &\Big\vert \cr 3 &\bullet &&&&&3 &\bullet &&&&&3 &\bullet \cr & &&&&& &\Big\vert &&&&& &\Big\vert \cr 5 &\bullet &&&&&5 &\bullet &&&&&5 &\bullet \cr & &&&&& & &&&&& &\Big\vert \cr & &&&&&8 &\bullet &&&&&8 &\bullet \cr & &&&&& & &&&&& & \cr & &&&&& & &&&&&13&\bullet \cr }$$
\vskip.5cm
\centerline{\bf Figure 6.}

\centerline{The three distinct Fibonacci sum graphs showing the paths}
\centerline{$P_3$ (on the left), $P_4$ (middle), and $P_5$ (right).}
\vskip.5cm
\hrule
\vskip.5cm
\endinsert

In Figure 6, the simple Fibonacci labelling scheme of  Theorem
2  produced  three  distinct sum graphs.  Because the same labels
are  re-used,  it  would  not  be possible to compare information
concerning  these  distinct  sum  graphs.   Stronger  theoretical
results follow:  results that will permit such comparison,  while
retaining the desirable asset of simultaneous display of  data at
different cartographic scales.
Consider, as a whole, the set of twelve nodes from  Figure 5.
Find  a  strategy  for  labelling  these  nodes that will produce
exactly the three paths of Figure 5 as  subgraphs of a single sum
graph.   Viewing  the  three  parts of Figure 5 as subgraphs of a
{\sl single\/}  sum  graph  will  guarantee  distinct  labels for
distinct nodes while retaining scale-shift characteristics.

One  way  to  achieve  such a labelling is as follows.  Assign
Fibonacci numbers consecutively (starting with 1) to the nodes of
one subgraph ($A$, in Figure 7).   Continue this scheme to a node
of subgraph $B$; thus, in Figure 7,  $A$ has nodes with labels 1,
2,  3  and  one  node in $B$ has label 5.  It might be natural to
label  the next node in $B$ with the next Fibonacci number --- 8.
However,  this  would introduce an unwanted edge between 3 and 5.
So,  label  the  next  node with one more than the next Fibonacci
number  ---  in  this  case  9  ---  to remove the possibility of
introducing  unwanted  edges.   Label  the remaining nodes in the
Fibonacci-style with 5 and 9 as the first two elements.  Continue
this  scheme  through to one node of subgraph $C$ (labels 14, 23,
and  37  are  thus  introduced).   The  second  node in the third
subgraph  must  not  be  labelled 60, or else an unwanted edge is
introduced  linking  23 to 37.  Call the label of the second node
61".   Continue  labelling  in the Fibonacci style using 37 and
61   as   the   first   two   elements   of   a   Fibonacci-style
label-generating  scheme.  In the case of Figure 7, all nodes are
now  labelled; a single extra node, which is a cataloging one, is
also  labelled.   All  paths of this single sum graph are exactly
those  desired.   The  label associated with the cataloging node,
416, is the catalogue number for the entire configuration;  other
labels  describe  the  local,  linear linkage patterns.  Distinct
labels  correspond  to  distinct  nodes  in  such a way that only
desired  paths  are  introduced  between  nodes.  A  single added
cataloging node permits associating information with a  label for
this  node  at  the  scale of the entire configuration --- in the
manner of object-oriented data structures.

\midinsert
\vskip.5cm
\hrule
\vskip.5cm
$$\matrix{ 1 &\bullet &&&&&5&\bullet &&&&&37 &\bullet \cr &\Big\vert &&&&& &\Big\vert &&&&& &\Big\vert \cr 2 &\bullet &&&&&9&\bullet &&&&&61&\bullet \cr &\Big\vert &&&&& &\Big\vert &&&&& &\Big\vert \cr 3 &\bullet &&&&&14&\bullet &&&&&98&\bullet \cr & &&&&& &\Big\vert &&&&& &\Big\vert \cr & &&&&&23&\bullet &&&&&159&\bullet \cr & &&&&& & &&&&& &\Big\vert \cr & &&&&& & &&&&&257&\bullet \cr & &&&&& & &&&&& & \cr & &&&&& & &&&&&416 &\bullet \cr }$$
\vskip.5cm
\centerline{\bf Figure 7.}

\centerline{A Fibonacci-style of labelling for a sum graph with one
cataloging node (416)}
\centerline{showing the paths $P_3$ (on the left),
$P_4$ (middle), and $P_5$ (right) as subgraphs.}
\vskip.5cm
\hrule
\vskip.5cm
\endinsert

Thus, two levels of variability in resolution  are  displayed ---
that of the linkage pattern within individual subgraphs, and that
of the weight of the entire graph, reflecting  to  some extent on
the size  of  the  data  set,  and the style of its subgraphs and
their  pattern  of  internal  connection (had the subgraph in the
middle  terminated  at  14,  the  subgraph  on the right (with an
added  edge)  would  have  begun with 23 and had an isolated node
with label 419).

Stronger  yet  would  be to construct a single sum graph from
which desired paths  would  emerge  (as in Figure 7) and in which
distinct paths would correspond to distinctly-labelled cataloging
nodes as in Figure 6.  The notion of  wanting one cataloging node
per  desired  path,  in  order  to  ensure greater variability in
resolution, motivates the following definition.

\noindent{\sl Definition 4\/}

Suppose  a set of $n$ nodes is partitioned into $t$ subsets.
Further suppose  $k$ of these subsets contain more than one node.
To  each  of  these  $k$  subsets  add a node.  The resulting $t$
subsets will be called constellations" (Figure 8).

\midinsert
\vskip.5cm
\hrule
\vskip.5cm
$$\matrix{ &\bullet &&&&& &\bullet &&&&& &\bullet \cr & &&&&& & &&&&& & \cr &\bullet &&&&& &\bullet &&&&& &\bullet \cr & &&&&& & &&&&& & \cr &\bullet &&&&& &\bullet &&&&& &\bullet \cr & &&&&& & &&&&& & \cr &\bullet &&&&& &\bullet &&&&& &\bullet \cr & &&&&& & &&&&& & \cr & &&&&& &\bullet &&&&& &\bullet \cr & &&&&& & &&&&& & \cr & &&&&& & &&&&& &\bullet \cr }$$
\vskip.5cm
\centerline{\bf Figure 8.}

\centerline{Three constellations, Left, Middle, and Right,
partition a distribution of nodes.}
\vskip.5cm
\hrule
\vskip.5cm
\endinsert

Now we return to the example of Figure 5,  with  three nodes
added to make three constellations (all  with more than one node,
as in Figure 6).
\noindent   We  seek  some  labelling  for  the  entire  set   of
constellation nodes  (Figure 8),  as nodes of a single sum graph,
that will

\item{1.}  produce the paths $P_3$, $P_4$, $P_5$;
\item{2.}  produce cataloging nodes within the subgraphs
containing $P_3$, $P_4$, $P_5$;
\item{3.}  make retrieval of path structure simple.

\noindent Because there are paths that are  to  be  retrieved  as
subgraphs  of  a  single  sum  graph,  some  sort of Fibonacci or
Fibonacci-style labelling will be needed (Theorem 2).  The labels
from Figure 5  cannot  be  chosen because under that circumstance
distinct nodes  do  not have distinct labels.  Theorem 1 suggests
that  distinctness  in  labelling  as  well  as retention of path
structure  is  achieved  by  multiplying  Fibonacci  numbers   by
constants. Thus,  the  issue  is to know what values to choose as
these  multipliers"  so  that  distinctness  of   node   labels
(required  by  Definition  1)  is  ensured.   Example  5,  below,
suggests  a  general  construction  that   will   satisfy   these
conditions.  It will be proved in full generality in Theorem 3.

\noindent{\sl Example 5\/}

1.  To ensure path structure, give the underlying Fibonacci label
pattern of 1,2,3,5; 1,2,3,5,8; 1,2,3,5,8,13 to, respectively, the
left, middle, and right constellations (Definition 4) in the node
pattern of Figure 8. To produce a set of suitable multipliers for
these nodes, proceed to step 2.

2.  Choose  the smallest prime number greater than the sum of the
largest  and  next  largest  numbers  used  in   the   underlying
Fibonacci pattern.  In this case,  13  is  the  largest number in
the underlying Fibonacci pattern and  8  is  the next largest, so
choose  23,   the  smallest  prime  number  larger  than  13+8=21
(choosing  21  would  introduce  an  unwanted edge).  This number
will  be  the  multiplier for one constellation (in this case, we
arbitrarily choose to use it for the left-hand constellation).

3.  Use  successive  powers  of  23  (23 functions therefore as a
base-multiplier) to label the nodes of successive constellations.
In this case, $23^2$ is used as the multiplier for the right-hand
constellation.  The nodes are now labelled as shown in Figure 9.

\topinsert
\vskip.5cm
\hrule
\vskip.5cm
$$\matrix{ x &\bullet &&&&&x^2 &\bullet &&&&&x^3 &\bullet \cr &\Big\vert &&&&& &\Big\vert &&&&& &\Big\vert \cr 2x&\bullet &&&&&2x^2&\bullet &&&&&2x^3&\bullet \cr &\Big\vert &&&&& &\Big\vert &&&&& &\Big\vert \cr 3x&\bullet &&&&&3x^2&\bullet &&&&&3x^3&\bullet \cr & &&&&& &\Big\vert &&&&& &\Big\vert \cr 5x&\bullet &&&&&5x^2&\bullet &&&&&5x^3&\bullet \cr & &&&&& & &&&&& &\Big\vert \cr & &&&&&8x^2&\bullet &&&&&8x^3&\bullet \cr & &&&&& & &&&&& & \cr & &&&&& & &&&&&13x^3 &\bullet \cr }$$
\vskip.5cm
\centerline{\bf Figure 9.}

\centerline{Sum graph derived from Figure 6 using the base
multiplier and its powers,}
\centerline{writing $x=23$ for brevity.}
\vskip.5cm
\hrule
\vskip.5cm
\endinsert

When this set of nodes is used as the set $S$ of  Definition
1, the  resulting  sum  graph  is  isomorphic to the union of the
three sum graphs  in  Figure  5.   The fact that three cataloging
nodes are introduced by this procedure  gives  an indication from
each  coefficient  of  the  cataloging  nodes of size, shape, and
connection  pattern  of  the  subgraph  it represents (as did the
single cataloging node of 416 for the entire graph in  Figure 7).
The set of steps in Example 5 may be stated more generally as  in
the Construction below.
\midinsert
\vskip.5cm
\hrule
\vskip.5cm
\centerline{\bf Construction:  Sum Graph Unification\/}

Given a set of nodes partitioned into constellations.  To  ensure
a  prescribed  path  structure  linking  the  nodes,  that can be
retrieved  electronically  entirely  (only)  from  the  numerical
characteristics  of  the  labels  for the nodes, assign labels in
the following manner.

1.  Label the nodes of each constellation with Fibonacci numbers,
in order, beginning with the label 1" in each constellation.

2.  Find  a  base multiplier for each Fibonacci label.  Form the
sum  of  the two largest labels from step 1.  The smallest prime
number  greater  than  this sum will serve as a multiplier.  Use
this prime  base  multiplier as the multiplier for labels of the
nodes in one constellation.

3.  Use  successive powers of the prime in step 2 as multipliers
for labels of the nodes in successive constellations.
\vskip.5cm
\hrule
\endinsert

\centerline{\bf 3.  Cartographic Application of Sum Graph Unification}

The  following  application  will  show  how  the  labelling
produced by the Sum Graph Unification Construction might be used.
Consider a set of  seven  North  American  cities  together  with
selected  suburbs  of  those  cities  (Table 1.1).   Column  1 in
Table 1.1  lists  these  cities  and  suburbs  in seven groups as
metropolitan areas (the latter  named in all upper case letters):
constellations.   To  consider  the  east-west  extent a proposed
metropolitan  mass  transit  system  might  need  to  cover,  the
longitude  is  also associated with each location (in column 2 of
Table 1.1).   The  sequential  ordering of cities and suburbs, by
longitude  from  east  to  west,  describes  a  path  within each
constellation  linking  these  nodes.   The  metro area node is a
cataloging  node  not  hooked into the path.  Column 3 associates
a  Fibonacci  number with each node of the entire distribution of
nodes (step 1  in the  Construction).  Column 4 shows weights for
the nodes by constellation; 37  is the base multiplier because it
is the  smallest  prime  greater than 21+13 (steps 2 and 3 in the
Construction).   Column  5  shows the product of columns 3 and 4;
distinct nodes have distinct labels.

Suppose  the  entire list is rearranged by longitude, independent
of  constellation;  positions  of  data  within  all  but the New
Orleans  constellation  remain  the  same.   In  the  New Orleans
constellation,  the  suburb  of  Metairie is shifted from the New
Orleans  constellation  to  the  St. Louis constellation (between
E. St. Louis  and  Lemay).  That Metairie jumps metropolitan area
is  evident  from  the  factored  weight  associated with it:  it
belongs  to constellation 7, that of New Orleans, as its exponent
in  the  factored  weight shows (Table 1.2).  Thus, the sum graph
node label shows that  it is out of regional order and provides a
direct  means  to  re-sort   it   back   into   regional   order.
Rank-ordering or other conventional  means  would not do so; rank
ordering does not show which city belongs in which constellation.
These sum graph node labels offer a way to  organize  data and to
retrieve predetermined  sequential  order  of  information from a
jumbled  data  set.   The  node  labels  are  somewhat  large  in
magnitude, but that is irrelevant in this particular application.
It may be important in others, and thus  it  is to this issue and
to the related one of data compression that  the remainder of the
material is directed.
\vskip.5cm
\centerline{\bf 4.  Sum Graph Unification:  Theory\/}
The  example  above may prove a useful source of mental reference
points on which to base the formal proof of the following  lemmas
needed to probe Theorem 3 below.  The first Lemma will prove that
there are no unwanted edges linking nodes  within  constellations
and  the  second  one  will prove that there are no edges linking
nodes between constellations.

For  the  most  part,  Theorem  3  is just a formalization of the
method  developed  in  the  example  based on Figure 9.  However,
additional  details  are  necessary  to  allow for constellations
of  a  single  node  (in  these cases no new node is added).  One
might  interpret  such  a  node  as a small city with no suburbs.
(Readers wishing to examine the  rigor of this method should read
Theorem 3 and associated material with care; others might wish to
skip to the next section.)

\noindent{\sl Lemma 3a\/}

Let $a$, $b$, $c$, $i$, $j$ be positive integers.   If $p > a+b$,
and $p > c$, it  is  impossible  for  $a\cdot p^i + b\cdot p^i = c\cdot p^j$ if $j\neq i$.

\noindent{\sl Proof\/}

Note that $a\cdot p^i+b\cdot p^i = (a+b)p^i < p^{i+1} \leq c\cdot p^j$ if $j>i$.  Similarly,  if $j a+b$.  Let $x$, $y$,
$z$  be  positive integers, $x\neq y$.  Then $a\cdot p^x + b\cdot p^y = c\cdot p^z$ is impossible.

\noindent{\sl Proof\/}:

Without loss of generality, assume $x < y$.   Then, $p^y < a\cdot p^x+b\cdot p^y < (a+b)p^y < p^{y+1}$.  Thus, for the equation  to
be possible, $z=y$. But then $a\cdot p^x \equiv 0(\hbox{mod} p)$,
which is impossible, since $ap^x < p^{x+1} \leq p^y$.

\noindent   We  now  formalize  the  ideas   exhibited   in   the
construction of Example 3.

\noindent{\sl Definition 5\/} (Harary, 1970)

A linear tree is a path.  A linear forest is a union of disjoint
linear trees.

{\sl Theorem 3\/} (Fibonacci sum graph unification)

Suppose we are given a set of $n$  nodes,  which  are partitioned
into $t$ subsets, $k$ of which contain more than a  single  node.
Then  there  is  a  set  $S$  of  $n+k$  suitably chosen positive
integers whose sum graph $G^+(S)$ consists of $t$  isolates  ($k$
additional  nodes  and  $t-k$  nodes  from  single-node  subsets)
together with a linear forest of $k$ nontrivial paths.

\noindent{\sl Proof\/}:

Suppose that the $n$ original nodes are $a_1$, $a_2$, $\ldots$, $a_n$.  Divide these into the $t$ desired subsets
$$\{x_{11}, x_{12}, \ldots x_{1n_1}\}$$
$$\{x_{21}, x_{22}, \ldots x_{2n_2}\}$$
$$\ldots$$
$$\{x_{t1}, x_{t2}, \ldots x_{tn_t}\}$$
where $n_1+n_2+\cdots +n_t = n$.   Let $N = 2 + \hbox{max} \{n_1, n_2, \ldots , n_t\}$.  Let $p$ be the smallest prime greater than
$F_N$,  the  $N$th  Fibonacci  number.  Now  label $n+k$ nodes as
follows:

\item{1.}  If $n_i = 1$, label $x_{i1}$ with $p^i$ (subsets  with
exactly one node).
\item{2.}  If  $n_i \neq 1$,  label $x_{i1}$ with $p^i$, $x_{i2}$
with $2p^i,\ldots x_{in_i}$  with  $p^iF_{n_i}$,  and  a new node
$y_i$ with $p^iF_{(1+n_i)}$ (subsets with more than one node).

\noindent   Follow this procedure for all $i$, $1 \leq i \leq t$.
Let  $S$  consist  of  the  original nodes together with the  new
$y_i$s.   Now  consider  constellations  consisting  of the nodes
labelled $x_i$ if $i=1$ and the  nodes $\{x_{i1},\ldots , x_{in}, y_i\}$ is $i\neq 1$.  Then  Theorems  1  and  2 assure that there
are  Fibonacci  paths  $x_{i1}, x_{i2}, \ldots x_{in}$  and  that
$y_i$  is  not  adjacent  to $x_{ia}$ for any $a$ ($1 \leq a \leq n_i$).   Lemma  3a  assures  that  there  are  no  edges within a
constellation  other  than  the Fibonacci path.  Lemma 3b assures
that  there  are  no  edges  between  constellations.   Thus, the
theorem is proved.
\vskip.5cm
\centerline{\bf 5.  Logarithmic Sum Graphs}

The  procedure  displayed  in  the  Construction,  and proved in
Theorem  3,  meets the criteria of producing desired paths, from
the labelling scheme alone, each with a corresponding cataloging
node,  as  subgraphs  of  a single sum graph.  In cases based on
large  data  sets,  the multipliers get very large very quickly.
However, if  the  logarithm  (using the base multiplier, $x$, as
the base of the logarithm) of each label is taken, this issue of
apparent  significance  vanishes  (Table 2).  In  the example on
which  Figure  9  was  based,  the  values  of  the  multipliers
transformed by the log base 23 display clearly the constellation
structure.  The nodes associated with all entries  with integral
part 1" are grouped in a constellation, all with integral part
2" in another, and all with integral part 3" in yet another.
The  integral  values  serve  as  a  data  key"  to  this data
structure.  The fractional values are, of course, the  same from
subset  to  subset,  exhibiting  the  same  underlying Fibonacci
linkage  pattern  from  subset  to subset.  The largest value in
each  subset  is  the cataloging node; if other nodes were to be
included  in,  for  example, the third constellation, those also
would have a  logarithmic  value greater than 3.8180367 but less
than  4.   Thus,  independent  of  how many nodes there are in a
single  constellation,  all the logarithmic labels are contained
in a band of real  numbers one unit wide:  3 is a greatest lower
bound (which is attained), and 4 is an upper bound for labels in
the   third   constellation.    Further,   the   logarithmically
- transformed labels increase  additively:  there  are  only  as
many different data keys as there are different constellations.

\midinsert
\vskip.5cm
\hrule
\vskip.5cm
$$\matrix{ 1 &\bullet &&&&&2 &\bullet &&&&&3 &\bullet \cr &\Big\vert &&&&& &\Big\vert &&&&& &\Big\vert \cr 1.22&\bullet &&&&&2.22&\bullet &&&&&3.22&\bullet \cr &\Big\vert &&&&& &\Big\vert &&&&& &\Big\vert \cr 1.35&\bullet &&&&&2.35&\bullet &&&&&3.35&\bullet \cr & &&&&& &\Big\vert &&&&& &\Big\vert \cr 1.51&\bullet &&&&&2.51&\bullet &&&&&3.51&\bullet \cr & &&&&& & &&&&& &\Big\vert \cr & &&&&&2.66&\bullet &&&&&3.66&\bullet \cr & &&&&& & &&&&& & \cr & &&&&& & &&&&&3.81&\bullet \cr }$$
\vskip.5cm
\centerline{\bf Figure 10.}

\centerline{Logarithmic sum graph}
\vskip.5cm
\hrule
\vskip.5cm
\endinsert

When these logarithmic labels are attached to  the  nodes  of the
graph  in  Figure  9  we  refer  to  the  resulting  graph  as  a
logarithmic  sum  graph" (Figure 10).  Note, however, that even
though this graph is  isomorphic to the sum graph of Figure 9, it
is not itself a sum  graph (in much the way that a truncated cone
is not itself a cone, even though it is derived from a cone).

From  a purely theoretical standpoint, it is possible to identify
the  constellation  to  which a node belongs very simply from its
assigned multiplier.   For, if $p$ is the base multiplier, a node
whose  multiplier  is  $N=a\cdot p^k$ has $k\leq \hbox{log}_p\, N \leq k+1$,  since  $a < p$.   Thus,  a  node  with multiplier $N$
belongs  to  constellation  $k$ if and only if $[\hbox{log}_p\,N] =k$ (where brackets denote the greatest integer function).  (From
a computer standpoint, one must be  careful,  since  occasionally
computational    error    might    make   $\hbox{log}_p\,p^k < k$
computationally.  Adding a suitably small amount to $\hbox{log}_p \,N$  before  determining  its  constellation  should  avert this
difficulty.)  In fact, it seems easier computationally  to  store
$\hbox{log}_p\, N$ rather than $N$  as a multiplier,  since  then
much smaller numbers can be stored.  This motivates the following
formal characterization of logarithmic sum graphs.

\noindent{\sl Definition 6}

Let  $S$ be a set of $n$ distinct positive integers, $p$ a prime.
Define the {\sl logarithmic sum graph\/}, relative to $p$,
$G^+(\hbox{log}_p\, S)$ as follows:

\item{1.}  $G^+(\hbox{log}_p\, S)$  has $n$ nodes, labelled with
the  $n$  different labels $\{\hbox{log}_p\, x \quad \vert \quad x \in S \}$.
\item{2.}  there  is  an edge between two nodes labelled $a$ and
$b$ if $p^a + p^b \in S$.

\noindent  Logarithmic  sum  graphs  retain  all  the advantages
afforded by Theorem 3, and they make it possible to handle large
data sets more easily.
\vskip.5cm
\centerline{\bf 6.  Reversed Sum Graphs.}

In  the  procedure  of  Theorem  3,  and in the logarithmic
modification of that procedure to  accommodate  large data sets,
the  cataloging  nodes  all have the largest labels within their
subgraph.   It  might  be  useful,  in  some situations, for the
cataloging  nodes  to  have  the  smallest  labels  within their
subgraphs.   For  this  purpose,  we  define  the  notion  of  a
reversed" sum graph.

\noindent{\sl Definition 7}

Let  $S$  be  a  set of positive integers such that the sum
graph $G^+(S)$  [logarithmic  sum graph $G^+(\hbox{log}_p\, S)$]
is  partitioned  into constellations such as those of Theorem 3.
Define  the  {\sl  reversed  sum  graph\/}  ${}^{+}G(S)$   [{\sl
reversed  logarithmic  sum  graph\/} $^{+}G(\hbox{log}_p\, S)$],
isomorphic to  $G^+(S)$ [$G^+(\hbox{log}_{p}\, S)$], as follows.
If the nodes in  a  given constellation have labels $a_1 < a_2 < \ldots < a_p$,  relabel them $a_p, a_{p-1}, \ldots , a_1$.  That
is,  the node labelled $a_i$ is given the new label $a_{p+1-i}$.
(Note that single-node constellations are not affected.)

\noindent{Example 6\/}

Let $S_4 = \{1,2,3,5,8,13\}$.  The graphs $G^+(S)$, $^+G(S)$
are  displayed  in Figure 11.  (As in the case of the logarithmic
sum  graph,  note that a reversed sum graph (Definition 7) is not
itself a sum graph.)

\midinsert
\vskip.5cm
\hrule
\vskip.5cm
$$\matrix{ 1 &\bullet &&&&&13 &\bullet \cr &\Big\vert &&&&& &\Big\vert \cr 2 &\bullet &&&&&8 &\bullet \cr &\Big\vert &&&&& &\Big\vert \cr 3 &\bullet &&&&&5 &\bullet \cr &\Big\vert &&&&& &\Big\vert \cr 5 &\bullet &&&&&3 &\bullet \cr &\Big\vert &&&&& &\Big\vert \cr 8 &\bullet &&&&&2 &\bullet \cr & &&&&& & \cr 13 & &&&&&1 &\bullet \cr }$$
\vskip.5cm
\centerline{\bf Figure 11.}

\centerline{A Fibonacci sum graph $G^+(S)$ (left)}
\centerline{and  its reversed sum graph $^+G(S)$ (right).}
\vskip.5cm
\hrule
\vskip.5cm
\endinsert

\noindent As Definition 7  suggests,  logarithmic  sum graphs may
also  be  reversed.  Figure 12 shows the logarithmic sum graph of
Figure 10  and  its reversed logarithmic sum graph.  Reversed sum
graphs,  logarithmic  or  not, always assign an integer, the data
key,  to  the  cataloging  node.   This  feature  is particularly
important  in  the  case  of the logarithmic representation, when
data  might  be  added  to or deleted from a single subgraph, all
with  integral  part  of  their  labels  identical to that of the
cataloging label.
\midinsert
\vskip.5cm
\hrule
\vskip.5cm
$$\matrix{ 1 &\bullet &&&&&2 &\bullet &&&&&3 &\bullet \cr &\Big\vert &&&&& &\Big\vert &&&&& &\Big\vert \cr 1.22&\bullet &&&&&2.22&\bullet &&&&&3.22&\bullet \cr &\Big\vert &&&&& &\Big\vert &&&&& &\Big\vert \cr 1.35&\bullet &&&&&2.35&\bullet &&&&&3.35&\bullet \cr & &&&&& &\Big\vert &&&&& &\Big\vert \cr 1.51&\bullet &&&&&2.51&\bullet &&&&&3.51&\bullet \cr & &&&&& & &&&&& &\Big\vert \cr & &&&&&2.66&\bullet &&&&&3.66&\bullet \cr & &&&&& & &&&&& & \cr & &&&&& & &&&&&3.81&\bullet \cr & &&&&& & &&&&& & \cr & &&&&& & &&&&& & \cr 1.51&\bullet &&&&&2.66&\bullet &&&&&3.81&\bullet \cr &\Big\vert &&&&& &\Big\vert &&&&& &\Big\vert \cr 1.35&\bullet &&&&&2.51&\bullet &&&&&3.66&\bullet \cr &\Big\vert &&&&& &\Big\vert &&&&& &\Big\vert \cr 1.22&\bullet &&&&&2.35&\bullet &&&&&3.51&\bullet \cr & &&&&& &\Big\vert &&&&& &\Big\vert \cr 1 &\bullet &&&&&2.22&\bullet &&&&&3.35&\bullet \cr & &&&&& & &&&&& &\Big\vert \cr & &&&&&2 &\bullet &&&&&3.22&\bullet \cr & &&&&& & &&&&& & \cr & &&&&& & &&&&&3 &\bullet \cr }$$
\vskip.5cm
\centerline{\bf Figure 12.}

\centerline{Logarithmic sum graph (top) and reversed logarithmic
sum graph (bottom).}
\vskip.5cm
\hrule
\vskip.5cm
\endinsert

\vskip.5cm
\centerline{\bf 7.  Augmented Reversed Logarithmic Sum Graphs}

Reversed logarithmic sum graphs single out cataloging  nodes
as  the  only  nodes  with  integral labels.  It may be useful to
consider  linkages  within  the  set  of  cataloging nodes and to
augment"  the  reversed  logarithmic  sum  graph   with   edges
displaying these linkages (Figure 13).

\midinsert
\vskip.5cm
\hrule
\vskip.5cm
$$\matrix{ 1.51&\bullet &&&&&2.66&\bullet &&&&&3.81&\bullet \cr &\Big\vert &&&&& &\Big\vert &&&&& &\Big\vert \cr 1.35&\bullet &&&&&2.51&\bullet &&&&&3.66&\bullet \cr &\Big\vert &&&&& &\Big\vert &&&&& &\Big\vert \cr 1.22&\bullet &&&&&2.35&\bullet &&&&&3.51&\bullet \cr & &&&&& &\Big\vert &&&&& &\Big\vert \cr 1 &\bullet &&&&&2.22&\bullet &&&&&3.35&\bullet \cr & &&&&& & &&&&& &\Big\vert \cr & &&&&&2 &\bullet &&&&&3.22&\bullet \cr & &&&&& & &&&&& & \cr & &&&&& & &&&&&3 &\bullet \cr }$$
\vskip.5cm
\centerline{\bf Figure 13.}

\centerline{ARL  sum graph derived from a  reversed  logarithmic sum
graph}
\centerline{\bf Reader  should  draw  edges  joining  nodes  1 and 2,
2 and 3, and 1 and 3}.
\vskip.5cm
\hrule
\vskip.5cm
\endinsert

\noindent {\sl Definition 8\/}

The {\sl augmented reversed logarithmic  sum graph,  ARL  sum
graph\/}, denoted $^+A(\hbox{log}_p\, S)$, consists  of the nodes
and  edges of $^+G(\hbox{log}_p\, S)$  together  with  all  edges
linking the nodes with integer labels in $^+G(\hbox{log}_p\, S)$.
Thus,  $^+A(\hbox{log}_p\, S)$ $=$ $^+G(\hbox{log}_p\, S)$ $\cup$
$\{$ complete graph on nodes with integer labels in
$^+G(\hbox{log}_p\, S)\}$.

\noindent  If  $m$ is  the number of nodes with integer labels in
$^+G(\hbox{log}_p\, S)$,  this  augmentation  adds   $m\choose 2$
edges to the  reversed  sum  graph $^+G(\hbox{log}_p\, S)$.   The
ARL sum graph is not itself a sum graph.

\noindent Augmented  reversed  logarithmic  sum graphs retain all
the  characteristics  of  Theorem  3,  have   the   computational
advantage of logarithmic sum graphs in  handling large data sets,
permit the reverse  sum  graph  strategy of integral labelling of
the  cataloging  node,  and have the added  feature of displaying
the  complete  linkage  pattern  among cataloging nodes.  Linkage
patterns  emerge  both  at the local scale and at the more global
cataloging scale.
\vskip.7cm
\centerline{\bf 8.  Cartographic Application of ARL Sum Graphs}

The  labels  of  Table  1.1,  derived  from  the  Sum  Graph
Unification  Construction,  offer  a  way to organize data and to
retrieve  predetermined  sequential  order  of information from a
jumbled  data  set.   The  relative  sizes of the weights for the
nodes  in  Table 1.1  are,  however,  awkward.   A  simple way to
overcome  this  awkwardness  is to take the logarithm of the node
weights (to the base of the  base multiplier).  Thus, in Table 3,
column  6  shows  the  $\hbox{log}_{37}$  of  each  node   weight
determined  in  Table 1.1  (listed  in column 5 of Table 3).  The
constellation number is easily  read off  as the integral part of
the  logarithm  and  all  entries  for a single constellation are
contained within a band of values one unit wide.  When the labels
are  reversed,  the  integral label corresponds to the cataloging
node.   This  reversed  logarithmic  sum  graph  (represented  by
Table 3) retains  the  favorable characteristics of Table 1.1 for
sorting  of  data;  the  node  labelling  scheme  of  Table 3 is,
however, easy to handle.

The   augmentation   afforded  by  ARL  sum  graphs  permits
significant compression of data, particularly in large data sets,
as  it  retains  the  favorable  characteristics  of the reversed
logarithmic sum graph noted above. To illustrate this capability,
we present the following application.

Consider  the  set  of  39  cities and metropolitan regions
labelled in Table 1.1.  One  set of data that is often stored is
distances  between  places  (distance" is used as an example).
Generally  this  set  is  stored  in  a square array, or better,
sometimes in an upper- or lower-triangular matrix.

Sum  graphs  can reduce greatly the number of entries that
need  to  be  stored.   Table  4.0  shows  a  complete  set  of
great-circle  distances  between   metropolitan   areas.   Each
metropolitan area is assigned the latitude and longitude of the
city  for  which  it  is  named.   Thus,  particular  sets   of
geographic  coordinates  are  viewed  simultaneously   at   two
different  scales.   Tables  4.1  to  4.7 show complete sets of
great-circle  distances  among  the cities in each of the seven
metropolitan areas (constellations).

The distance between Livonia and Scarborough (for example),
which  does  not appear directly in any of the set of Tables in
Table 4,  may  nonetheless be obtained by summing the distances
from Livonia to  DETROIT, from DETROIT to TORONTO, from TORONTO
to Scarborough  (Figure 14).  The algorithm displayed in Figure
14  shows how to use the reversed logarithmic node label of two
arbitrary nodes to determine  the  distance  between them using
only  the  entries  in  Table  4.0,  between metropolitan areas
(constellations), and in Tables 4.1-4.7 (showing local linkages
within  each  constellation).   The distance so-obtained is not
itself  a  great-circle  distance but it may well be a distance
more     realistically    representing    current    air-travel
circumstances.
\vskip.5cm
\hrule
\vskip.5cm
\midinsert
$$\matrix{ \hbox{DETROIT}&\longrightarrow &&&&&&&&&\hbox{TORONTO} \cr \Big\uparrow & &&&&&&&&&\Big\downarrow \cr \hbox{Livonia}&\longrightarrow &&&&&&&&&\hbox{Scarborough} \cr }$$
\centerline{\bf Figure 14.}

\noindent Commutative diagram showing distance calculation scheme
using  Table 4;  algorithm  showing  how  to find distance within
Table 4 using the data key provided by the  reversed  logarithmic
sum graph label.
\endinsert
\vskip.5cm
\centerline{\bf Algorithm}

\noindent\item{1.}  Assumption:  the cataloging city is also  the
city with the lowest non-integral label in its constellation.

\noindent\item{2.}   Find  the  distance  from a city with a node
with reversed logarithmic sum graph label $j.x$ to one with label
$k.y$, $j\leq k$ (and $x < y$ if $j=k$)

\item{a.}  If  $j=k$,  use  Table $4.j$ to find the distance from
$j.x$ to $j.y$.

\item{b.}  If $j < k$,

\item\item{i.}  use Table 4.0 to find distance between cataloging
cities $j$ and $k$.

\item\item{ii.}  use Table $4.j$ to find distance from $j$.lowest
to $j.x$.

\item\item{iii.} use Table $4.k$ to find distance from $k$.lowest
to $k.y$.

\noindent  Add the results of i, ii, and iii to find the required
distance.

\vskip.5cm
\hrule
\vskip.5cm

There   are  32  different  cities  in  this   example.   An
upper-triangular  32  by  32  matrix   of   ${32\choose 2} = 496$
different entries would normally be required to find between-city
distances.  Using the sum graph method, shown in the algorithm of
Figure  14,  requires  the use of 8 smaller Tables: Table 4.0 for
distances   between   cataloging  node  cities  and  Table $4.i$,
$1\leq i\leq 7$,  for  distances  of  cities in constellation $i$
from  cataloging  city  $i$.  The  latter  procedure, composed of
smaller matrices, requires storing (from each matrix) a total  of
$${7\choose 2} + {6\choose 2} + {4\choose 2} + {5\choose 2} + {6\choose 2} + {5\choose 2} + {3\choose 2} + {3\choose 2}$$
$= 21 + 15 + 6 + 10 + 15 + 10 + 3 + 3 = 83$ separate entries.  In
this case, sum graph methods afford a compression ratio of  about
6 to 1 over traditional methods.

With  larger  data  sets, the compression ratio becomes much
more  substantial.   Given  a  data  set  of 10,000 entries to be
partitioned  into  100  constellations  of  100   entries   each,
traditional  methods  using  an  upper  triangular  matrix  would
require that ${10,000\choose 2} = 49,995,000$  entries be stored.
Sum graph methods would require storing ${100\choose 2}$  entries
for Table 4.0 and ${100\choose 2}$  entries  for  each  of Tables
$4.i$, $1 \leq i \leq 100$, for a total of $101 \cdot {100\choose 2} = 499,950$ entries.  In  this  case  the  compression ratio is
100 to 1.  If  instead  the  10,000  entries are partitioned in a
different manner, different  compression  ratios result.  If 1000
constellations  of  10  entries  each are used, the corresponding
compression ratio is 91.8 to 1; if 10 constellations of 1000 each
are used, the  compression  ratio  is  10.09  to  1.  Clearly the
manner in which the partition  is  selected is important.  Larger
data sets  bring  even  larger  compression ratios:  if 1,000,000
data  points  are  considered,  and  are  partitioned  into  1000
constellations of 1000 each, the corresponding compression  ratio
is 1000 to 1.

Any  process  of  this  sort  also  needs to accommodate the
insertion  of  new  data; when it does so without having to alter
existing  structure,   it   is   dynamic."    The   Sum   Graph
Unification Construction  is dynamic to an extent.  Table 5 shows
part  of  the  data  set  of  Table 3 with Ann Arbor added to the
Detroit  metro  area.    Only   the   one   constellation   needs
relabelling; all others  remain undisturbed.  If, however, enough
new entries had been added to force an increase in the prime base
multiplier, then  a  global  change  would have been required for
that  single  entry  (generally  easy to achieve electronically).
None of the formul{\ae} would have required alteration.

Dynamic"  tables  of  this  sort might see application as
on-board  mapping  systems in cars or buses giving optimum route
displays  in  an  interactive  mode  (so-called  IVHS  or  other
commonly-used acronyms).  So data becomes accurate more  quickly
in  response  to  changing  traffic  patterns transmitted to the
vehicle in some sort of interactive fashion.  Advances in theory
can bring advances in technology to the level of affordable cost
and widespread application.  The application of sum graphs might
be one effort in that direction.
\vskip.5cm
\centerline{\bf 9.  Summary}

We have taken a tool from graph theory and specialized it in
a  number  of  directions  in order to deal with various types of
problems  that  often  arise  with  data  structures.    Table  6
organizes  these  specializations in capsule format.  Independent
of  how  the  sum  graph  is  specialized  to  adapt  to  various
difficulties  in  data  management,  however, the linkage pattern
between nodes in a sum graph is  determined by node weight alone,
which  is  derived  from  whether  or  not  one node is linked to
another.  There is no reliance on geographic  direction or on any
sort of other relative ordering based on the  underlying space in
which  the  nodes  are  embedded.   Hence,  the  sum  graph  data
structure  has  a theoretical base free from directional bias and
is  perhaps   therefore,   translation   invariant.   Determining
whether or not this theoretical data structure offers a graphical
application at  the level of GIS theory--as in the quadtree) that
permits translational invariance of the structure (independent of
pixel shape)  under  GIS  constraints, appears a significant next
step in bringing theory into practice.
\vfill\eject
\centerline{\bf References}

\ref Bergstrand, D., F. Harary, K. Hodges, G. Jennings,
L. Kuklinski, and J. Wiener.  1989.
The sum number of a complete graph.
Malaysian Mathematical Society, {\sl Bulletin\/}.
Second Series, 12, no. 1, 25-28.

\ref Bergstrand, D., F. Harary, K. Hodges, G. Jennings,
L. Kuklinski, and J. Wiener.  1992.
Product graphs are sum graphs.
{\sl Mathematics Magazine\/}, 65, no. 4, 262-264.

\ref Harary, F.  1969.  {\sl Graph Theory\/}.
Reading: Addison-Wesley.

\ref Harary, F.  1970.  Covering and packing in graphs, I.
{\sl Annals\/}, New York Academy of Sciences, 175, 198-205.

\ref Harary, F.  1990.  Sum graphs and difference graphs.
{\sl Congressus Numerantium\/}, 72, 101-108; {\sl Proceedings\/},
of the Twentieth Southeastern Conference on Combinatorics,
Graph Theory, and Computing (Boca Raton, FL 1989).

\ref Niven, I., and H. S. Zuckerman.  1960.  {\sl An
Introduction to the Theory of Numbers\/}.  New York:  Wiley.

$^*$ Sandra L. Arlinghaus, Institute of Mathematical Geography,
2790 Briarcliff, Ann Arbor, MI 48105;
William C. Arlinghaus, Lawrence Technological University,
Southfield, MI 48075
Frank Harary, New Mexico State University,
Las Cruces, NM 88003.

\vfill\eject
\centerline{\bf TABLE 1.1:}
\centerline{\bf Analysis according to
sum graph unification construction}
\vskip.2cm
\settabs\+&E. St. Louis\quad &ITUDE\quad &NACCI\quad &MULTI-\quad
&FACTORED\quad &474659385665\quad &ORDER \cr
\+&City        &LONG-  &FIBO-  &BASE   &FACTORED&NODE   &RANK \cr
\+&Suburb      &ITUDE  &NACCI  &MULTI- &WEIGHT  &WEIGHT &ORDER\cr
\+&METRO       &west   &LABEL  &PLIER  &        &       &     \cr
\vskip.2cm
\+&Salem        &70 54 &1  &$37$ &$1\cdot 37$ &37          &1 \cr
\+&Lynn         &70 57 &2  &$37$ &$2\cdot 37$ &74          &2 \cr
\+&Quincy       &71 00 &3  &$37$ &$3\cdot 37$ &111         &3 \cr
\+&Brockton     &71 01 &5  &$37$ &$5\cdot 37$ &185         &4 \cr
\+&Cambridge    &71 07 &8  &$37$ &$8\cdot 37$ &296         &5 \cr
\+&Boston       &71 07 &13 &$37$ &$13\cdot 37$ &481        &6 \cr
\+&BOSTON       &      &21 &$37$ &$21\cdot 37$ &777        &7 \cr
\+&Longueuil    &73 30 &1 &$37^2$ &$1\cdot 37^2$ &1369     &8 \cr
\+&Verdun       &73 34 &2 &$37^2$ &$2\cdot 37^2$ &2738     &9 \cr
\+&Montreal     &73 35 &3 &$37^2$ &$3\cdot 37^2$ &4107     &10\cr
\+&Laval        &73 44 &5 &$37^2$ &$5\cdot 37^2$ &6845     &11\cr
\+&MONTREAL     &      &8 &$37^2$ &$8\cdot 37^2$ &10952    &12\cr
\+&Camden       &75 06 &1 &$37^3$ &$1\cdot 37^3$ &50653    &13\cr
\+&Philadelphia &75 13 &2 &$37^3$ &$2\cdot 37^3$ &101306   &14\cr
\+&Upper Darby  &75 16 &3 &$37^3$ &$3\cdot 37^3$ &151959   &15\cr
\+&Norristown   &75 21 &5 &$37^3$ &$5\cdot 37^3$ &253265   &16\cr
\+&Chester   &75 22 &8 &$37^3$ &$8\cdot 37^3$ &405224      &17\cr
\+&PHILADELPHIA &   &13 &$37^3$ &$13\cdot 37^3$ &658489    &18\cr
\+&Scarborough&79 12 &1 &$37^4$ &$1\cdot 37^4$ &1874161    &19\cr
\+&Toronto   &79 23 &2 &$37^4$ &$2\cdot 37^4$ &3738322     &20\cr
\+&North York&79 25 &3 &$37^4$ &$3\cdot 37^4$ &5622483     &21\cr
\+&York      &79 29 &5 &$37^4$ &$5\cdot 37^4$ &9370805     &22\cr
\+&Etobicoke &79 34 &8 &$37^4$ &$8\cdot 37^4$ &14993288    &23\cr
\+&Mississauga&79 37 &13 &$37^4$ &$13\cdot 37^4$ &24364093 &24\cr
\+&TORONTO    &     &21 &$37^4$ &$21\cdot 37^4$ &39357381  &25\cr
\+&Windsor   &83 00 &1 &$37^5$ &$1\cdot 37^5$ &69343957    &26\cr
\+&Warren    &83 03 &2 &$37^5$ &$2\cdot 37^5$ &138687914   &27\cr
\+&Detroit   &83 10 &3 &$37^5$ &$3\cdot 37^5$ &208031871   &28\cr
\+&Dearborn  &83 15 &5 &$37^5$ &$5\cdot 37^5$ &346719785   &29\cr
\+&Livonia   &83 23 &8 &$37^5$ &$8\cdot 37^5$ &554751656   &30\cr
\+&DETROIT   &      &13 &$37^5$ &$13\cdot 37^5$ &901471441 &31\cr
\+&E. St. L. &90 10 &1 &$37^6$ &$1\cdot 37^6$ &2565726409  &32\cr
\+&St. Louis &90 15 &2 &$37^6$ &$2\cdot 37^6$ &5131452818  &33\cr
\+&Lemay     &90 17 &3 &$37^6$ &$3\cdot 37^6$ &7697179227  &34\cr
\+&ST. LOUIS &      &5 &$37^6$ &$5\cdot 37^6$ &12828632045 &35\cr
\+&New Orleans&90 05 &1 &$37^7$ &$1\cdot 37^7$ &94931877133&36\cr
\+&Marrero   &90 06 &2 &$37^7$ &$2\cdot 37^7$&189863754266 &37\cr
\+&Metairie  &90 11 &3 &$37^7$ &$3\cdot 37^7$&284795631399 &38\cr
\+&NEW ORLEANS&     &5 &$37^7$ &$5\cdot 37^7$&474659385665 &39\cr
\vfill\eject
\hrule
\vskip.5cm
\centerline{\bf TABLE 1.2:}
\centerline{\bf Analysis according to
sum graph unification construction}
\centerline{\bf Two constellations ordered
from east to west by longitude}
\vskip.5cm
\settabs\+&E. St. Louis\quad &ITUDE\quad &NACCI\quad &MULTI-\quad
&FACTORED\quad &474659385665\quad &ORDER \cr
\+&City        &LONG-  &FIBO-  &BASE   &FACTORED&NODE   &RANK \cr
\+&Suburb      &ITUDE  &NACCI  &MULTI- &WEIGHT  &WEIGHT &ORDER\cr
\+&METRO       &west   &LABEL  &PLIER  &        &       &     \cr
\vskip.5cm
\+&New Orleans  &90 05 &1 &$37^7$ &$1\cdot 37^7$ &94931877133 &36\cr
\+&NEW ORLEANS  &      &5 &$37^7$ &$5\cdot 37^7$&474659385665 &39\cr
\+&Marrero      &90 06 &2 &$37^7$ &$2\cdot 37^7$&189863754266 &37\cr
\+&E. St. Louis &90 10 &1 &$37^6$ &$1\cdot 37^6$ &2565726409  &32\cr
\+&Metairie     &90 11 &3 &$37^7$ &$3\cdot 37^7$&284795631399 &38\cr
\+&St. Louis    &90 15 &2 &$37^6$ &$2\cdot 37^6$ &5131452818  &33\cr
\+&ST. LOUIS    &      &5 &$37^6$ &$5\cdot 37^6$ &12828632045 &35\cr
\+&Lemay        &90 17 &3 &$37^6$ &$3\cdot 37^6$ &7697179227  &34\cr
\vskip.5cm
\hrule
\vskip1cm
\hrule
\vskip.5cm
\centerline{\bf TABLE 2:}
\centerline{\bf Multipliers and their logarithms to the base}
\centerline{\bf of the base multiplier of 23}
\centerline{\bf for the example of Figure 7.}
\settabs\+\indent\qquad\qquad\quad&Multiplier
\qquad\qquad\qquad & Logarithm, base 23 \cr
\vskip.5cm
\+&Multiplier                 &Logarithm, base 23 \cr
\vskip.5cm
\+&$1\cdot 23 = 23$           &1\cr
\+&$2\cdot 23 = 46$           &1.2210647\cr
\+&$3\cdot 23 = 69$           &1.3503793\cr
\+&$5\cdot 23 = 115$          &1.5132964\cr
\+&$1\cdot 23^2 = 529$        &2\cr
\+&$2\cdot 23^2 = 1058$       &2.2210647\cr
\+&$3\cdot 23^2 = 1587$       &2.3503793\cr
\+&$5\cdot 23^2 = 2645$       &2.5132964\cr
\+&$8\cdot 23^2 = 4232$       &2.6631942\cr
\+&$1\cdot 23^3 = 12167$      &3\cr
\+&$2\cdot 23^3 = 24344$      &3.2210647\cr
\+&$3\cdot 23^3 = 36501$      &3.3503793\cr
\+&$5\cdot 23^3 = 60835$      &3.5132964\cr
\+&$8\cdot 23^3 = 97336$      &3.6631942\cr
\+&$13\cdot 23^3 = 158171$    &3.8180367\cr
\vskip.5cm
\hrule
\vfill\eject
\centerline{\bf TABLE 3:}
\centerline{\bf Table 1.1
labelled as a reversed logarithmic sum graph}
\vskip.2cm
\settabs\+&E. St. Louis\quad &ITUDE\quad &NACCI\quad
&MULTI-\quad
&FACTORED\quad &474659385665\quad &ORDER
\cr
\+&City        &LONG-  &FIBO- &BASE   &FACTORED&NODE  &LOG    \cr
\+&Suburb      &ITUDE  &NACCI &MULTI- &WEIGHT  &WEIGHT&BASE   \cr
\+&METRO       &       &LABEL &PLIER  &        &      &37 NODE\cr
\vskip.2cm
\+&Salem        &70 54 &21 &$37$ &$21\cdot 37$&777&1.746657\cr
\+&Lynn         &70 57 &13 &$37$ &$13\cdot 37$&481&1.629043\cr
\+&Quincy       &71 00 &8  &$37$ &$8\cdot 37$&296&1.509974\cr
\+&Brockton     &71 01 &5  &$37$ &$5\cdot 37$&185&1.394708\cr
\+&Cambridge    &71 07 &3  &$37$ &$3\cdot 37$&111&1.269430\cr
\+&Boston       &71 07 &2  &$37$ &$2\cdot 37$&74&1.169991\cr
\+&BOSTON       &      &1  &$37$ &$1\cdot 37$&37&1\cr
\+&Longueuil  &73 30 &8  &$37^2$ &$8\cdot 37^2$&10952&2.509974\cr
\+&Verdun     &73 34 &5  &$37^2$ &$5\cdot 37^2$&6845&2.394708\cr
\+&Montreal   &73 35 &3  &$37^2$ &$3\cdot 37^2$&4107&2.269430\cr
\+&Laval      &73 44 &2  &$37^2$ &$2\cdot 37^2$&2738&2.169991\cr
\+&MONTREAL   &        &1  &$37^2$ &$1\cdot 37^2$&1369&2\cr
\+&Camden   &75 06 &13 &$37^3$ &$13\cdot 37^3$&658489&3.629043\cr
\+&Phil. &75 13 &8  &$37^3$ &$8\cdot 37^3$&405224&3.509974\cr
\+&U. Darby  &75 16 &5  &$37^3$ &$5\cdot 37^3$&253265&3.394708\cr
\+&Norris.   &75 21 &3  &$37^3$ &$3\cdot 37^3$&151959&3.269430\cr
\+&Chester   &75 22 &2  &$37^3$ &$2\cdot 37^3$&101306&3.169991\cr
\+&PHILADELPHIA&       &1  &$37^3$ &$1\cdot 37^3$&50653&3\cr
\+&Scar.  &79 12 &21 &$37^4$ &$21\cdot 37^4$&39357381&4.746657\cr
\+&Toronto&79 23 &13 &$37^4$ &$13\cdot 37^4$&24364093&4.629043\cr
\+&NYork   &79 25 &8  &$37^4$ &$8\cdot 37^4$&14993288&4.509974\cr
\+&York     &79 29 &5  &$37^4$ &$5\cdot 37^4$&9370805&4.394708\cr
\+&Etobicoke&79 34 &3  &$37^4$ &$3\cdot 37^4$&5622483&4.269430\cr
\+&Missi.   &79 37 &2  &$37^4$ &$2\cdot 37^4$&3748322&4.169991\cr
\+&TORONTO  &          &1  &$37^4$ &$1\cdot 37^4$&1874161&4\cr
\+&Wind. &83 00 &13 &$37^5$ &$13\cdot 37^5$&901471441&5.629043\cr
\+&Warren &83 03 &8  &$37^5$ &$8\cdot 37^5$&554751656&5.509974\cr
\+&Detroit&83 10 &5  &$37^5$ &$5\cdot 37^5$&346719785&5.394708\cr
\+&Dearb. &83 15 &3  &$37^5$ &$3\cdot 37^5$&208031871&5.269430\cr
\+&Livonia&83 23 &2  &$37^5$ &$2\cdot 37^5$&138687914&5.169991\cr
\+&DETROIT&            &1  &$37^5$ &$1\cdot 37^5$&69343957&5\cr
\+&ESLou&90 10 &5  &$37^6$ &$5\cdot 37^6$&12828632045&6.394708\cr
\+&SLou &90 15 &3  &$37^6$ &$3\cdot 37^6$&7697179227&6.269430\cr
\+&Lemay&90 17 &2  &$37^6$ &$2\cdot 37^6$&5131452818&6.169991\cr
\+&ST. LOUIS&          &1  &$37^6$ &$1\cdot 37^6$&2565726409&6\cr
\+&NOrl&90 05 &5  &$37^7$ &$5\cdot 37^7$&474659385665&7.394708\cr
\+&Marr&90 06 &3  &$37^7$ &$3\cdot 37^7$&284795631399&7.269430\cr
\+&Meta&90 11 &2  &$37^7$ &$2\cdot 37^7$&189863754266&7.169991\cr
\+&NEW ORLEANS&       &1  &$37^7$ &$1\cdot 37^7$&94931877133&7\cr
\vfill\eject
\centerline{\bf TABLE 4.0:  Distances between all metro areas}
\settabs\+&NEW ORLEANS\quad & BOS\quad & MONT \quad &PHIL \quad
&TOR\quad &DET \quad &1034 \quad &1349\cr
\+&            &BOS &MONT &PHIL &TOR  &DET   &SL    &NO  \cr
\+&BOSTON      &0   &255  &263  &429  &615   &1034  &1349\cr
\+&MONTREAL    &    &0    &388  &312  &523   &974   &1394\cr
\+&PHIL        &    &     &0    &331  &444   &808   &1086\cr
\+&TORONTO     &    &     &     &0    &211   &662   &1112\cr
\+&DETROIT     &    &     &     &     &0     &452   &936 \cr
\+&ST LOUIS    &    &     &     &     &      &0     &596 \cr
\+&NEW ORLEANS &    &     &     &     &      &      &0   \cr
\centerline{\bf TABLE 4.1:  Boston-area cities}
\settabs\+&Quincy\quad &Salem\quad & Lynn\quad &Quincy \quad
&Brock.\quad &Cambr.\quad &Boston\cr
\+&            &Salem &Lynn &Quincy &Brock.  &Cambr.   &Boston\cr
\+&Salem       &0     &4.29 &19.1   &31.6    &15.3     &21.4  \cr
\+&Lynn        &      &0    &15.1   &27.8    &10.2     &17.2  \cr
\+&Quincy      &      &     &0      &12.6    &10.9     &5.96  \cr
\+&Brock.      &      &     &       &0       &22.4     &13.6  \cr
\+&Cambr.      &      &     &       &        &0        &9.21  \cr
\+&Boston      &      &     &       &        &         &0     \cr
\centerline{\bf TABLE 4.2:  Montreal-area cities}
\settabs\+&Longueuil\quad &Longue.\quad
& Verdun\quad &Laval \quad &Mont.\cr
\+&            &Longue. &Verdun &Laval &Mont.\cr
\+&Longueuil   &0       &6.6    &11.3  &4.64 \cr
\+&Verdun      &        &0      &9.29  &3.54 \cr
\+&Laval       &        &       &0     &7.35 \cr
\+&Montreal    &        &       &      &0    \cr
\centerline{\bf TABLE 4.3:  Philadelphia-area cities}
\settabs\+&Philadelphia\quad &Camden\quad & Chester\quad
&U. Darby \quad &Norris.\quad &Phila.\cr
\+&            &Camden &Chester &U Darby &Norris.  &Phila.\cr
\+&Camden      &0      &15.2    &9.12    &18.3    &7.7    \cr
\+&Chester     &       &0       &9.64    &18.4    &13.0   \cr
\+&Upper Darby &       &        &0       &11.2    &3.5    \cr
\+&Norristown  &       &        &        &0       &10.7   \cr
\+&Philadelphia&       &        &        &        &0      \cr
\centerline{\bf TABLE 4.4:  Toronto-area cities}
\settabs\+&Mississauga \quad &Scar.\quad & Miss.\quad &N. York
\quad &York\quad &Etob.\quad &Tor.\cr
\+&            &Scar. &Miss. &N. York &York  &Etob.   &Tor.\cr
\+&Scarborough &0     &24.3  &11.0    &14.8  &19.5    &10.8\cr
\+&Mississauga &      &0     &17.9    &10.4  &6.27    &13.5\cr
\+&North York  &      &      &0       &7.66  &11.8    &8.23\cr
\+&York        &      &      &        &0     &4.75    &5.12\cr
\+&Etobicoke   &      &      &        &      &0       &9.23\cr
\+&Toronto     &      &      &        &      &        &0   \cr
\centerline{\bf TABLE 4.5:  Detroit-area cities}
\settabs\+&Dearborn\quad &Windsor\quad &Warren\quad &Dear. \quad
&Livonia\quad &Detroit\cr
\+&            &Windsor &Warren &Dear. &Livonia  &Detroit  \cr
\+&Windsor     &0       &16.3   &12.8  &20.7     &9.18     \cr
\+&Warren      &        &0      &20.0  &19.3     &13.9     \cr
\+&Dearborn    &        &       &0     &10.5     &6.27     \cr
\+&Livonia     &        &       &      &0        &11.5     \cr
\+&Detroit     &        &       &      &         &0        \cr
\vfill\eject
\centerline{\bf TABLE 4.6:  St. Louis-area cities}
\settabs\+&E. St. Louis\quad &E. St. L.\quad
& Lemay\quad &St. Louis \cr
\+&            &E. St. L. &Lemay &St. Louis \cr
\+&E. St. Louis&0         &6.29  &4.49      \cr
\+&Lemay       &          &0     &1.79      \cr
\+&St. Louis   &          &      &0         \cr
\centerline{\bf TABLE 4.7:  New Orleans-area cities}
\settabs\+&New Orleans\quad & Met.\quad &Mar. \quad &New O.\cr
\+&            &Met. &Mar. &New O. \cr
\+&Metairie    &0    &7.61 &5.98   \cr
\+&Marrero     &     &0    &5.84   \cr
\+&New Orleans &     &     &0      \cr
\vfill\eject
\hrule
\vskip.5cm
\centerline{\bf TABLE 5:}
\centerline{\bf New data added --- Ann Arbor}
\vskip.5cm
\settabs\+&E. St. Louis\quad &ITUDE\quad &NACCI\quad
&MULTI-\quad
&FACTORED\quad &474659385665\quad &ORDER
\cr
\+&City        &LONG-  &FIBO- &BASE   &FACTORED&NODE  &LOG    \cr
\+&Suburb      &ITUDE  &NACCI &MULTI- &WEIGHT  &WEIGHT&BASE   \cr
\+&METRO AREA  &       &LABEL &PLIER  &        &      &37 NODE\cr
\vskip.5cm
\+&Wind. &83 00 &21 &$37^5$ &$21\cdot 37^5$&1456223097&5.746657\cr
\+&Warren &83 03 &13  &$37^5$ &$13\cdot 37^5$&901471441&5.629043\cr
\+&Detroit&83 10 &8  &$37^5$ &$8\cdot 37^5$&554751656&5.509974\cr
\+&Dearb. &83 15 &5  &$37^5$ &$5\cdot 37^5$&346719785&5.394708\cr
\+&Livonia&83 23 &3  &$37^5$ &$3\cdot 37^5$&208031871&5.269430\cr
\+&Ann Arbor&83 45 &2&$37^5$ &$2\cdot 37^5$&138687914&5.169991\cr
\+&DETROIT &           &1  &$37^5$ &$1\cdot 37^5$&69343957&5\cr
\vskip.2cm
\+&ESLou&90 10 &5  &$37^6$ &$5\cdot 37^6$&12828632045&6.394708\cr
\+&SLou &90 15 &3  &$37^6$ &$3\cdot 37^6$&7697179227&6.269430\cr
\+&Lemay&90 17 &2  &$37^6$ &$2\cdot 37^6$&5131452818&6.169991\cr
\+&ST. LOUIS&          &1  &$37^6$ &$1\cdot 37^6$&2565726409&6\cr
\vskip.2cm
\+&NOrl&90 05 &5  &$37^7$ &$5\cdot 37^7$&474659385665&7.394708\cr
\+&Marr&90 06 &3  &$37^7$ &$3\cdot 37^7$&284795631399&7.269430\cr
\+&Meta&90 11 &2  &$37^7$ &$2\cdot 37^7$&189863754266&7.169991\cr
\+&NEW ORLEANS&       &1  &$37^7$ &$1\cdot 37^7$&94931877133&7\cr
\vskip.5cm
\hrule
\vfill\eject
\hrule
\vskip.5cm
\centerline{\bf TABLE 6:}
\centerline{\bf Specializations of sum graphs}
\vskip.5cm
\settabs\+\indent\qquad&Augmented reversed logarithmic\qquad\qquad
\qquad&intermediate and global scales.\cr
\vskip.5cm
\+&Type of graph              &Characteristics                \cr
\vskip.5cm
\+&Sum graph                  & Variable resolution at        \cr
\+&(Figure 7)                 &local and global scales, only. \cr
\+&                           &Shape, size, and connection    \cr
\+&                           &pattern of parts to whole      \cr
\+&                           &suggested by global label.     \cr
\+&Sum graph with base multiplier &Variable resolution at     \cr
\+&(Figure 9)                 &intermediate and global scales.\cr
\+&                           &Relative shape, size, and      \cr
\+&                           &connection pattern of parts    \cr
\+&                           &to whole suggested by multiple \cr
\+&                           &labels associated with split   \cr
\+&                           ®ions.                       \cr
\+&Logarithmic sum graph      &Confines sum graph labels to   \cr
\+&(Figure 10)                &a single unit for each         \cr
\+&                           &subgraph.  Deals well with     \cr
\+&                           &split regions; is not itself   \cr
\+&                           &a sum graph.  Label on         \cr
\+&                           &cataloging node suggests       \cr
\+&                           &relative shape, size, and      \cr
\+&                           &connection pattern of parts    \cr
\+&                           &to the whole                   \cr
\+&Reversed sum graph         &Not itself a sum graph.  Sole  \cr
\+&(Figure 11)                &function is to assign an       \cr
\+&                           &integral value to the          \cr
\+&                           &cataloging node of each        \cr
\+&                           &subgraph.                      \cr
\+&Augmented reversed logarithmic & Combines characteristics  \cr
\+&\quad sum graph            &of logarithmic and reversed    \cr
\+&(Figure 13)                &sum graphs.  Added edges       \cr
\+&                           &join cataloging nodes.         \cr
\+&                           &Linkage patterns are           \cr
\+&                           &suggested at local,            \cr
\+&                           &intermediate, and global       \cr
\+&                           &levels of resolution.          \cr
\vskip.5cm
\hrule
\vfill\eject

\centerline{\bf 5.  SAMPLE OF HOW TO DOWNLOAD THE ELECTRONIC FILE}

\centerline{\bf  BACK ISSUES OF {\sl SOLSTICE\/} NOW AVAILABLE ON FTP}

\noindent This section shows the exact set of commands that  work
to  download {\sl Solstice\/} on  The  University  of  Michigan's
Xerox  9700.   Because different universities will have different
installations  of {\TeX},  this  is  only a rough guideline which
{\sl might\/} be of use to the reader. (BACK   ISSUES   AVAILABLE
using anonymous ftp to open um.cc.umich.edu, account  GCFS;  type
cd GCFS after  entering system;  then type ls to get a directory;
then type get solstice.190 (for example) and download it or  read
it according to local constraints.) Back issues will be available
on this account; this account is ONLY for back issues;  to  write
Solstice,  send   e-mail   to   Solstice@UMICHUM.bitnet   or   to
Solstice@um.cc.umich.edu .   Issues  from  this  one  forward are
available on FTP on account IEVG (substitute IEVG for GCFS above).

First  step  is  to  concatenate  the  files  you  received   via
bitnet/internet.   Simply  piece  them together in your computer,
one  after  another,  in  the  order  in which they are numbered,
starting with the number, 1."

The  files  you  have received are ASCII files;  the concatenated
file  is  used  to  form  the  .tex file from which the .dvi file
(device  independent)  file is formed.  The words percent-sign"
and backslash" are written out in the example  below;  the user
should type them symbolically.

\noindent
ASSUME YOU HAVE SIGNED ON AND ARE AT THE SYSTEM PROMPT, \#.
\smallskip
\# create -t.tex

\# percent-sign t from pc c:backslash words backslash
solstice.tex to mts -t.tex char notab

(this command sends my file, solstice.tex, which I did as
a WordStar (subdirectory, words") ASCII file to the
mainframe)

\# run *tex par=-t.tex

(there may be some underfull (or certain over) boxes that
generally  cause  no  problem;  there should be no other
error"  messages  in  the  typesetting--the  files you
receive were already tested.)

\# run *dvixer par=-t.dvi

\# control *print* onesided

\# run *pagepr scards=-t.xer, par=paper=plain
\vfill\eject
\centerline{\bf 6.  SOLSTICE--INDEX, VOLUMES I, II, AND III}
\smallskip
\noindent {\bf Volume III, Number 2, Winter, 1992}
\smallskip
\noindent {\bf 1.}  A Word of Welcome from A to U.
\smallskip
\noindent {\bf 2.}  Press clippings--summary.
\smallskip
\noindent {\bf 3.}  Reprints:
\smallskip
\noindent {\bf A.}  What Are Mathematical Models and What
Should They Be? by Frank Harary, reprinted from {\sl Biometrie -
Praximetrie\/}.
\smallskip \noindent {\sl
1.  What Are They?  2.  Two Worlds:  Abstract and Empirical
3.  Two Worlds:  Two Levels  4.  Two Levels:  Derviation and
Selection  5.  Research Schema  6.  Sketches of Discovery
7.  What Should They Be?
\/}
\smallskip
{\bf B.}  Where Are We?  Comments on the Concept of
Center of Population, by Frank E. Barmore, reprinted from
{\sl The Wisconsin Geographer\/}.
\smallskip \noindent {\sl
1.  Introduction  2.  Preliminary Remarks  3.  Census Bureau
Center of Population Formul{\ae}  4.  Census Bureau Center of
Population Description  5.  Agreement Between Description and
Formul{\ae}  6.  Proposed Definition of the Center of
Population  7.  Summary  8.  Appendix A  9.  Appendix B
10.  References
\/}
\smallskip
\noindent {\bf 4.}  Article:
\smallskip
The Pelt of the Earth:  An Essay on Reactive Diffusion,
by Sandra L. Arlinghaus and John D. Nystuen.
\smallskip \noindent {\sl
1.  Pattern Formation:  Global Views  2.  Pattern Formation:
Local Views  3.  References Cited  4.  Literature of Apparent
Related Interest.
\/}
\smallskip
\noindent {\bf 5.}  Feature
Meet new{\sl Solstice\/} Board Member William D. Drake;
comments on course in Transition Theory and listing of
student-produced monograph.
\smallskip
\noindent {\bf 6.} Downloading of Solstice.
\smallskip
\noindent {\bf 7.} Index to Solstice.
\smallskip
\noindent {\bf 8.} Other Publications of IMaGe.
\smallskip
\noindent {\bf Volume III, Number 1, Summer, 1992}
\smallskip
\noindent{\bf 1.  ARTICLES.}
\smallskip\noindent
{\bf Harry L. Stern}.
\smallskip\noindent
{\bf Computing Areas of Regions With Discretely Defined Boundaries}.
\smallskip\noindent
1. Introduction 2. General Formulation 3. The Plane 4.  The Sphere
5.  Numerical Example and Remarks.  Appendix--Fortran Program.
\smallskip
\noindent{\bf 2.  NOTE }
\smallskip\noindent
{\bf Sandra L. Arlinghaus, John D. Nystuen, Michael J. Woldenberg}.
\smallskip\noindent
{\bf  The Quadratic World of Kinematic Waves}
\smallskip
\noindent{\bf 3.  SOFTWARE REVIEW}
\smallskip
RangeMapper$^{\hbox{TM}}$  ---  version 1.4.
Created  by {\bf Kenelm W. Philip},  Tundra Vole Software,
Fairbanks, Alaska.  Program and Manual by  {\bf Kenelm W. Philip}.
\smallskip
Reviewed by {\bf Yung-Jaan Lee}, University of Michigan.
\smallskip
\noindent{\bf 4.  PRESS CLIPPINGS}
\smallskip
\noindent{\bf 5.  INDEX to Volumes I (1990) and II (1991) of
{\sl Solstice}.}
\smallskip
\noindent {\bf Volume II, Number 1, Summer, 1991}
\smallskip
\noindent 1.  ARTICLE

Sandra L. Arlinghaus, David Barr, John D. Nystuen.
{\sl The Spatial Shadow:  Light and Dark --- Whole and Part\/}

This account of some of the projects of sculptor David Barr
attempts to place them in a formal, systematic, spatial  setting
based  on  the  postulates  of  the  science of space of William
Kingdon Clifford (reprinted in {\sl Solstice\/}, Vol. I, No. 1.).
\smallskip
\smallskip
\noindent 2.  FEATURES

\item{i}  Construction Zone --- The logistic curve.
\item{ii.} Educational feature --- Lectures on Spatial Theory"
\smallskip
\noindent {\bf Volume II, Number 2, Winter, 1991}
\smallskip
\noindent 1.  REPRINT

Saunders Mac Lane, Proof, Truth, and Confusion."  Given as the
Nora and Edward Ryerson Lecture at The University of Chicago in
1982.  Republished with permission of The University of Chicago
and of the author.

I.  The Fit of Ideas.  II.  Truth and Proof.  III.  Ideas and Theorems.
IV.  Sets and Functions.  V.  Confusion via Surveys.
VI.  Cost-benefit and Regression.  VII.  Projection, Extrapolation,
and Risk.  VIII.  Fuzzy Sets and Fuzzy Thoughts.  IX.  Compromise
is Confusing.

\noindent 2.  ARTICLE

Robert F. Austin.  Digital Maps and Data Bases:
Aesthetics versus Accuracy."

I.  Introduction.  II. Basic Issues.  III. Map Production.
IV.  Digital Maps.  V.  Computerized Data Bases.  VI.  User
Community.

\noindent 3.  FEATURES

Press clipping; Word Search Puzzle; Software Briefs.
\smallskip
\smallskip
\smallskip
\noindent{\bf INDEX to Volume I (1990) of {\sl Solstice}.}
\vskip.5cm
\noindent{\bf Volume I, Number 1, Summer, 1990}

\noindent 1.  REPRINT

William Kingdon Clifford, {\sl Postulates of the Science of Space\/}

This reprint of a portion of  Clifford's  lectures  to  the
Royal  Institution in the 1870's suggests many geographic topics
of concern in the last half of the twentieth century.   Look for
connections  to  boundary  issues,  to  scale problems, to self-
similarity and fractals, and to non-Euclidean  geometries  (from
those based on denial of Euclid's parallel  postulate  to  those
based on a sort of mechanical polishing").  What else did,  or
might, this classic essay foreshadow?

\noindent 2.  ARTICLES.

Sandra L. Arlinghaus, {\sl Beyond the Fractal.}

An original article.  The fractal notion of  self-similarity
is  useful  for  characterizing  change  in  scale;  the  reason
fractals are effective in the geometry of central  place  theory
is  because  that  geometry  is hierarchical in nature.  Thus, a
natural place to look for other connections of this  sort  is to
other geographical concepts that are also hierarchical.   Within
this fractal context, this article examines the case of  spatial
diffusion.

When the idea of diffusion is extended to see adopters" of
an innovation as attractors" of new adopters,  a  Julia set is
introduced as a possible axis against which to measure one class
of geographic phenomena.   Beyond the fractal  context,  fractal
concepts,  such  as  compression"  and  space-filling"   are
considered in a broader graph-theoretic setting.
\smallskip
\smallskip
William C. Arlinghaus, {\sl Groups, Graphs, and God}

An original article based on a talk given  before  a MIdwest
GrapH TheorY (MIGHTY) meeting.  The author,  an  algebraic  graph
theorist, ties his research interests to a broader  philosophical
realm,  suggesting  the  breadth  of  range  to  which  algebraic
structure might be applied.

The  fact  that  almost  all  graphs  are rigid (have trivial
automorphism groups) is exploited to argue probabilistically  for
the  existence  of  God.  This  is  presented  with the idea that
applications  of  mathematics  need  not be limited to scientific
ones.
\smallskip
\noindent 3.  FEATURES
\smallskip
\item{i.}  Theorem Museum --- Desargues's  Two  Triangle  Theorem
from projective geometry.
\item{ii.} Construction Zone --- a centrally symmetric hexagon is
derived from an arbitrary convex hexagon.
\item{iii.} Reference Corner --- Point set theory and topology.
\item{iv.}  Educational Feature --- Crossword puzzle on spices.
\item{v.}   Solution to crossword puzzle.
\smallskip
\noindent 4.  SAMPLE OF HOW TO DOWNLOAD THE ELECTRONIC FILE
\smallskip
\noindent{\bf Volume I, Number 2, Winter, 1990}
\smallskip
\noindent 1.  REPRINT

John D. Nystuen (1974), {\sl A City of Strangers:  Spatial Aspects
of Alienation in the Detroit Metropolitan Region\/}.

This paper examines the urban shift from people space" to
machine space" (see R. Horvath,  {\sl Geographical Review\/},
April, 1974) in the Detroit metropolitan  region  of 1974.   As
with Clifford's {\sl Postulates\/}, reprinted in the last issue
of {\sl Solstice\/}, note  the  timely  quality  of many of the
observations.

\noindent 2.  ARTICLES

Sandra Lach Arlinghaus, {\sl Scale and Dimension: Their Logical
Harmony\/}.

Linkage  between  scale  and  dimension  is made using the
Fallacy of Division and the Fallacy of Composition in a fractal
setting.
\smallskip
\smallskip
Sandra Lach Arlinghaus, {\sl Parallels between Parallels\/}.

The earth's sun introduces a symmetry in the perception of
its trajectory in the sky that naturally partitions the earth's
surface  into  zones  of  affine  and hyperbolic geometry.  The
affine zones, with  single  geometric  parallels,  are  located
north and south of the  geographic  parallels.   The hyperbolic
zone, with multiple geometric parallels, is located between the
geographic  tropical  parallels.   Evidence  of  this geometric
partition is suggested in the geographic environment --- in the
design of houses and of gameboards.
\smallskip
\smallskip
Sandra L. Arlinghaus, William C. Arlinghaus, and John D. Nystuen.
{\sl The Hedetniemi Matrix Sum:  A Real-world Application\/}.

In a recent paper, we presented an algorithm for finding the
shortest distance between any two nodes in a network of $n$ nodes
when  given  only  distances between adjacent nodes [Arlinghaus,
Arlinghaus, Nystuen,  {\sl Geographical  Analysis\/}, 1990].  In
that  previous   research,  we  applied  the  algorithm  to  the
generalized  road  network  graph surrounding San Francisco Bay.
Here,  we  examine consequent changes in matrix entires when the
underlying  adjacency pattern of the road network was altered by
the  1989  earthquake  that closed the San Francisco --- Oakland
Bay Bridge.
\smallskip
\smallskip
Sandra Lach Arlinghaus, {\sl Fractal Geometry  of Infinite Pixel
Sequences:  Su\-per\--def\-in\-i\-tion" Resolution\/}?

Comparison of space-filling qualities of square and hexagonal
pixels.
\smallskip
\noindent 3.  FEATURES
\item{i.}       Construction  Zone ---  Feigenbaum's  number;  a
triangular coordinatization of the Euclidean plane.
\item{ii.}  A three-axis coordinatization of the plane.
\smallskip
\vfill\eject
\centerline{\bf 7.  OTHER PUBLICATIONS OF IMaGe}
\centerline{\bf MONOGRAPH SERIES}
\centerline{Scholarly Monographs--Original Material, refereed}

Prices exclusive of shipping and handling;
payable in U.S. funds on a U.S. bank, only.
All monographs are \$15.95, except \#12 which is \$39.95.

Monographs are printed by Digicopy.

1.  Sandra L. Arlinghaus and John D. Nystuen.  Mathematical
Geography and Global Art:  the Mathematics of  David Barr's
Four Corners Project,'' 1986.

This monograph contains Nystuen's  calculations,  actually  used
by Barr to position his abstract  tetrahedral  sculpture  within
the earth. Placement of the sculpture vertices in Easter Island,
South Africa, Greenland, and Indonesia was chronicled in film by
The Archives of American Art for The Smithsonian Institution. In
addition to the archival material, this  monograph also contains
Arlinghaus's solutions to broader theoretical questions ---  was
Barr's  choice  of  a  tetrahedron  unique  within  his  initial
constraints, and, within the set of Platonic solids?

2.  Sandra L. Arlinghaus.  Down the Mail Tubes:  the Pressured
Postal Era, 1853-1984, 1986.

The  history  of the pneumatic post, in Europe and in the United
States,  is  examined  for  the  lessons  it  might offer to the
technological scenes of the late twentieth century. As Sylvia L.
Thrupp, Alice Freeman Palmer Professor Emeritus of  History, The
University  of  Michigan,  commented  in her review of this work
Such  brief  comment  does  far  less  than  justice  to   the
intelligence and the stimulating quality of the author's writing,
or to the breadth of her reading.  The detail of her accounts of
the interest of American private enterprise,  in  New  York  and
other  large  cities  on  this   continent,   in   pushing   for
construction  of  large  tubes  in  systems  to be leased to the
government,  brings  out  contrast between American and European
views  of  how  the  new technology should be managed.  This and
many  other  sections  of  the monograph will set readers on new
tracks of thought.''

3.  Sandra L. Arlinghaus.   Essays on Mathematical Geography,
1986.

A  collection  of  essays intended to show the range of power in
applying pure mathematics to human systems.  There are two types
of essay: those which employ traditional mathematical proof, and
those which do not. As mathematical proof may itself be regarded
as art, the former style of essay might represent traditional''
art, and the latter, surrealist'' art.  Essay titles are:
The   well-tempered  map  projection,''  Antipodal graphs,''
Analogue clocks,''  Steiner  transformations,''  Concavity
and  urban  settlement  patterns,''  Measuring  the   vertical
city,'' Fad and permanence in human systems,''   Topological
exploration in geography,'' A space for thought,'' and Chaos
in human systems--the Heine-Borel Theorem.''

4.  Robert F. Austin, A Historical Gazetteer of Southeast Asia,
1986.

Dr. Austin's Gazetteer draws geographic coordinates of Southeast
Asian place-names together with references to these  place-names
as they have appeared in historical and literary documents. This
book  is   of  obvious  use  to  historians  and  to  historical
geographers specializing in Southeast Asia.  At a  deeper level,
it might serve as a valuable source in  establishing  place-name
linkages which have remained previously unnoticed, in  documents
describing trade or other communications connections, because of
variation in place-name nomenclature.

5.  Sandra L. Arlinghaus, Essays on Mathematical Geography--II,
1987.

Written in the same format as IMaGe Monograph \#3, that seeks to
use pure'' mathematics in  real-world  settings,  this  volume
contains the following material:  Frontispiece -- the Atlantic
Drainage Tree,'' Getting a Handel on Water-Graphs,''  Terror
in Transit: A Graph Theoretic Approach to the Passive Defense of
Urban  Networks,''  Terrae Antipodum,''  Urban  Inversion,''
Fractals:    Constructions,  Speculations,   and   Concepts,''
Solar  Woks,''   A  Pneumatic  Postal  Plan:  The  Chambered
Interchange and ZIPPR Code,'' Endpiece.''

6.  Pierre Hanjoul, Hubert Beguin, and Jean-Claude Thill,
Theoretical Market Areas Under Euclidean Distance, 1988.
(English language text; Abstracts written in French and
in English.)

Though  already  initiated  by Rau in 1841, the economic  theory
of the shape of  two-dimensional market areas has long  remained
concerned  with  a  representation  of  transportation  costs as
linear in distance.  In the general gravity model, to which  the
theory   also   applies,   this   corresponds  to  a  decreasing
exponential   function    of    distance    deterrence.    Other
transportation  cost  and  distance  deterrence  functions  also
appear in the literature, however.  They  have  not  always been
considered from the viewpoint  of  the shape of the market areas
they generate,  and  their  disparity  asks the question whether
other types of functions would not be  worth being investigated.
There is thus a need for a general theory  of market areas:  the
present work aims at filling this gap,  in the case of a duopoly
competing  inside  the  Euclidean  plane  endowed with Euclidean
distance.

(Bien   qu'\'ebauch\'ee   par   Rau  d\es  1841,  la  th\'eorie
\'economique  de  la forme des aires de march\'e planaires s'est
longtemps  content\'ee  de l'hypoth\ese de co\^uts de transport
proportionnels  \a  la  distance.   Dans le mod\ele gravitaire
g\'en\'eralis\'e, auquel on peut \'etendre cette th\'eorie, ceci
correspond au  choix  d'une  exponentielle  d\'ecroissante comme
fonction  de  dissuasion  de la distance.  D'autres fonctions de
co\^ut de transport ou de dissuasion de la distance apparaissent
cependant dans la  litt\'erature. La forme des aires de march\'e
qu'elles  engendrent  n'a pas toujours \'et\'e \'etudi\'ee ; par
ailleurs,  leur  vari\'et\'e am\ene \a se demander si d'autres
fonctions encore ne m\'eriteraient pas d'\^etre examin\'ees.  Il
para\^it donc utile de disposer d'une th\'eorie g\'en\'erale des
aires de march\'e : ce \a  quoi  s'attache ce travail en cas de
duopole,  dans  le  cadre  du plan euclidien muni d'une distance
euclidienne.)

7.  Keith J. Tinkler, Editor, Nystuen---Dacey Nodal Analysis,
1988.

Professor  Tinkler's  volume  displays  the  use  of  this graph
theoretical  tool  in  geography, from  the original Nystuen ---
Dacey article, to a  bibliography of uses, to  original  uses by
Tinkler.  Some reprinted  material is  included,  but by far the
larger  part  is  of  previously  unpublished material.  (Unless
otherwise   noted,   all   items  listed  below  are  previously
unpublished.)  Contents:
 Foreward' " by Nystuen, 1988;  Preface" by Tinkler,  1988;
Statistics for Nystuen --- Dacey Nodal Analysis,"  by Tinkler,
1979; Review of Nodal Analysis literature by Tinkler (pre--1979,
reprinted with permission; post---1979, new as of 1988); FORTRAN
program listing for Nodal Analysis  by Tinkler; A graph theory
interpretation of nodal regions'' by John D. Nystuen and Michael
F. Dacey, reprinted with  permission, 1961; Nystuen---Dacey data
concerning telephone  flows  in  Washington and Missouri,  1958,
1959 with comment by Nystuen, 1988;  The expected distribution
of nodality  in  random  (p, q)  graphs  and  multigraphs,''  by
Tinkler, 1976.

8.  James W. Fonseca, The Urban Rank--size Hierarchy:
A Mathematical Interpretation, 1989.

The  urban  rank--size  hierarchy  can  be  characterized  as an
equiangular spiral of the form
$r=ae^{\theta \, \hbox{cot}\alpha}$.
An equiangular spiral can also be constructed  from a  Fibonacci
sequence. The urban rank--size hierarchy is thus shown to mirror
the properties derived from Fibonacci  characteristics  such  as
rank--additive properties. A new method of structuring the urban
rank--size hierarchy  is  explored  which  essentially parallels
that  of the traditional rank--size  hierarchy  below  rank  11.
Above rank 11 this method may help  explain the frequently noted
concavity of the rank--size  distribution  at  the upper levels.
The research suggests that  the  simple rank--size rule with the
exponent equal to 1 is not  merely a  special case, but rather a
theoretically justified norm  against which deviant cases may be
measured. The spiral distribution model allows conceptualization
of a new view of the  urban  rank--size  hierarchy  in which the
three largest cities share functions in a Fibonacci hierarchy.

9.  Sandra L. Arlinghaus,  An Atlas of Steiner Networks, 1989.

A  Steiner  network  is a tree of minimum total length joining a
prescribed, finite,  number  of  locations;  often new locations
are introduced into the prescribed set to  determine the minimum
tree.  This Atlas explains the mathematical  detail  behind  the
Steiner construction for  prescribed  sets  of $n$ locations and
displays the steps, visually, in a series of Figures.  The proof
of  the  Steiner  construction is by mathematical induction, and
enough  steps  in  the early part of the induction are displayed
completely  that  the  reader  who is well--trained in Euclidean
geometry,  and  familiar  with  concepts  from  graph theory and
elementary  number  theory,  should  be  able  to  replicate the
constructions for full as well as for degenerate Steiner trees.

10.  Daniel A. Griffith, Simulating $K=3$ Christaller Central
Place Structures:  An Algorithm Using A Constant Elasticity of
Substitution Consumption Function, 1989.

An  algorithm  is  presented that uses BASICA or GWBASIC on  IBM
compatible machines.  This algorithm simulates Christaller $K=3$
central place structures,  for  a four--level  hierarchy.  It is
based upon earlier published work by the author.  A  description
of  the  spatial  theory,  mathematics,  and  sample output runs
appears  in  the monograph.  A digital version is available from
the author, free of charge, upon request; this  request  must be
accompanied by a 5.25--inch formatted diskette.   This algorithm
has  been  developed  for  use  in  Social   Science   classroom
laboratory situations, and is designed to
(a) cultivate a deeper understanding of central place theory,
(b) allow parameters of a central place system to be altered and
then  graphic  and  tabular  results  attributable  to these
changes viewed, without experiencing the tedium  of  massive
calculations, and
(c) help  promote  a  better  comprehension  of the complex role
distance plays in the space--economy.  The algorithm also should
facilitate  intensive  numerical  research  on   central   place
structures;  it  is  expected  that  even  the sample simulation
results will reveal interesting  insights  into abstract central
place theory.

The background spatial theory concerns demand and competition in
the  space--economy;  both linear and non--linear spatial demand
functions are discussed.  The mathematics is concerned with
(a)  integration  of  non--linear  spatial  demand  cones  on  a
continuous  demand  surface,  using  a  constant  elasticity  of
substitution consumption function,
(b) solving for roots of polynomials,
(c) numerical approximations to integration and root extraction,
and
(d) multinomial   discriminant   function   classification    of
commodities into central place hierarchy levels.  Sample  output
is  presented  for  contrived  data   sets,   constructed   from
artificial and empirical information, with the wide range of all
possible  central  place  structures  being   generated.   These
examples should facilitate implementation testing.  Students are
able  to  vary  single  or  multiple  parameters of the problem,
permitting  a  study  of how certain changes manifest themselves
within  the  context  of  a theoretical central place structure.
Hierarchical  classification  criteria  may  be  changed, demand
elasticities may or may not vary and can take on a wide range of
non--negative  values,  the uniform transport cost may be set at
any positive  level, assorted fixed costs and variable costs may
be  introduced,  again  within  a  rich  range  of non--negative
possibilities,  and  the  number  of commodities can be altered.
Directions  for  algorithm  execution  are summarized.  An ASCII
version  of  the  algorithm,  written  directly from GWBASIC, is
included in an appendix; hence, it is free of typing errors.

11.  Sandra L. Arlinghaus and John D. Nystuen,
Environmental Effects on Bus Durability, 1990.

This  monograph  draws  on the authors' previous publications on
Climatic" and Terrain" effects on bus durability.   Material
on  these  two  topics  is  selected,  and reprinted, from three
published  papers  that  appeared  in  the  {\sl  Transportation
Research Record\/} and in the {\sl Geographical Review\/}.   New
material  concerning  congestion"  effects  is examined at the
national  level,  to  determine  dense,"  intermediate," and
sparse"  classes  of  congestion,  and  at  the local level of
congestion  in  Ann  Arbor  (as suggestive of how one  might use
local data). This material is drawn together in a single volume,
along  with  a  summary of the consequences of all three effects
simultaneously,  in  order  to suggest direction for more highly
automated studies that should  follow naturally with the release
of the 1990 U. S. Census data.

12.  Daniel A. Griffith, Editor.
Spatial Statistics:  Past, Present, and Future,  1990.

Proceedings  of  a  Symposium of the same name held at Syracuse
University  in  Summer,  1989.   Content  includes a Preface by
Griffith and the following papers:

Brian Ripley, Gibbsian interaction models";

J. Keith Ord, Statistical methods for point pattern data";

Luc Anselin, What is special about spatial data";

Robert P. Haining, Models in human geography:
problems in specifying, estimating, and validating models
for spatial data";

R. J. Martin,
The role of spatial statistics in geographic modelling";

Daniel Wartenberg,
Exploratory spatial analyses:  outliers,
leverage points, and influence functions";

J. H. P. Paelinck,
Some new estimators in spatial econometrics";

Daniel A. Griffith,
A numerical simplification for estimating parameters of
spatial autoregressive models";

Kanti V. Mardia,
Maximum likelihood estimation for spatial models";

Ashish Sen, Distribution of spatial correlation statistics";

Sylvia Richardson,
Some remarks on the testing of association between spatial
processes";

Graham J. G. Upton, Information from regional data";

Patrick Doreian,
Network autocorrelation models:  problems and prospects."

Each chapter is preceded by an Editor's Preface" and followed
by a Discussion and, in some cases, by an author's Rejoinder to
the Discussion.

13.  Sandra L. Arlinghaus, Editor.  Solstice --- I,  1990.

14.  Sandra L. Arlinghaus, Essays on Mathematical Geography
--- III, 1991.

A continuation of the series.  Essays in this volume are:
Table  for  central  place  fractals;  Tiling  according to  the
Administrative" Principle; Moir\'e maps; Triangle partitioning;
An  enumeration  of  candidate  Steiner  networks; A topological
generation gap; Synthetic centers of gravity:  A conjecture.

15.  Sandra L. Arlinghaus, Editor, Solstice --- II, 1991.

16.  Sandra L. Arlinghaus, Editor, Solstice --- III, 1992.

DISCUSSION PAPERS--ORIGINAL
Editor, Daniel A. Griffith
Professor of Geography
Syracuse University

1.  Spatial Regression Analysis on the PC:
Spatial Statistics Using Minitab.  1989.
Cost:  \$12.95, hardcopy. DISCUSSION PAPERS--REPRINTS Editor of MICMG Series, John D. Nystuen Professor of Geography and Urban Planning The University of Michigan 1. Reprint of the Papers of the Michigan InterUniversity Community of Mathematical Geographers. Editor, John D. Nystuen. Cost: \$39.95, hardcopy.

Contents--original editor:  John D. Nystuen.

1.  Arthur Getis, Temporal land use pattern analysis with the
use of nearest neighbor and quadrat methods."  July, 1963

2.  Marc Anderson, A working bibliography of mathematical
geography."  September, 1963.

3.  William Bunge, Patterns of location."  February, 1964.

4.  Michael F. Dacey, Imperfections in the uniform plane."
June, 1964.

5.  Robert S. Yuill, A simulation study of barrier effects
in spatial diffusion problems."  April, 1965.

6.  William Warntz, A note on surfaces and paths and
applications to geographical problems."  May, 1965.

7.  Stig Nordbeck, The law of allometric growth."
June, 1965.

8.  Waldo R. Tobler, Numerical map generalization;"
and Waldo R. Tobler, Notes on the analysis of geographical
distributions."  January, 1966.

9.  Peter R. Gould, On mental maps."  September, 1966.

10.  John D. Nystuen, Effects of boundary shape and the
concept of local convexity;"  Julian Perkal, On the length
of empirical curves;" and Julian Perkal, An attempt at
objective generalization."  December, 1966.

11. E. Casetti and R. K. Semple, A method for the
stepwise separation of spatial trends."  April, 1968.

12.  W. Bunge, R. Guyot, A. Karlin, R. Martin, W. Pattison,
W. Tobler, S. Toulmin, and W. Warntz, The philosophy of maps."
June, 1968.

Reprints of out-of-print textbooks.
Printer and obtainer of copyright permission:  Digicopy Corp.
Inquire for cost of reproduction---include class size

1.  Allen K. Philbrick.  This Human World.

2.  John F. Kolars and John D. Nystuen.  Human Geography.

Publications of the Institute of Mathematical Geography have
been reviewed or noted in

1.  The Professional Geographer published
by the Association of American Geographers;

2.  The Urban Specialty Group Newsletter
of the Association of American Geographers;

3.  Mathematical Reviews published by the
American Mathematical Society;

4.  The American Mathematical Monthly published
by the Mathematical Association of America;

5.  Zentralblatt fur Mathematik,  Springer-Verlag, Berlin

6.  Mathematics Magazine, published by the Mathematical
Association of America.

7.  Science, American Association for the Advancement of Science

8.  Science News.

9.  Harvard Technology Window.

10.  Graduating Engineering Magazine.

11.  Newsletter of the Association of American Geographer.

12.  Journal of The Regional Science Association.
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