Volume IV, Number 1.  Summer, 1993.
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               {\ifodd\pageno\righthead \else\lefthead\fi}\fi}
 \def\righthead{\sl\hfil SOLSTICE }
 \def\lefthead{\sl Summer, 1993 \hfil}
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 \centerline{\big SOLSTICE:}
 \centerline{\bf SUMMER, 1993}
 \centerline{\bf Volume IV, Number 1}
 \centerline{\bf Institute of Mathematical Geography}
 \centerline{\bf Ann Arbor, Michigan}
 \centerline{\bf SOLSTICE}
 \line{Founding Editor--in--Chief: 
      {\bf Sandra Lach Arlinghaus}. \hfil}
 \centerline{\bf EDITORIAL BOARD}
 \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}
 \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}
 \line{{\bf Engineering Applications} \hfil}
 \line{{\bf William D. Drake},
        University of Michigan, \hfil}
 \line{{\bf Education} \hfil}
 \line{{\bf Frederick L. Goodman},
        University of Michigan, \hfil}
 \line{{\bf Business} \hfil}
 \line{{\bf Robert F. Austin, Ph.D.} \hfil}
 \line{President, Austin Communications Education Services \hfil}
       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. 
 \noindent {\bf Send all correspondence to:
 Institute of Mathematical Geography, 2790 Briarcliff,
 Ann Arbor, MI 48105-1429, (313) 761-1231, IMaGe@UMICHUM,
 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).
       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.
       {\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.}
 \copyright Copyright, June, 1993 by the
 Institute of Mathematical Geography.
 All rights reserved.
 {\bf ISBN: 1-877751-55-3}
 {\bf ISSN: 1059-5325} 
 \centerline{\bf TABLE OF CONTENT}
 \noindent{\bf  1.  WELCOME TO NEW READERS}
 \noindent{\bf  2.  PRESS CLIPPINGS---SUMMARY}
 \noindent{\bf 4.  ARTICLES}

 \noindent{\bf Electronic Journals: 
 Observations Based on Actual Trials, 1987-Present}

 \noindent {\bf Sandra L. Arlinghaus and Richard H. Zander}.

   Content issues;  
   Production issues;
   Archival issues;

 {\bf Wilderness As Place}

 \noindent {\bf John D. Nystuen}

   Visual paradoxes;
   Wilderness defined;
   Conflict or synthesis;
   Wilderness as place;
   Suggested readings;
   Visual illusion authors.

 {\bf The Earth Isn't Flat.  And It Isn't Round Either:
 Some Significant and Little Known Effects of the
 Earth's Ellipsoidal Shape}  

 {\bf Frank E. Barmore}

 reprinted from {\sl The Wisconsin Geographer\/}.

    The Qibla problem;
    The geographic center;
    The center of population;

 {\bf Microcell Hex-nets?}

 \noindent {\bf Sandra Lach Arlinghaus}  

    Microcell hex-nets;

 {\bf Sum Graphs and Geographic Information}

 {\bf Sandra L. Arlinghaus, William C. Arlinghaus, Frank Harary}

    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;
 \smallskip\noindent {\bf 5.  DOWNLOADING OF SOLSTICE}

 \noindent{\bf 6.  INDEX to Volumes I (1990),  II (1991),  and
 III (1992) of {\sl Solstice}.}

 \noindent{\bf 7.  OTHER PUBLICATIONS OF IMaGe }

 \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,
 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

 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.
 \centerline{\bf 2.  PRESS CLIPPINGS---SUMMARY}

 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.
 \noindent 2. {\bf Science News}, ``Math for all seasons"
 by Ivars Peterson, January 25, 1992, Vol. 141, No. 4.
 \noindent 3.  {\bf Newsletter of the Association of American
 Geographers}, June, 1992.
 \noindent 4. {\bf American Mathematical Monthly},
 ``Telegraphic Reviews" --- mentioned as
 ``one of the World's first electronic journals using {\TeX}," 
 September, 1992.
 \noindent 5. {\bf Harvard Technology Window}, 1993.
 \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.
 \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.
 \centerline{\bf 4.  ARTICLES}
 \centerline{\bf ELECTRONIC JOURNALS:}
 \centerline{\bf Sandra L. Arlinghaus and Richard H. Zander.$^*$}

 \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.

 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:

 An inexpensive way to share information,  quickly,  with  a  large
 number of individuals;
 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;
 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;
 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. 

 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:
 ``Genuine" electronic journals.
 Mere computerized versions of hardcopy titles.
 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

 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.
 \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:

 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

 standard  reviews  that  are  specific  to the  discipline  of the
 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);
 news media, including field-specific  conferences  and meetings as
 well as mass media; 
 standard  book/journal  registers of documents  using conventional
 book/journal codes (such as ISBN and ISSN); and,
 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.

 The  editor  should  consider  the  unusual  to  boost  regard and
 readership for this mode of journal transmission, such as:

 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
 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);
 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;  
 the use of novel typesetting or other electronic capabilities that
 display the power of the vehicle of transmission (Horstmann 1991);
 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:

 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:

 Copyright  a  hard  copy  of  the  original  transmission (thereby
 placing it in the Library of Congress);
 Advertise that the original computerized version or a hard copy of
 the  original  transmission  is  available  (on-demand)  to  those
 wishing it--including libraries;
 Store single hard copies in selected libraries (including  that of
 the author's institution);
 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;
 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
               \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.
 Copy-protect  diskettes  (using  some  sort  of seal unique to the
 journal) to prevent unthinking abuse.

 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. 
 \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.  

 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.

 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.
 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).
 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.
 {\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. 
 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.
 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. 
 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.  
 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.
 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.
 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."

 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.

 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.
 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.
 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.
 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

 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
 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.  
 Do not  spell any  words with all capital letters (this  may  make
 it difficult to search for them; it also looks bad).
 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

 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
 \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.

 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: 

 Reading  or leaving messages. These are typed while online and may
 be public or private (readable only by the addressee).  
 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.     
 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.  

 A bulletin  board  can be of interest to scholars in the following

 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.  
 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.

 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.
 \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.

 Availability --- the  E-journal should be archived indefinitely in
 an  institution  willing  to  provide  copies or the equivalent on
 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).
 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

 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.

 \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

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

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

 \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.
 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. 
 \centerline{\bf WILDERNESS AS PLACE}
 \centerline{\bf John D. Nystuen $^*$}

 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.

 \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.

 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.

 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.  

 \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

 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

 \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?

 \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 

 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
 \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.
 \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.
 \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
 $^*$ John D. Nystuen,
   Professor of Geography and Urban Planning,
   College of Architecture and Urban Planning,
   The University of Michigan,
   Ann Arbor, Michigan, 48109.
 \centerline{\bf Frank E. Barmore $^*$}
 \centerline{\bf Reprinted, with permission, from}
 \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
 \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)
 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
 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.

 \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.
           &Qibla Error\quad&min.arc\cr %sample line
 \+&&{\bf Place}&
   &\qquad{\bf Qibla}&
   &\qquad{\bf Qibla Error}&\cr

 \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:
 \centerline{\sl GEOLOGICAL MARKER\/}
 {\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,
 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,

 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

 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.
 \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). 
 \centerline{Center of Population}
 \settabs\+\qquad&In three dimensions for an ellipsoid\quad
           &latitude\quad &longitude\quad &\cr %sample line
 \+&Method of computation&label&latitude&longitude&depth\cr
 \+&Bureau of the Census formul{\ae}
 \+&In three dimensions for a sphere
           &39.1823&90.3477&165 km\cr
 \+&In three dimensions for an ellipsoid
           &39.1887&90.3469&165 km\cr
 \+&On the surface of a sphere
 \+&On the surface of an ellipsoid
 \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.
 \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)

 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.).

 \centerline{\bf Appendix B}
 \centerline{\bf Half-way Point Calculation.}
 \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:
 {1\over 2}\,\,
 \int_0^{\pi /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:
 {\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:
    + {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''.
 \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.)
 Frank E. Barmore,
   Department of Physics, Cowley Hall,
   University of Wisconsin, La Crosse,
   La Crosse, WI 54601 
 \centerline{\bf MICROCELL HEX-NETS?}
 \centerline{\bf Sandra Lach Arlinghaus $^*$}

 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

 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.

 \centerline{Table:  Comparison of fractal dimensions}
 \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
 \+&(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

 \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.
 Sandra Lach Arlinghaus,
   Institute of Mathematical Geography,
   2790 Briarcliff, Ann Arbor, MI 48105.
 \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.

    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

    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.
 \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.
     &              &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
 \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} 

 \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.

    $2$ &\bullet   \cr
        &\Big\vert \cr 
    $3$ &\bullet   \cr
        &          \cr 
    $5$ &\bullet   \cr
        &          \cr 
    $6$ &\bullet   \cr
        &          \cr 
    $10$&\bullet   \cr
 \centerline{\bf Figure 2.}

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

 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}

 \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

     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.

    $1$ &\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
 \centerline{\bf Figure 4.}

     A Fibonacci sum graph containing a path and an isolate} 

 \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
    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.
     &\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
 \centerline{\bf Figure 5.}

 \centerline{Three graphs, Left, Middle, and Right, representing
 serial linkage of data.}

    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

    $1$ &\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
 \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).}

    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.

    $1$ &\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
 \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.}

 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).
     &\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
 \centerline{\bf Figure 8.}

 \centerline{Three constellations, Left, Middle, and Right,
 partition a distribution of nodes.}

      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.

    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
 \centerline{\bf Figure 9.}

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

      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.
 \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.

 \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.
 \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}\}
     \{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

 \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.
 \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.

    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
 \centerline{\bf Figure 10.}

 \centerline{Logarithmic sum graph}

 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.
 \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.)

    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
 \centerline{\bf Figure 11.}

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

 \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.
    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
 \centerline{\bf Figure 12.}  

 \centerline{Logarithmic sum graph (top) and reversed logarithmic
 sum graph (bottom).}

 \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).

    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
 \centerline{\bf Figure 13.}  

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

 \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.
 \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
   \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.
 \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


      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.
 \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.
 \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.

 \centerline{\bf TABLE 1.1:}
 \centerline{\bf Analysis according to
                 sum graph unification construction}
 \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
 \+&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
 \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}
 \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
 \+&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
 \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.}
            \qquad\qquad\qquad & Logarithm, base 23 \cr
 \+&Multiplier                 &Logarithm, base 23 \cr
 \+&$1\cdot 23 = 23$           &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
 \centerline{\bf TABLE 3:}                                                
 \centerline{\bf Table 1.1
                 labelled as a reversed logarithmic sum graph} 
 \settabs\+&E. St. Louis\quad &ITUDE\quad &NACCI\quad 
           &FACTORED\quad &474659385665\quad &ORDER 
 \+&City        &LONG-  &FIBO- &BASE   &FACTORED&NODE  &LOG    \cr     
 \+&Suburb      &ITUDE  &NACCI &MULTI- &WEIGHT  &WEIGHT&BASE   \cr     
 \+&METRO       &       &LABEL &PLIER  &        &      &37 NODE\cr     
 \+&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
 \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
 \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
 \centerline{\bf TABLE 5:}                                                
 \centerline{\bf New data added --- Ann Arbor} 
 \settabs\+&E. St. Louis\quad &ITUDE\quad &NACCI\quad 
           &FACTORED\quad &474659385665\quad &ORDER 
 \+&City        &LONG-  &FIBO- &BASE   &FACTORED&NODE  &LOG    \cr     
 \+&Suburb      &ITUDE  &NACCI &MULTI- &WEIGHT  &WEIGHT&BASE   \cr     
 \+&METRO AREA  &       &LABEL &PLIER  &        &      &37 NODE\cr     
 \+&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
 \+&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
 \centerline{\bf TABLE 6:}                                                
 \centerline{\bf Specializations of sum graphs} 
 \settabs\+\indent\qquad&Augmented reversed logarithmic\qquad\qquad
           \qquad&intermediate and global scales.\cr                    
 \+&Type of graph              &Characteristics                \cr
 \+&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


 \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, 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 .   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.
  \# 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
 \# 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
 \centerline{\bf 6.  SOLSTICE--INDEX, VOLUMES I, II, AND III}
 \noindent {\bf Volume III, Number 2, Winter, 1992}
 \noindent {\bf 1.}  A Word of Welcome from A to U.
 \noindent {\bf 2.}  Press clippings--summary.
 \noindent {\bf 3.}  Reprints:
 \noindent {\bf A.}  What Are Mathematical Models and What
 Should They Be? by Frank Harary, reprinted from {\sl Biometrie -
 \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?
           {\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
 \noindent {\bf 4.}  Article:
 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.
 \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.
 \noindent {\bf 6.} Downloading of Solstice.
 \noindent {\bf 7.} Index to Solstice.
 \noindent {\bf 8.} Other Publications of IMaGe.
 \noindent {\bf Volume III, Number 1, Summer, 1992}
 \noindent{\bf 1.  ARTICLES.}
 {\bf Harry L. Stern}. 
 {\bf Computing Areas of Regions With Discretely Defined Boundaries}.
 1. Introduction 2. General Formulation 3. The Plane 4.  The Sphere
 5.  Numerical Example and Remarks.  Appendix--Fortran Program.
 \noindent{\bf 2.  NOTE }
 {\bf Sandra L. Arlinghaus, John D. Nystuen, Michael J. Woldenberg}.  
 {\bf  The Quadratic World of Kinematic Waves}
 \noindent{\bf 3.  SOFTWARE REVIEW}
 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}.
 Reviewed by {\bf Yung-Jaan Lee}, University of Michigan.
 \noindent{\bf 4.  PRESS CLIPPINGS}
 \noindent{\bf 5.  INDEX to Volumes I (1990) and II (1991) of
             {\sl Solstice}.}
 \noindent {\bf Volume II, Number 1, Summer, 1991}
 \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.).
 \noindent 2.  FEATURES

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

 \noindent 3.  FEATURES

 Press clipping; Word Search Puzzle; Software Briefs.
 \noindent{\bf INDEX to Volume I (1990) of {\sl Solstice}.}
 \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
     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.
 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
 \noindent 3.  FEATURES
 \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.
 \noindent{\bf Volume I, Number 2, Winter, 1990}
 \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 

 \noindent 2.  ARTICLES

 Sandra Lach Arlinghaus, {\sl Scale and Dimension: Their Logical

      Linkage  between  scale  and  dimension  is made using the 
 Fallacy of Division and the Fallacy of Composition in a fractal
 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.
 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.
 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
 \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.
 \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,

 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,
 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,

 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

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

 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,
 (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

 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.

 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.

 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.