Volume V, Number 2.  Winter, 1994.
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 \centerline{\big SOLSTICE:}

 \centerline{\bf WINTER, 1994}

 \centerline{\bf Volume V, Number 2}
 \centerline{\bf Institute of Mathematical Geography}

 \centerline{\bf Ann Arbor, Michigan}
 \centerline{\bf SOLSTICE}
 \line{Founding Editor--in--Chief: 
      {\bf Sandra Lach Arlinghaus} \hfil}
 \line{Institute of Mathematical Geography and University of Michigan \hfil}
 {\bf GOPHER:  on Arizona State University Department
                of Mathematics gopher}
 \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.\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:}
 Sandra Arlinghaus, Institute of Mathematical Geography,
 2790 Briarcliff, Ann Arbor MI 48105.
 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).
 Concern for manuscript security.  Prospective authors  sometimes
 worry, quite reasonably, that an article published in electronic
 format might easily be altered by a subscriber and then  re-sent
 to others.  Of  course, the  same sorts  of  concern might arise 
 with conventional publishing, given the availability of machines
 that  seem able  to photocopy,  cut, paste, and bind, all in one
 step.  The  strategy  at IMaGe, with {\sl Solstice\/} is roughly
 the  following:    we use  an  oriental  rug  as  the  model.  A
 {\sl Solstice \/}  document  is  viewed  as  a  weaving of words
 against  the  warp  and  the  woof  of  columns  and  rows in an
 electronic  matrix.   The spacing of the document, though right-
 justified, is done by  hand (rather than by word-processor).  In
 that way, an original pattern of blank spaces is created that is
 difficult  to  mimic  -- an  important  feature in defeating any
 significant alteration.  Indeed,  deliberate  spacing  ``errors"
 that hopefully do not  detract  from  the overall visual effect,
 are introduced at the whim  of the  Editor.  Any  re-transmitted
 document that does not match the document originally transmitted
 from IMaGe (when the two copies  are  superimposed) is therefore
 an altered, bogus copy. 

       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, December, 1994 by the
 Institute of Mathematical Geography.
 All rights reserved.
 {\bf ISBN: 1-877751-56-1}
 {\bf ISSN: 1059-5325} 
 \centerline{\bf TABLE OF CONTENT}
 \noindent{\bf  2.  PRESS CLIPPINGS---SUMMARY}
 \noindent{\bf 3.  ARTICLES}
 \noindent{\bf The Paris Metro:  Is its Graph Planar? }
 \noindent{\bf Sandra L. Arlinghaus,
               William C. Arlinghaus,
               Frank Harary}
 Transmitted as part 2 of 9.
     Planar graphs; 
     The Paris Metro;
     Planarity and the Metro;
     Significance of lack of planarity.
 \noindent{\bf Interruption! }
 \noindent{\bf Sandra Lach Arlinghaus}
 Transmitted as part 3 of 9.
     Classical interruption in mapping;
     Abstracts variants on interruption and mapping;
     The utility of considering various mapping surfaces--GIS;
     Future directions.
 \noindent{\bf 4.  REPRINT} 
 \noindent {\bf Imperfections in the Uniform Plane}.
 \noindent{\bf Michael F. Dacey}
 \noindent {\bf Forewords by John D. Nystuen}
 Forewords transmitted as part 4 of 9; 
 article transmitted as parts 5 and 6 of 9;
 tables transmitted as part 7 of 9. 
 Reprinted from {\sl Michigan Inter-university Community of 
 Mathematical Geographers\/}, Papers, John D. Nystuen, Editor.
 Reprinted here with permission.
    Original (1964) Nystuen Foreword;
    Current (1994) Nystuen Foreword;
    The Christaller spatial model;
    A model of the imperfect plane;
    The disturbance effect;
    Uniform random disturbance;
    Definition of the basic model;
    Point to point order distances;
    Locus to point order distances;
    Summary description of pattern;
    Comparison of map pattern;
    Theoretical order distances;
    Analysis of the pattern of urban places in Iowa;
    Almost periodic disturbance model;
    Lattice parameters;
    Disturbance variables;
    Scale variables;
    Comparison of $M_2$ and Iowa;
 \noindent{\bf  5.  FEATURES}
 \noindent{\bf Construction Zone: 
 The Braikenridge-MacLaurin Construction}
 Transmitted as part 8 of 9.
 \noindent{\bf Population Environment Dynamics:  Course and Monograph}
 \noindent{\bf William D. Drake}
 Transmitted as part 8 of 9.
 \noindent{\bf 6.  DOWNLOADING OF SOLSTICE}

 \noindent{\bf 7.  INDEX to Volumes I (1990),  II (1991),  
 III (1992),  IV (1993) and V (1994, part 1) of {\sl Solstice}.}

 \noindent{\bf 8.  OTHER PUBLICATIONS OF IMaGe }
 All transmitted as part 9 of 9.

 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

      Thanks much to subscribers who have offered input.  Helpful
 suggestions  are  important  in trying to keep abreast, at least
 somewhat, of the constantly  changing  electronic  world.   Some
 suggestions from readers  have  already been implemented; others
 are being worked on.  Indeed,  it is particularly  helpful  when
 the reader  making  the suggestion  becomes actively involved in
 carrying it out.  We hope you continue to enjoy
 {\sl Solstice\/}.
 \centerline{\bf 2.  PRESS CLIPPINGS---SUMMARY}

     Volume 72,  Number 4,  October 1993  issue of {\sl Papers in
 Regional   Science:    The  Journal  of  the  Regional   Science
 Association\/}  carried  an article by Gunther Maier and Andreas
 Wildberger  entitled ``Wide Area Computer Networks and Scholarly
 Communication  in Regional Science."  Maier and Wildberger noted
 that ``Only one  journal  in this directory can be considered to
 be related to  Regional  Science,  {\sl Solstice:  An Electronic
 Journal of Geography and Mathematics\/}."
      Beyond that,  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.

 \noindent 7.  {\bf Earth Surface Processes and Landforms},
 18(9), 1993, p. 874.

 \noindent 8.  {\bf On Internet}, 1994.

 If  you  have  read about {\sl Solstice\/} elsewhere, please let
 us know the correct citations (and add to those above).  Thanks.
 We  are happy to  share information  with all  and are delighted
 when others share with us, as well.  

 Publications of the Institute of Mathematical Geography have,
 in addition, been reviewed or noted in 
 1.  {\sl The Professional Geographer\/} published
 by the Association of American Geographers;

 2.  The {\sl Urban Specialty Group Newsletter\/}
 of the Association of American Geographers;

 3.  {\sl Mathematical Reviews\/} published by the
 American Mathematical Society;

 4.  {\sl The American Mathematical Monthly\/} published
 by the Mathematical Association of America;

 5.  {\sl Zentralblatt\/} fur Mathematik,  Springer-Verlag, Berlin

 6.  {\sl Mathematics Magazine\/}, published by the Mathematical
 Association of America.

 7.  {\sl Newsletter\/} of the Association of American Geographer.

 8.  {\sl Journal of The Regional Science Association\/}.

 9.  {\sl Journal of the American Statistical Association\/}.

 \centerline{\bf 3. ARTICLES}
 \centerline{\bf The Paris Metro:  Is Its Graph Planar?}
 \centerline{\bf Sandra L. Arlinghaus,
                 William C. Arlinghaus, and
                 Frank Harary}
 \centerline{The University of Michigan,}
 \centerline{Lawrence Technological University, and}
 \centerline{New Mexico State University.}
 \centerline{``Over the river and through the woods,}
 \centerline{To Grandmother's house we go.}
 \centerline{The horse knows the way to carry the sleigh}
 \centerline{Through the white and drifting snow."}
 Song of unknown origin
 \noindent{\bf To appear in
 {\sl Structural Models in Geography\/}
 by this set of authors.}
 The reader should read this article with a map of the Paris Metro
 in hand.

       In the Euclidean plane, crossing lines intersect at a point
 in  the plane;  the line segment determined by $X$, $Y$ and $X'$,
 $Y'$ intersect at a point $Z$ (Figure 1). The graph that includes
 the four nodes  $X$, $Y$, $X'$, $Y'$  and the two edges $XY'$ and
 $X'Y$  does {\sl not}  have  a  fifth  node at any other location 
 (Figure 2).  To  make  this  viewpoint consistent with our narrow
 Euclidean mindset,  think  of  stretching  the edge $X'Y$ so that
 there is no visual hint of ``intersection" -- the horse  knows to
 go  over  the  river  even  though  an  aerial view of the wintry
 landscape  sees  the  river  and road as two ``intersecting" dark
 tracings across the white, snowy backdrop. 

 \midinsert\vskip 2.0in
 \noindent{\bf Figure 1.}  Draw nodes $X$ and $Y$, left to right,
 horizontally.  Draw nodes $X'$ and $Y'$, left to right,
 horizontally below the  first set.  Join $X$ to $Y'$ and join
 $X'$ to $Y$ using straight segments. 
 Label their intersection as $Z$.

 \midinsert\vskip 2.0in
 \noindent{\bf Figure 2.}  Draw nodes $X$ and $Y$, left to right,
 horizontally.  Draw nodes $X'$ and $Y'$, left to right,
 horizontally below the  first set.  Join $X$ to $Y'$ and join
 $X'$ to $Y$ using straight segments.   

 \noindent{\bf Planar graphs}

      To capture this idea more formally, we introduce the concept
 of embedding; the approach and material in  this section  follows
 closely  that  of  Harary  (1969,  pp.  102-113).    A  graph  is
 {\sl embedded\/} in a surface when it is drawn on that surface in
 such a way that no two edges intersect (geometrically). The graph
 in Figure 2 has not been embedded in the plane:  the  edges $X'Y$
 and $XY'$ intersect.  The graph in Figure 3 has been  embedded in
 the plane.  The connection pattern of the graphs in Figures 2 and
 3   is   identical:    topologically,  they  are   said   to   be
 {\sl homeomorphic\/}.   They  are  equivalent  structural models.
 Thus, we distinguish between a planar graph and a plane graph.  A
 graph is {\sl planar\/} if it can be embedded  in the  plane  (as
 can Figure 2);  a  graph  is {\sl plane\/} if it has already been
 embedded in the plane (as has Figure 3).  The graph in  Figure  2
 is planar but not plane; the graph in Figure 3 is both planar and

 \midinsert\vskip 3.0in
 \noindent{\bf Figure 3.}  Draw nodes $X$ and $Y$, left to right,
 horizontally.  Draw nodes $X'$ and $Y'$, left to right,
 horizontally below the first set.  Join $X$ to $Y'$ using a
 straight segment and join $X'$ to $Y$ using a curved line
 that does not pass through the segment joining $X$ to $Y'$.   

     A graph  that  cannot be embedded in the plane is called {\sl
 nonplanar\/}.   There  are  two  nonplanar  graphs  of particular
 importance.  One is the graph composed of two sets of three nodes:
 think of one set of three nodes arranged horizontally  and of the
 other set as arranged horizontally below the  first  set.   Edges
 join  each node of the top set to each node of the bottom set:  a
 total  of  nine  edges  (Figure 4  shows the detail of labeling). 
 This set is denoted as $K_{3,3}$.  The other  critical  nonplanar
 graph is denoted as $K_5$.  It is composed  of a pentagon and all
 edges joining the nodes  (Figure 5 shows detail).

 \midinsert\vskip 3.0in
 \noindent{\bf Figure 4.}  Draw nodes $X_1$, $X_2$, $X_3$ from
 left to right as one set of nodes arranged horizontally. 
 Draw nodes $Y_1$, $Y_2$, $Y_3$ from left to right as another set
 of nodes arranged horizontally, below the first set.  Draw 
 edges $X_1Y_1$, $X_1Y_2$, $X_1Y_3$; $X_2Y_1$, $X_2Y_2$, $X_2Y_3$;
 $X_3Y_1$, $X_3Y_2$, $X_3Y_3$ to form the nonplanar $K_{3,3}$ graph.

 \midinsert\vskip 3.0in
 \noindent{\bf Figure 5.}  Draw nodes $X_1$, $X_2$, $X_3$, $X_4$,
 and $X_5$ arranged as nodes of a regular
 pentagon.  Join the nodes as a pentagon:  along edges
 $X_1X_2$, $X_2X_3$, $X_3X_4$, $X_4X_5$, $X_5X_1$. 
 Join the remaining nodes: along edges
 $X_1X_3$, $X_1X_4$, $X_2X_4$, $X_2X_5$, $X_3X_1$, $X_3X_5$.

 Generally,  one might look at  geometric intersections to suggest
 whether or not a given graph is planar:  simple-looking geometric
 intersection patterns can often be  unscrambled  in the  plane to
 eliminate any geometric intersections (as was Figure 2 in  Figure
 3).  More complicated geometric intersection patterns ($K_{3,3}$,
 $K_5$)  cannot be undone (Harary, 1969).  As with the  four color
 problem,  and  as  is often the case, what is a simple problem to
 consider is in fact  a difficult one to solve.  It was  not until
 1930 that Kuratowski finally solved the long-standing problem  of 
 characterizing  planar graphs.  The statement of the  theorem  is
 simple; its proof is not (see Harary, 1969, for proof).

 \noindent{\sl Kuratowski's Theorem}

 A graph is planar  if and only if it has no subgraph homeomorphic
 to $K_5$ or to $K_{3,3}$.

 The  $K$  in  the notation honors Kuratowski for his achievement.
 With  this  elegant  theorem  in  hand,  we now  turn to consider 
 planarity in the geographic world.

 \noindent{\bf The Paris Metro}

      The Paris Metro is a subway system that, for the  most part,
 under the  streets  of Paris,  links  the classical  ``Portes" --
 City ``Gates"-- to each other as the  many routes criss-cross the
 Seine  in  association  with the various bridges (Figure 6).  The
 Paris Metro map is a graph; there are numerous nodes representing
 local  stations  along  a  single  train  route as well as larger
 stations  at  which  one  can  transfer  from  one metro route to
 another.  There  are directed arcs, forming a cycle, in the south
 west  of  the  map  leading  to  the  Porte d'Auteuil, and in the
 northwest leading to Pr\'e St. Gervais.  All other arcs represent
 two-way  Metro  linkages.  The  map is complicated in appearance;
 subway lines  often follow surface traffic patterns.  Pedestrians
 need access  to  subway routes from sidewalks.  Indeed, the Paris
 Metro map  reflects  the  surface pattern of the numerous rotary,
 star-shaped  intersections and tortuous  ``rues" that add much to
 Parisian  charm.  The  Metro graph is strongly  connected; choose
 any two  metro  stops -- they  are  mutually reachable within the
 entire  system,  although  a  transfer  might  be  required.  Any
 well-designed  mass transit system should clearly have this style
 of  connectedness,  lest  passengers  be  stranded.   There are a
 number  of  nodes  with indegree and outdegree in excess of four. 
 Anyone  who has  traversed  the  maze  of  possible  transfers at
 Montparnasse-Bienvenue,  for  example,  will  be  aware  of   how
 complicated  a  trip  from  ``here"  to ``there" can be.  Because
 there are quite a few transfer nodes  with  a  number of incident
 edges,  it is natural to consider whether or not a $K_{3,3}$ or a
 $K_5$ might be contained as a subgraph of the Metro graph.  David
 Singmaster  has  shown that the London Underground is non-planar;
 is the Metro graph planar?
 \topinsert\vskip 6.0in
 \noindent{\bf Figure 6.}  Map of the Paris Metro.

 \noindent{\bf Planarity and the Metro}

      Indeed,  the Metro  is not planar,  either;  when the map is
 strictly considered as a digraph, it is an easy matter to  choose
 six nodes and a set of edges to form a $K_{3,3}$.  If one wishes,
 however,  to  eliminate  the  possibility  of a transfer from one
 train  to  the  other,  in  order  to  have  direct geo-graphical
 adjacency as well as graphical adjacency, it is  also possible to
 find a $K_{3,3}$ under these tighter constraints.

      The Metro stops of Etoile (``star") and Nation are joined on
 the  north  by  a single Metro route arching across the  northern
 part of the city; they are joined on  the south by a  single arch
 paralleling the southern perimeter of Paris; and, they are joined
 across a diametral route, through Ch\^atelet  as a  ``center," by
 a single Metro route passing under the Champs Elys\'ees, Concord,
 Palais  Royal,  H\^otel  de  Ville   and   the  Bastille.    When
 Montparnasse-Bienvenue and Stalingrad are  chosen  also, as nodes
 intermediate on these southern and northern  arches,  along  with
 Gare de l'Est as a final node, this set of nodes can be joined in
 a  $K_{3,3}$  with  only direct geographic  linkage (requiring no
 transfers) between pairs of nodes along distinct edges. Label the
 nodes as follows (Figure 7):

 \item{1.}  Etoile
 \item{2.}  Montparnasse-Bienvenue
 \item{3.}  Nation
 \item{4.}  Ch\^atelet
 \item{5.}  Gare de l'Est
 \item{6.}  Stalingrad

 \topinsert\vskip 6.0in
 \noindent{\bf Figure 7.}  Metro map with labeled nodes and
 distinguished edges linking the nodes.

      Each odd-numbered node is joined to each even-numbered  node
 along  distinct  edges,  as  required for a $K_{3,3}$.  Thus  the
 Paris  Metro,  viewed  as  a  structural  model, is nonplanar; to
 travel from Montparnasse-Bienvenue to the Gare de l'Est requires,
 when represented as a map in the plane, that the edge from node 2
 to node 5 cross at least one of the other edges of the $K_{3,3}$.
 The  geographical  and  social  implications  of  this  lack   of
 planarity are significant.

 \noindent {\bf Significance of lack of planarity}

      One  might  imagine  a  subway  system  to exist in a  plane
 parallel  to  the  plane  of surface traffic, some number of feet
 below  the  surface.  Experience  with even simple subway systems
 defeats  this  notion;  trains  run on elevated tracks in regions
 with high water tables or on landfill; their elevation is altered
 to cross natural barriers such  as rivers.  There is considerable
 topographic relief in most  subway systems.  Natural difficulties
 can  force  a  subway  system out of a planar environment.  Thus,
 collisions  between  trains  on  different  routes,  in different
 (intersecting)  planes,  must  be  considered;  the separation of
 routes  into  different  layers  (planes)  offers protection from
 collision--except where the planes intersect.

      In  the  case  of  Paris, there are Metro lines at different
 levels; trains enter selected stations at different depth levels.
 Passengers  trying  to switch from one Metro route to another  at
 Montparnasse-Bienvenue may recall running up or down  stairs  and
 through connecting tunnels to execute a transfer.  The Metro  map
 shows the route north from Montparnasse-Bienvenue, toward Od\'eon,
 Ch\^atelet, the Gare de l'Est,  and the  Porte de Clignancourt to
 ``cross"  routes  12  (from  Mairie  d'Issy  to  the  Porte de la
 Chapelle) and 10 (from the Gare d'Orleans-Austerlitz to the Porte
 d'Auteuil).  If these crossings were ``real,"  rather  than over-
 or under-passes, there could be serious metro collisions at them. 
 Map evidence suggests that it is the  Orleans/Clignancourt  route
 that  is  at a  different  level as routes 10 and 12 intersect at
 nearby S\`evres Babylon station.  A lack of planarity can be used
 to advantage by engineers planning new  stations or new routes in
 a tightly-packed transport system.

 \noindent{\bf References}

 \ref Harary, F. 1969.  {\sl Graph Theory\/}.  Reading, Mass., 

 \ref Kuratowski, K.  1930.  Sur le probl\`eme des courbes
 gauches en topologie.  {\sl Fund. Math.\/}, {\bf 15}, 271-283.

 \centerline{\bf Interruption!}
 \centerline{\bf Sandra Lach Arlinghaus}
 \centerline{The University of Michigan}

 {\sl Interruption\/}, from  the Latin--{\sl rumpere\/} (to break)
 plus {\sl inter\/} (between, among), means  literally  ``to break
 into (between)."  The concept of ``interruption"  can be employed
 to guide research direction between apparently  disparate objects
 of  study; ``interruption"  is  a meta-concept  like ``symmetry," 
 ``duality,"  and  a  host  of  others.  We  are all familiar with
 flat  maps  of  the Earth that are interrupted.  Indeed, all flat
 maps of the Earth are interrupted; the one-point compactification
 of  the  sphere  guarantees  that  this  is so from a topological
 standpoint.  From a more pragmatic standpoint, we know that it is
 not  possible  to  remove  the  peel  from an orange and place it
 flatly in the plane -- the peel will rip.      

 \noindent{\bf Classical interruption in mapping}

     It  is this pragmatic view of mapping the Earth into the plane 
 that  conjures  up  most  visual  images  of an ``interrupted" map
 projection -- one in which some cuts have  been made (typically in
 the  oceans)  in  order  to  preserve  some  degree of a desirable
 property, such as conformality or equality of  area.   Philbrick's
 (1963) Sinu-Mollweide has the northern hemisphere  continuous with
 slits in the oceans in the southern hemisphere; Goode's Homolosine
 Equal Area projection (Goode, various years) has  interruptions in
 oceans in both hemispheres.  Either of these  projections would be
 viewed, clearly, as an ``interrupted" projection.       
     However,  would  all  who see these as interrupted also view a
 cylindrical  projection  (Miller,  for example) as ``interrupted"?  
 Of course it is, for once the sphere is projected onto the surface
 of  the  cylinder,  the  cylinder  must  then  be  ``developed" or
 unrolled into a section of the plane. The development of a surface
 in the plane is a cut -- a form of breaking into the cylinder -- an
 interruption.  The difference is that the interruption in a Miller
 cylindrical projection often determines the boundary of the map in
 the plane -- our eye seeks closure and when the cut coincides with
 the map boundaries we use for  closure,  the visual effect is less
 jarring; the interruption is masked by the boundary.

 \noindent{\bf Abstract variants on interruption and mapping}

      Going  farther  abstractly,  one might consider rather than a
 map  on  a  cylinder,  a  map  on  a M\"obius strip; Tobler (1961)
 described  a  scheme  in  which  a  pin,  poked through a map on a 
 M\"obius  strip,  emerges  at  its  antipodal  point.   When  this
 procedure is continued a finite number of times, the boundaries of
 a region and its antipodal region are traced out simultaneously on
 this one-sided map.  This  novel  approach suggests ways to  trace
 out partial, discrete, boundaries.  Spilhaus (1979) suggests  that
 to  construct  a  continuous map of the antipodes one ``show which
 land is opposite other land $\ldots $ by taking a  pair of maps of
 two hemispheres and putting them back to back  with the North Pole
 covering the South Pole."  Neither construction  touches on deeper 
 non-Euclidean  aspects  of this style of construction (Arlinghaus,
   From the viewpoint of interruption, however, what is interesting
 is  the  mere  idea of considering a map on a M\"obius strip.  The
 cylinder  and  the M\"obius strip are both developable surfaces in
 the plane  and  they  are but  two  members  of a  broader  class.
 Because developable surfaces, when  interrupted and  placed in the
 plane, are those  whose  boundaries  can easily  mask the cuts  of
 interruption,  they are  a class  of  particular  interest.   This
 broader  class  of  surface  may  be  viewed  as  composed  of two
 structurally  parallel  sequences of transformations -- one easily
 visualized  and  the other visualized easily only by analogy  with
 the first (Figure 1).  (This sort of characterization is common in
 a variety of books  that  deal  with  elementary  topology, as for 
 example in Courant and Robbins, 1941.)

 \midinsert\vskip 4.0in
 \noindent{\bf Figure 1.}  Two sequences:  on the left, a
 rectangle is rolled up into a cylinder, and then the 
 cylinder is joined, end-to-end, to form a torus.  On the
 right, a rectangle, given a half-twist, is rolled up 
 into a M\"obius strip, and then joined (with another
 half twist), end-to-end, to form a Klein bottle.

 \noindent{\bf Visual sequence:}

 \item{1.}  A plane  rectangle  may  be  rolled into a cylinder by
 gluing together the upper left to the upper right corners and the
 lower left to the lower right corners.  The  result is a cylinder
 with diameter that of the length of the top of the rectangle.

 \item{2.}  A  cylinder  may  be rolled into a torus by gluing one
 circular end of the cylinder to the other -- the seam along which
 gluing  takes  place  is  the circle that matches the ends of the
 straight line seam along the length of the cylinder.

 \noindent{\bf Abstract sequence:}

 \item{1.}  A plane rectangle may be rolled into a M\"obius  strip
 by gluing together  the upper  left to the lower right corners of
 the rectangle and the lower left to the  upper  right  corners of
 the rectangle.  The result is a M\"obius strip; the gluing action
 imparts a half-twist to the rectangular strip.

 \item{2.}  A  M\"obius strip may be rolled into a Klein bottle by
 gluing  one  ``circular"  end of the M\"obius strip to the other,
 as with the torus.

 What  can  be  glued  can  be  unglued  (in  this context); thus,
 cylinder, torus, M\"obius strip, and Klein bottle are developable
 surfaces in the plane.  One can view each of them as a surface on
 which to map; difficulty in such  an approach is encountered only
 when  the  need  to  visualize  physical  objects is relied upon. 
 Conceptually, from a structural viewpoint,  the M\"obius strip is
 no  more  difficult  to  consider than is the cylinder; the Klein
 bottle no more difficult than is the torus.  

 \noindent{\bf The utility of considering various mapping surfaces--GIS}

      A current  maxim  of  those concerned with the protection of
 various  elements  of the environment is ``to think globally, act
 locally."  While  this  may  have  fine implications for landfill
 management, it is a dangerous cartographic practice.  Globally we
 should  think  of  a  sphere  or  some other approximation of the
 Earth's  surface  that is topologically equivalent (homeomorphic)
 to the sphere.  Locally we tend to think of our immediate part of
 the Earth as flat; recently, Barmore (1992; 1994) has  shown  the
 difficulty in determining  geographic  centers  of  various sorts
 when concerns for curvature are not involved in policy decisions. 
 In earlier times, this sort of lack of  tying  knowledge  of  the
 earth as a sphere to a local plane environment was evident:  from
 Eratosthenes'  measurement  of  the  Earth  to  the great voyages
 undertaken  at  the  end  of the Middle Ages and beginning of the
 Renaissance in Western Europe.  
     Most  mapping  is done from the global/spherical viewpoint to
 the local/planar  viewpoint; it need not be, and when the mapping
 is  from  developable  surface to plane, or from sphere to object
 homeomorphic to the sphere, then maps that hide interruption  can
 be constructed.  One place where this issue  has,  for  the  most
 part, not been addressed at all, is in the electronic environment
 of the Geographic Information Systems (GISs).  In a recent paper,
 Tobler (1993) speaks to this  issue  at some length and notes, in
 particular, that of the hundreds  of  GISs  available,  ``The one
 exception, explicitly designed to  consider the spheroidal earth,
 is  the  `Hipparchus'  system  developed  by  Hrvoje  Lukatela of
 Calgary, Alberta (Lukatela 1987)."   GISs such as this apparently
 offer  a  way   to  make   maps  directly  from  spherical  data,
 eliminating the middle step of imitating the traditional drafting
 processes of the human arm  and the planar  decisions  associated
 with those.  This sort of idea seems quite natural--why should we
 use  the computer to imitate the classical  drafting process; why
 not  use  it  to  take  advantage  of the underlying mathematical
 characteristics of the real problems of dealing with surfaces?    

     Another route to this sort of end  might  be to construct data
 structures  in  the  environment  of  the mathematics of the Klein
 bottle, torus, M\"obius  strip,  or cylinder,  and then to develop
 (as in ``unroll") the mathematics to make plane maps.   Either way
 -- from sphere to sphere homeomorph, or from  developable  surface
 to  plane,  one  might  look forward to more elegantly constructed
 electronic  programs  for  executing  mapping -- with  the   usual
 hoped-for consequence  that  elegance in theory leads to  leaps in

 \noindent Future directions 

     What  is  important  to  consider  for  maps  is important to
 consider  for  other  representations  of  the  earth's  surface.
 Cartographic considerations can guide disparate research projects
 of spatial character.

      Structural models (Harary, Norman, and Cartwright, 1965), one
 form of abstract graphs (Harary 1969), can  offer  yet another way
 to map the Earth.  These abstract graphs serve as ``maps" whenever
 any discrete set of real-world locations and flows can be captured
 in channels linking locations:  the  locations  serve as the nodes
 for  the  graph  and  the  channels serve as  edges linking nodes. 
 Thus, a set of cities and the railroad tracks  joining them may be
 represented visually as a structural model -- the cities are nodes
 and  the  tracks  are  edges  of  the  model.  Indeed,  a  set  of
 individuals, at least some of whom share a common belief, may also
 be represented as a structural model;  the  individuals are  nodes
 and  the  belief,  if  shared,  is represented along edges linking
 appropriate  individuals.   There  are numerous examples one might
 construct.  What is important is that these models,  as are  maps,
 are  also  subject  to  interruption.   Because  it  is abstractly
 preferable  to  avoid  or to mask interruption, it is important to
 know how it arises. 


 \noindent{\bf References}

 \ref Arlinghaus, Sandra L.  1987.  Terrae Antipodum.  In
 {\sl Essays on Mathematical Geography -- II\/}, Monograph
 \#5, Institute of Mathematical Geography, Ann Arbor, MI, 33-40.

 \ref Barmore, Frank.  1992.  Where are we?  Comments on the
 concept of the ``center of population."  {\sl Solstice:  An
 Electronic Journal of Geography and Mathematics\/}. 
 Monograph \#16, Institute of Mathematical Geography, Ann Arbor,
 MI, 22-38.

 \ref Barmore, Frank.  1994.  Center here; center there; center, 
 center everywhere.  {\sl Solstice:  An Electronic Journal of
 Geography and Mathematics\/}.  Monograph \#18,
 Institute of Mathematical Geography, Ann Arbor, MI, 12-25.

 \ref Courant, R. and Robbins, H. 1941. {\sl What Is Mathematics?\/}
 Oxford University Press, London.

 \ref {\sl Goode's School Atlas\/}.  Various editions.  Rand McNally.
 First copyrighted, 1922; Golden Anniversary Edition, Espenshade,
 E. B. , Jr. and Morrison, J. L.  {\sl Rand McNally World Atlas\/}.
 Rand Mc Nally, Chicago.

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

 \ref Harary, F., Norman R., and Cartwright, D.  1965.  
 {\sl Structural Models:  An Introduction to the Theory of
 Directed Graphs\/}.  Wiley, New York.

 \ref Lukatela, H.  1987.  Hipparchus Geopositioning Model:  
 An Overview, Proceedings, AutoCarto 8:87-96, ASPRS \& ACSM,

 \ref Philbrick, Allen K.  1963. {\sl This Human World\/}.
 Wiley, New York.  Reprinted, Institute of Mathematical Geography.

 \ref Spilhaus, Athelstan.  1979.  To see the oceans, slice up
 the land.  {\sl Smithsonian Magazine\/}, Nov. 1979, 116-122.

 \ref Tobler, Waldo R.  1961.  World map on a M\"obius strip.
 {\sl Surveying and Mapping\/}, XXI, p. 486.

 \ref Tobler, Waldo R.  1993.  Global spatial analysis.
 In Tobler, {\sl Three Presentations on Geographical Analysis
 and Modeling\/}, National Center for Geographic Information
 and Analysis, Technical Report 93-1.
 \ref Webster's Seventh New Collegiate Dictionary.  1965.
 G. \& C. Merriam Co., Springfield Mass. 

 \centerline{\bf 4. REPRINT}

 \centerline{\bf Imperfections in the Uniform Plane}
 \centerline{\bf Michael F. Dacey}
 \centerline{\bf with Forewords by John D. Nystuen, 
                                   The University of Michigan}

 In this section, {\sl Solstice\/} Board member,  John D. Nystuen,
 selects  a  paper  from  the  collected  papers  of  the Michigan
 Inter-University Community of  Mathematical  Geographers (MICMOG)
 (of which  he is Editor) to reprint here, some 30 years after its
 initial  presentation.   In  addition  to  the reprint of work of
 Michael  Dacey,  Nystuen's original Foreword, and introduction of
 Dacey  and  his  work  to  the  assembled  MICMOG  group, is also
 reprinted. In addition, a new Foreword by Nystuen takes a look at
 the  Dacey  paper  in  retrospect.  The  paper  is reprinted with
 permission of Nystuen, on behalf of the Michigan Inter-University
 Community of Mathematical Geographers.

 \centerline{\bf Foreword, December, 1994}
 \centerline{\bf John D. Nystuen}
     Thirty years ago Michael Dacey contributed to the development
 of spatial statistics in highly original ways.  Many of the ideas
 he  used  and  introduced  to the literature in the 1960s are now 
 part of generally accepted spatial theory.   For example,  he was
 one of the first to use the idea of  a dimensional transformation
 to  permit  evaluations  of  the spatial association of point and 
 area phenomena.  The transformational approach proved useful as a
 general   concept   as  Keith  Clarke  has  demonstrated  in  his
 interesting book  (Clarke, 1990).   Arthur Getis,  a colleague of
 Dacey's, and Barry Boots used many of Dacey's ideas in their book
 (Getis and Boots, 1978) about modelling spatial process.
     Today,  vigorous  effort  is  being expended on incorporating
 spatial analysis  functions  into  Geographic Information Systems
 (GIS) software.   We are  re-issuing one of Dacey's seminal works
 to  bring to the attention  of contemporary scholars an important
 source of many of the concepts now becoming accessible to general
 uses of GIS technology.   Dacey's  work  now  speaks  to  another
 \noindent {\bf References}
 \ref Clarke, Keith C. 1990. {\sl Analytical and Computer
 Cartography\/}, Prentice-Hall, Englewood Cliffs, NJ.

 \ref Getis, A. and Boots, B.  1978.  {\sl Models of Spatial
 Processes\/}, Cambridge University Press, Cambridge.

 \centerline{\bf Foreword, May, 1964}
 \centerline{\bf John D. Nystuen}

 We are pleased  to  present  to our readers a  paper by Professor
 Michael F. Dacey.  Many of us are aware, if  only vaguely, of his
 provocative  and  voluminous   writings.    Professor  Dacey  has 
 penetrated deeply into realms where few, if any, have gone before. 
 He  travels  alone  and has left but a thin trail of mimeographed
 papers  as  scent.  The track is now long and difficult to follow
 and he does  not rest.  He has allowed one of his works to become
 discussion paper \#4 of our series.  We hope this will expose his
 activities to a wider audience.  Some may be inspired to join him
 in the new work  that he is doing.  I hope so.  Certainly we must
 keep in contact with him.  Regrettably many of his results depend
 upon  his  previous  statements  now difficult to obtain.  I will
 attempt  in  this  foreword a short review of the pertinent ideas
 by  way  of a summary of this paper.  I have also added, with his
 permission, a glossary of symbols at the end of the paper.

     Michael Dacey has for several years explored abstract spatial
 patterns  using  probabilistic  methods.   This paper is one of a
 series  of  such  studies.   Most  of the work provides empirical
 examples of the concepts. The contrast in methodologies displayed
 between discussion paper \#3  (W. Bunge, ``Patterns of Location")
 and  this  one  is  marked.   Professor  Bunge  turns  away  from
 probabilistic formulations (see page 3 of ``Patterns of Location")
 and  Professor  Dacey  rejects  deterministic  models (see page 1 
 below).   I  believe  the  relative  worth  of  these  two  broad
 approaches   to   abstract   geography  will  receive  increasing
 attention in the literature.  There is much precedent for concern
 over this question in  other  disciplines.  Clearly Dacey accepts
 the value of a probabilistic approach.

 It may aid the reader if the paper is viewed as consisting of six

 \item{1.}   Professor  Dacey first describes an abstract model of
 imperfections  in  a  uniform plane.  The characteristics of this
 model  are  specified in a general way.  I believe that Professor
 Dacey  is  the  first to suggest models where non-random patterns
 are disturbed by random variables (see Dacey and Tung, 1962).
 \item{2.}   The  point  pattern  which  results  from  the  above 
 mentioned  model  is  to  be  summarized quantitatively in such a
 fashion that it can be compared with some actual geographic point
 pattern.   Professor  Dacey  calls  upon  his  previous extensive
 investigations of nearest neighbor statistics to do this job $^1$. 
 He specifies  how  measures  of the distances to the 1st nearest,
 2nd  nearest,  $\ldots $  kth  nearest  neighbors  of a sample of
 points  in  the  point  pattern may be used to describe the point
 pattern  by  probability  distributions  of  these  lengths.  The
 strategy is to then compare the  probability distributions of the
 model  with  a  geographic  pattern  using  a  simple  ${\chi}^2$
 Professor  Dacey  is  aware  that nearest neighbor methods may be
 used to compare point-to-area relations as well as point-to-point
 relations.  A point pattern  is not  simply a set of points.  The
 points occupy a space for which a  metric is defined.  The metric
 makes  possible  distance  measures between the points.  The fact
 that  there  is  a  space creates the boundary problems mentioned
 in the text. The original purpose of these statistics was to test
 if points were more clustered or more even than random. Imagine a
 study area which is mostly empty but  has  in one small region an
 even distribution of points.  Measuring  distances between points
 and  using  the  nearest  neighbor  test  would  indicate a point
 pattern more even than random.  In one  sense,  however, they are
 clustered for they occupy only a small section of the study area. 
 There is a strategy for this situation.  Use another point set to
 represent  the  area.   This  may  be  done  by  using   an  even
 distribution of points in the area or by assigning  points to the
 area at random. The second set of points now represents the study
 area.  The area has been abstracted into a  point pattern and the
 nearest  neighbor  method may be used.   Measures between the two
 point sets now reveals the original point pattern to be clustered. 
 The  decision  concerning  which  method  to  employ depends upon
 whether  the  phenomenon studied has a  postulated interaction of
 point-to-point or point-to-area. The text indicates the procedure
 for using either method.

 \item{3.}  Theoretical order distances are specified by equations
 (16)  and (17).  The probability functions are made more explicit
 and  operational  by  assuming each lattice point is disturbed by
 the  same  two  dimensional  normal variate.  Professor Dacey has
 ample  evidence  that these  particular probability distributions
 are useful for this purpose.$^2$

 \item{4.}   Solutions of  the  equations  in the previous section
 would  yield  an  analytic  solution  regarding  expected   order
 distances  for  various  disturbance  models.    However,   these
 equations  prove  very  difficult  to  evaluate.   Recourse  to a 
 simulated solution is sought. An {\sl almost periodic disturbance
 model\/} is postulated. Its parameters are estimated from data on
 an actual pattern of urban places in Iowa. Using these parameters,
 a set of points conforming  to  the  structure of the theoretical
 model  is  generated  with  random  digits  and  tables of normal
 deviates.   This  artificial  pattern  is  one  of  many possible
 representations of the theoretical pattern.  It  is  presumed  to
 display the type of pattern expected from an analytic solution if
 one could be found.

 \item{5.}   The  author  now  has two patterns:  one, a simulated
 theoretical pattern which conforms to the structure of the model;
 and the other,  an  actual  urban place pattern in Iowa.  He also
 is able to make the appropriate nearest neighbor  measures  which
 characterize each pattern.  The frequency distributions  are then
 compared using the ${\chi}^2$ statistic.

 \item{6.}  In an addendum, the author presents further testing of
 his  model  by  taking  advantage  of  a  computer  program which
 generates the distance measures required.  The paper ends.

 It  must  be  clear to the reader from the contents of this paper
 that Michael  Dacey has indeed traveled over much ground.  He has
 previously  developed  many  of the results needed in this study. 
 Many  of  his  solutions  and  applications  are   ingenious.  He
 exhibits an understanding of the theoretical implications of  his
 work.  He has a wide knowledge of the  literature on  probability
 and is able to adopt simulation methods  and  computer technology
 to his purpose.  All he lacks is someone to talk to.


 \noindent{\bf Endnotes}

 \noindent 1.  Examples of his statements on nearest neighbor
 measures include:  ``Analysis of Central Place Patterns by
 Nearest Neighbor Method," Seattle, May 1959, mimeographed;
 ``Analysis of Central Place and Point Patterns by a Nearest
 Neighbor Method,"  {\sl Proc. of IGU Symposium in Urban
 Geography\/}, Lund, 1960, pp. 55-75; ``Identification of 
 Randomness in Point Patterns," (with Tze-hsiung Tung), Philadelphia,
 June 1962; mimeographed.  (Dacey and Tung is now forthcoming in
 the {\sl Journal of Regional Science\/}, v. 4.

 \noindent 2.  See references at the end of the paper and also:
 ``Order Neighbor Statistics for a Class of Random Patterns in
 Multidimensional Space," {\sl Annals\/}, Association of 
 American Geographers, v. 53 (Dec. 1963):  505-515, ``Certain
 Properties of Edges on a Polygon in a Two Dimensional
 Aggregate of Polygons Having Randomly Distributed Nuclei,"
 Philadelphia, June 1963, mimeographed. 

 \centerline{\bf Imperfections in the Uniform Plane}
 \centerline{\bf Michael F. Dacey} 
 \centerline{Wharton School of Finance and Commerce}
 \centerline{University of Pennsylvania}
 \centerline{See end of article for additional information}

 A statistical formulation of the spatial  properties  of  central
 place system is proposed. Currently, the theoretical locations of
 central places are specified by geometric or algebraic quantities. 
 This  type  of  statement  leads  to certain rejection of central
 place  models,  for it is inconceivable that any observed pattern
 of central  places corresponds exactly to the specified geometry. 
 A  probabilistic  formulation is preferred for empirical analysis
 because  deviations  from  the  precise  locations  are contained
 within the statement of the model.

 In  the classical theory of Christaller (1933) and L\"osch (1939)
 central  places  form a honeycomb pattern or hexagonal lattice on
 the undifferentiated, unbounded plane.  A probabilistic statement
 of this location pattern incorporates deviations from the precise
 lattice locations, and the deviations  are subject to  stochastic
 processes.  This initial formulation  of a probabilistic  central
 place  distribution  uses  the  concept  of  imperfections in the
 uniform  plane  to define these deviations.  Imperfections may be
 combined with  the central place geometry in many ways.  Here one
 basic  formulation  and  two closely related models are proposed.
 The models  possess  some  properties  of the Christaller-L\"osch
 system  and  evidently  are  not  inconsistent with the spirit of
 central place theory.

 This  report  has  two  purposes.   First,  a  general  model  of 
 imperfections  in  the uniform plane is constructed.  Second, the
 application  of a particular model to a map pattern is evaluated.

 The  map pattern of urban places in Iowa has been selected for an
 initial  examination  of the imperfection concept.  The empirical
 test involves interpretation  of parameters of the model in terms
 of  phenomena  commonly  studied by geographers and estimation of
 these  parameters  from the Iowa map pattern.  Because the formal 
 statement  of  the model contains equations that are difficult to
 evaluate  analytically,  this initial study has used a simulation
 technique  to  obtain  summary  measures on theoretical patterns.
 Properties of a fabricated pattern are compared with the Iowa map
 pattern, and the level  of  agreement  is found acceptable to the
 first approximation.

 \noindent{\bf The Christaller Spatial Model}

 The  theoretical  distribution of central places may be expressed
 in terms of a plane  lattice.  Let $P$ represent a plane symmetry
 lattice.  Choosing  any  arbitrary  point  of  this lattice as an
 origin point  $O$,  the location of any other given lattice point
 can be defined with respect to this origin by a vector $T$ 
 T = u t_1 + v t_2 \eqno(1)
 where $u$ and $v$ are integers.  The vector notation implies that 
 the plane is constructed as a linear lattice having a translation
 period $t_1$ which is repeated periodically at an interval $t_2$.
 The  translation  periods  $t_1$  and  $t_2$  may  be regarded as 
 vectors  separated  by  the  angle  $g$.   Using  $K$ to denote a 
 collection, the lattice points of $P$ are defined by
 P=K T=K(u t_1 +v t_2). \eqno(2)

 Central  place theory conventionally uses a hexagonal lattice for
 which  the  translations  $t_1$  and  $t_2$ are of the same  unit
 length and the angle of periodic rotation is $g=\pi /3$.

 A more general discussion is obtained by not restricting attention
 to the hexagonal lattice.  In this report $P$ represents any plane
 lattice  which  may  have  a  three-, four-, or six-fold axis.  In
 applying  the  lattice  to  a  particular problem, the translation
 periods  $t_1$  and  $t_2$  and  the  angle  of  rotation $g$ need

 \noindent{\bf Types of Imperfections in the Uniform Plane}

 Three  types  of  imperfection in the uniform plane are studied in
 this  report.   These imperfections are closely related to certain
 kinds  of  imperfections  found  in  nearly  perfect crystals.  An
 introduction to crystal imperfections is found in Van Bueren (1961,
 especially Chapters 2-4) and an excellent synthesis of the concept
 of imperfection in the solid state is given by Seitz (1952).   The
 basic  principles  of  our  formulation draw heavily upon concepts
 used   in  the  study  of  crystals  and  the  solid  state;   the
 mathematical formulation is, however, quite different.

 The imperfections under consideration are identified as
 (i) dislocations or disturbances,
 (ii) vacant lattice sites and
 (iii) interstitial points. 
 These three types of  imperfections  are  most  easily defined by
 considering two maps containing point  symbols.  For  the present
 purposes assume the maps have identical area and number of points. 
 One map represents a finite domain of the lattice $P$.  The other
 map, called $S$, may show fabricated  locations  or the positions
 of actual objects.  Figure 1 is ``good" map  $S$  overlaid  on  a
 square $P$.

 \item{i.}  The  term dislocation is more descriptive of the first
 imperfection,  but  it  has a definite meaning in crystallography
 and  solid  state  physics;  so  we  shall call this imperfection
 a disturbance.  A disturbance occurs when the location of a point
 is not exactly at a theoretical lattice site but is `sufficiently'
 close so that with high degree of certainty a  disturbed point is
 correctly associated with its theoretical location.

 \item{ii.}  A vacant lattice site occurs where no point is `close'
 to a theoretical lattice site.  Where two or more points occur in
 the vicinity of a lattice site, it is not called a vacant lattice
 site even though the one  point  correctly  associated with  that
 theoretical location may not be identifiable.

 \item{iii.}   An interstitial  imperfection occurs in the uniform
 plane  where  a  point  is  not identified with any lattice site. 
 Interstitial  locations occur where a point is too distant from a
 theoretical  location  to  be  associated  with  high  degree  of
 certainty with a particular lattice  site,  or where  two or more
 points  are  located  `close' to a lattice site and the one point 
 correctly  assigned  to  that   theoretical   location   is   not

 These  imperfections  are  not  given  precise  definitions.   In
 constructing  the imperfection model more precise definitions are

 \noindent {\bf A Model of the Imperfect Plane}

 One  basic formulation and two modifications are described.   All
 imperfections  under  consideration  are the result of stochastic
 processes,  in  the  space  rather  than  the  more  common  time
 dimension.  The principal feature of an imperfection model is the
 imperfection in pattern related to  disturbances  or  shocks from
 geometrically exact locations (Figure 1).  While this single type
 of  imperfection  is  adequate  for  many physical systems, it is
 probably too restrictive to encompass patterns formed by economic,
 social or cultural systems.  To handle complex map  patterns  two
 additional  types  of two dimensional stochastic  processes  were
 studied.  One type of imperfection generates  interstitial points
 and  is  defined by a two dimensional,  uniform, random variable. 
 The other type of imperfection  generates  clusters of points and
 is  defined  by  spatially  contiguous probability distributions. 
 Because  the  pattern  of  urban  places  in  Iowa  is relatively
 homogeneous  and  contains  no  examples  of  large  metropolitan
 centers, it was not necessary to incorporate a contagious process
 in a model for the Iowa map  pattern.  For  this reason, only the
 first two types of imperfections are discussed in this report.

 \topinsert\vskip 6.5in
 \noindent{\bf Figure 1.}  Map of imperfection model. 
 Most symbols show disturbance effect on a square lattice. 
 There are two vacant lattice sites, and two examples of
 interstitial points.  Most map patterns are, of course,
 not this regular.  This figure shows a six by four square
 lattice which has been altered as suggested.

 \noindent{The Disturbance Effect}

 Each  lattice  point  of  $P$  is  associated  with a  stochastic 
 variable $\xi$. The $\xi$ is the disturbance variable and defines
 the realized location of a point with respect  to its theoretical
 lattice site.  It is convenient  to separate  $\xi$  into its two
 polar  components:   a  distance  $\rho $  and  a  rotation angle
 $\theta $.  So, $\xi \equiv (\rho, \theta)$.

 The  displacement  of  the  point  $s_{ab}$  from its equilibrium
 position  $(at_1 + bt_2)$  is  given  by  the   random   variable
 $\xi _{ab}$.  So, the disturbed position of this point is
 s_{ab} = at_1 +bt_2 + \xi_{ab}.  \eqno(3)
 It  is  assumed  that the  same stochastic variable is associated
 with each lattice site.  Then, if a point is disturbed  from each
 lattice site the collection of randomly disturbed points is
 S_1=K(ut_1 + vt_2 + \xi_{ab}),  \eqno(4)
 $u$  and  $v$  integers.  This notation indicates that $\xi $ has
 translation  period  $t_1$  which  is repeated periodically at an
 interval $t_2$.  In this sense the stochastic variable is carried
 through space and is  associated  in turn with each lattice site. 
 Accordingly, in  point set $S_1$ each lattice site $(at_1 + bt_2)$
 has exactly one corresponding disturbed point $s_{ab}$.

 \noindent {Vacant Lattice Sites}

 It is not necessary to apply a disturbance to each lattice  site. 
 Instead a lattice site and the variable  $\xi_{ab}$ may  be taken
 in conjunction with a binary or on-off operator  which  nullifies
 the  vectors  defining  some  disturbed   points  so   that   the
 corresponding lattice sites are vacant.  As a  consequence, there
 is  a  sparser  network  of  disturbed points than lattice sites. 
 Because a disturbed  point  is  not associated with  each lattice
 site,  the  disturbance  term  is  said  to  be  repeated  almost
 periodically.  A more precise definition of the  almost  periodic
 disturbance is given.

 A  binary operator to produce vacant lattice sites is defined for
 $(at_1 + bt_2)$,  denoted  in  symbols by $\beta_{ab}$, such that
 for $0 \leq \lambda \leq 1$,
 \beta_{ab}=1, \quad \hbox{with probability}\,\,\lambda
 \beta_{ab}=0, \quad \hbox{with probability}\,\,1-\lambda.  \eqno(5)
 The vectors defining location of the disturbed point $s_{ab}$ are
 multiplied  by  $\beta_{ab}$  so  that  the  disturbed  point  is 
 realized  with  probability  $\lambda $  and  is not defined with

 probability  $(1-\lambda)$.   In  more precise form, the location
 of the disturbed point having equilibrium position $(at_1 + bt_2)$
 s_{ab} = \beta_{ab}(at_1 + bt_2 + \xi_{ab}) \eqno(6)
 with the usual convention that $s_{ab}=0$ does not define a point
 at the lattice site 0. So, for $\beta_{ab}=0$ the disturbed point
 $s_{ab}$  does  not  exist,  while for $\beta_{ab}=1$ location is
 found precisely in the manner for the period disturbance.

 Each lattice site is associated with the same stochastic variable
 and with the same binary operator.  Accordingly, the relation (6)
 is  carried  through space with translation period $t_1$ repeated
 periodically  at  interval   $t_2$.   The  collection  of  points
 generated by the almost periodic disturbance is
 S_2=K(\beta_{uv}(ut_1 + vt_2 +\xi_{uv}) \eqno(7)
 $u$  and $v$ integers.  The $S_2$ is completely identified by the 
 underlying  lattice  $P$,  the  probability  $\lambda$,  and  the 
 parameters  specifying  the components $\rho$ and $\theta$ of the
 stochastic variable $\xi$.  It is summarized by the parameter set
 $S(t_1,t_2;\lambda , \xi )$.

 \noindent Uniform Random Disturbance

 This collection of points, denoted by $R$, is a random point set. 
 To make the definition explicit, an arbitrary origin is  selected
 and  the  lattice  point  $O$  of  $P$ is convenient.  The $R$ is
 specified by the theoretical frequency of points within  distance
 $r$  of  the  origin.   Where  the  parameter  $\gamma $  is  the
 expectation that a unit area contains  a  point belonging to $R$,
 p=\pi \gamma r^2 \eqno(8)
 where  $\gamma > 0$.   The  frequency $p$ describes any arbitrary
 disk  of  radius  $r$,  so  that  the   distribution   $\xi $  is
 independent of the specified origin.   It  is a property of  $R$,
 Feller (1957) that the distribution conforms to a Poisson process. 
 The probability of finding exactly $j$ points of $R$  within  any
 disk of radius $r$ is $p^je^{-p}/j!$.

 \noindent  Definition of the Basic Model

 The  model  to  be  considered  in  this report is defined by the
 combination of an $S$ and the $R$ point sets; call this model $M$
 M=S \cup R. \eqno(9)
 This model is summarized by the parameter set $M(t_1,t_2;\lambda ,
 \xi ;  \mu )$,  where  $\mu = (\lambda + \gamma )$.   For a model
 containing $S$ and $R$ points only, $\mu $ is the mean density of
 points per unit area.

 Several  interesting  formulations  of $M$ are defined by special
 values of the parameters $\lambda $ and $\gamma $. 
 The {\sl periodic disturbance model\/} $M_1$ is given by $\lambda
 = 1$, for one disturbed  point  is  associated with each  lattice
 site.  A  {\sl  complete  periodic  disturbance model\/} also has
 $\gamma = 0$, for each point is disturbed from a lattice site and
 there are no random points from $R$.

 The  {\sl almost  periodic disturbance model\/}, called $M_2$, is
 given  by $0<\lambda < 1$.  The magnitude of $\gamma $ determines
 if  $M_2$  has  a  one-to-one correspondence of points to lattice
 sites or if $M_2$ has more or less points than lattice sites.  If
 $\gamma  =  1 - \lambda $  the  theoretical  density  of   points
 belonging to $S_2$ and $R$ equals the density of  lattice  sites. 
 If $\gamma > 1 - \lambda $ the expected number of  points exceeds
 the number of lattice sites, while the expected  number of points
 is less for $\gamma < 1 - \lambda $.

 The  point set given for $\lambda = 0$ is a random point pattern. 
 It  is  of  course  recognized that $R$ is only one of many point
 sets that could be combined with $S_1$ or $S_2$ disturbed points.

 \noindent{\bf Description of Pattern}

 The  disturbance  models are  described by the underlying lattice
 $P$,  the  density  measures  $\lambda $  and  $\gamma $  and the
 disturbance  process $\xi $.  The combination of these parameters
 produce  disturbed  and  interstitial  points  and vacant lattice
 sites  in  the  uniform  plane.   In  a  formal  sense a model is
 completely  specified  by  the lattice parameters and the several
 probability  functions.   This specification of a model does not,
 however, describe  or  summarize  in any useful fashion the point
 pattern generated  by a particular model.  But, numerical summary
 of point pattern  $M$  is  prerequisite to test of the hypothesis
 that  an  observed  map  pattern  is  similar  to an imperfection

 To  measure  the  level  of  correspondence  between observed and
 theoretical patterns there is need for (i) measurements on one or
 more  properties  of  the  observed  pattern and (ii) theoretical
 values  for  the  same  properties  on the pattern defined by the 
 model.   In  addition,  if  parameter  values  for  the model are
 estimated from the observed pattern,  the  properties for test of
 similarity between observed and  theoretical  patterns  should be
 independent  of  the  properties  initially  used   to   estimate

 In  this  report  pattern  is  summarized by two classes of order
 distance  statistics.  The methods are described briefly and then
 their utility as descriptive measures of pattern are indicated.

 \noindent Point to Point Order Distances

 Let $i$ represent any arbitrary point in a point pattern $Q$. The
 measured  map  distance  from  $i$  to  the  $j$ nearest point is
 represented by $R_{ij}$.  $J$ measurements are taken from $i$ and
 are ordered to satisfy the inequalities
 R_{i1} 0
 f(\theta) = (2\pi)^{-1} \qquad 0 < \theta < 2\pi .
 It  seems  appropriate to accept that $f(\xi )$ is identical  for
 each  lattice  site  so  that the parameter $\sigma $ is constant
 throughout  the  lattice  space.   Using  (17) to define (14) and
 substituting the  resulting  probability transformation into (16)
 gives an expression for order statistics that, for me, is totally 

 Some  simplification  is gained by interpreting the normal law by
 the  bivariate or circular normal distribution.  In this case the
 distance  variable  $X$  has  a well known form.  It may be shown
 that the distribution function is
 F(x) = 1/2 \hbox{exp}(-t^2/2\eta^2)
 \int_0^{{(x/\eta)}^2} \,\, e^{-x/2} I_0(tx^{1/2}/\eta)\,\, dx
 \quad x>0 \eqno(18) 
 where  $\eta=2\sigma^2$ and $I_0(\bullet)$ is the modified Bessel
 function  of  the  first  kind of zero order.  This expression is
 recognized as the integral of the non-central ${\chi}^2$ with two
 degrees  of freedom.  In a slightly different form it occurs as a
 basic  distribution  function  in  bombing  or coverage problems,
 Germond  (1950).   By  substituting  (18)  for $F(\omega )$, (16)
 gives the $z$ crude moment of order statistics from a non-central
 ${\chi}^2$  distribution; however, tables of values have not been

 It  is  apparent that even the simplest imperfection model yields
 equations that are difficult to evaluate.  Where $\lambda \neq 1$ 
 and/or  $\gamma \neq 0$  the  equation systems are immensely more
 complex  and  numerical  evaluation  may  be  considered, for any
 practical  purpose  at  this  time,  impossible.   In  order   to
 circumvent these mathematical problems the imperfection model has
 been evaluated by  simulation  of  an equation system for a given
 set of parameter values.

 \noindent{\bf Analysis of the Pattern of Urban Places in Iowa}

 The  imperfection  models  were  designed  to  produce  types  of 
 patterns  and  distributions  studied  in  the  social  sciences. 
 Moreover, the particular class of patterns motivating the present 
 formulation are formed by map representations of urban places. As
 a partial evaluation of the adequacy of the imperfection model to
 replicate  town  and  city  patterns,  the  distribution of urban
 places in Iowa, 1950, is studied.

 Many parameters of the Iowa distribution are already available in 
 Dacey  (1963a).   These  data  provide  empirical   estimates  of 
 parameters for application of the imperfection model  to the Iowa
 pattern. Using estimated parameters, the degree of correspondence
 of $M_2$  with  the observed pattern of urban places is analyzed.
 Simulation is used to evaluate the theoretical imperfection model.

 \noindent{\bf Almost Periodic Disturbance Model}

 The almost periodic disturbance model $M_2$ is specified by three
 sets of parameters:

 $t_1$, $t_2$ and $g$ identify the underlying lattice $P$,

 $\xi $ specifies the disturbance term generating the point set
 $S_2$ and

 $\lambda $ and $\gamma $ are the scale densities for the point
 sets $S_2$ and $R$, respectively.

 These three  sets  of  parameters  are  given numerical values by
 relating the imperfection concept  to structural features of  the
 Iowa  map  pattern.   In  this  construction,  each  parameter is
 described  in  terms  of  the  corresponding property of the Iowa
 pattern.   Since  the  theoretical   pattern   is   synthetically
 fabricated, the definitions and interpretations of parameters are
 biased toward operational statements.

 \noindent Lattice Parameters

 The  $M_2$  is  fabricated  as a rectangular map space containing
 the domain of a square lattice. The domain is of dimensions 12 by
 18 and contains 96 points.  Thus, the parameters are $t_1 = t_2 =
 1$, $g=\pi /2$.

 The  primitive  cells  of  the  square  lattice  have an abstract
 correspondence  to  counties,  and in this context lattice points 
 represent the geographic  center  of  counties.  This lattice has
 some resemblance to the Iowa map.  In gross form Iowa is  roughly
 a rectangle  and  most counties in Iowa are approximately square. 
 However,  the counties do not form a square grid, largely because
 of  surveying  adjustments  for   the   earth's   curvature.   An
 alternative,  and  possibly  a  closer, approximation to the Iowa
 structure is the diamond lattice.

 The  lattice has 96 squares while Iowa has 99 counties.  There is 
 no formal  advantage to using a lattice of approximately the same
 dimensions as the study area.

 For specification of other parameters the following relations are
 established between $M_2$ and the Iowa map:

 \item{i.}   square  lattice  cells of $M_2$ are equated with Iowa

 \item{ii.}  lattice  points  of $M_2$ are equated with geographic
 centers of counties,

 \item{iii.} $S_2$ and $R$ points are equated with urban places.

 Using  this  dictionary ($\alpha $) the distribution function for
 distance from  lattice  site to $S_2$ point is estimated from the 
 observed distances from geographic center of counties to  nearest
 urban place and ($\beta $) the frequency  distribution  of points
 in  primitive  lattice  cells  is  estimated  from  the  observed 
 frequency  distribution  of urban  places in counties.  These two
 properties are evidently independent  of the order distances used
 to summarize observed and theoretical patterns.

 \noindent Disturbance Variables

 In  my  earlier  study  of  Iowa  it  was shown that for interior 
 counties  containing  an  urban  place  the  distance  from   the 
 geographic center to nearest urban place was closely approximated
 by the folded half-normal distribution, as defined for $f(\rho )$
 in (17), with scale parameter  $\sigma = 0.2286$.   Observed  and
 calculated frequency distributions are compared in Table 1.

 The angular component $\theta $ of the disturbance term is  taken
 as a uniform random variable, as defined in (17).  No evidence is
 presented for this assumption, so the uniform variable is entered
 into the model on the theoretical consideration that a completely
 chance  factor  occurs  in the  disturbance process.  However, in
 examining  the location of  places  with  respect  to  geographic
 centers I found no evidence of directional bias.

 On the basis of these estimates, the vector component $\rho $ and
 the angular component $\theta $ of the disturbance variable $\xi $
 are  defined  for  $M_2$  by  the   folded,   uniform   bivariate
 distribution (17).

 \noindent Scale Variables

 The  remaining  two  parameters of $M_2$ are the density measures
 $\lambda $ and $\gamma $.  Because $M_2$ contains only  $S_2$ and
 $R$ points, the density of all points is $\mu =  \lambda + \gamma
 $.  For the Iowa map pattern there are 93 places and 99 counties,
 so the estimated density of total points  in  $M_2$ is $(93/99) =
 \mu $.

 The individual densities $\lambda $ and $\gamma $ were  estimated
 from  the  frequency  distribution  of  urban  places  among Iowa
 counties,  Table 2.  A two parameter probability density function
 that gives a good fit to the observed frequencies has been stated
 by Dacey (1963b).  By assuming that each disturbed point in $S_2$
 is  always  located  in  the  primitive  cell  of its theoretical
 lattice  site  and  that  each  random  point in $R$ has an equal
 probability  of occurring in each primitive cell, the probability
 that a cell contains $x$ points is
 f(x; \lambda , \mu )
 (\gamma^{x+1} e^{-\gamma }/x!)
 + (x\lambda\gamma^{x-1}e^{-\gamma }/x!) \eqno(19)
 where  $\gamma  =  \mu  -  \lambda $  and  $x=0,1,\ldots $.  The 
 parameter $\lambda $ was estimated by the method of moments from
 the distribution of urban places among Iowa  counties.   Table 2
 compares  observed  and  expected frequencies for the parameters
 $\lambda = 0.74$, $\gamma  = 0.20$ and $\mu = 0.94 \cong 93/99$.

 \noindent{\bf Comparison of $M_2$ and Iowa}

 A  synthetic  pattern was constructed from the pattern $M_2$ for
 the parameters
 t_1=t_2=1 \quad g=\pi /2
 \sigma = 0.2286 \quad \lambda = 0.7396 \quad \gamma = 0.1979.
 These  parameters were applied to a  space  containing 96 lattice
 sites, so that $M_2$ contained 71 $S_2$ points and 19 $R$ points.
 Tables of random digits and standard normal deviates were used to
 generate  a  synthetic $M_2$.  Because of the small pattern size,
 random digits and normal deviates were tested for randomness.

 The  $M_2$ and Iowa patterns were described by (i) distances from
 origin points to the 10 nearest neighbors and (ii) distances from
 loci  to  the  10  nearest  points.   The boundary constraint was
 applied  so  that  the  number  of recorded measurements tends to
 decrease as the order of neighbor increases.

 Order  mean  distances  are  listed in Table 3 for point to point
 measurements and in Table 4 for locus to point measurements.  The
 tabulated  data  on  $M_2$  give  mean distances for the 10 lower
 order neighbors and the number of  recorded measurements for each
 order.  Distances  obtained  from  the Iowa map were standardized
 by multiplying  each  observed  mean order distance by the square
 root  of the density of urban places.  The tabulated data on Iowa
 give  the  standardized  mean distances and approximate miles for
 the  10  lower  order neighbors.  Also tabulated are the absolute
 and  percentage  differences  between the observed and calculated
 mean  order  distances.   Many other properties of $M_2$ and Iowa
 were collected but are not included in this report.

 There  are  many reasons for not conducting an elaborate analysis
 for goodness-of-fit of the $M_2$ data to the Iowa data. Important
 reasons  include  the  small  size  of  the  fabricated $M_2$ and 
 difficulty  in  transforming  frequency  distributions  into  the
 normal  form.   These  and  similar  problems  could, largely, be
 handled in a more careful experimental design.  More control  was
 not exercised because  I  wanted  a fast, crude evaluation  of an
 imperfection  model  to  determine  whether  it   possessed   any
 empirical reference, and, hence, merited detailed  consideration. 
 A  fair  test  of  the  imperfection  approach  to  urban systems
 requires a substantially more sophisticated model than $M_2$

 Though  recognizing  the  `imperfections'  in  $M_2$,  it   seems
 sufficiently  provocative  to  justify  release  of  this  highly
 preliminary  report.   While  statistical  methods  were  used to
 evaluate  hypotheses  of  no  difference  between  $M_2$ and Iowa
 (which  were  not  rejected  by  the  available data), reports on 
 levels of significance and other statistical findings do not seem
 particularly critical at this stage of development.

 \noindent{\bf Evaluation}

 The synthetic pattern $M_2$ reproduces with considerable fidelity
 the Iowa map pattern of urban places.  The correspondence between
 $M_2$  and  Iowa  is  a  statistical  rather  than a cartographic
 similarity.  This criterion of similarity  determines the type of
 conclusions that can be drawn from the present study.

 Both  patterns  were summarized by sets of distance measurements. 
 These    distances   represent,    however,    quite    different
 conceptualizations.   The  Iowa  pattern  refers  to  an observed
 distribution  that  exists  in the real world, and at a point  in
 time  a  study  area  has  a  single pattern of urban places.  In 
 contrast, the synthetic pattern  represents a probabilistic model
 that is an abstract construction.  This  model does not  describe
 one map pattern.  Instead, the model defines a set of theoretical
 values.  It is possible to interpret the model and  synthetically
 construct a pattern that is representative of the model; yet, the
 model generates  only  one  of an infinity of  different patterns
 that correspond precisely to the statement of the model.

 In more  formal terms, the reduction of the distribution of urban
 places  to  order  distances  in  a  one-to-one  mapping  but the
 reduction of the  model  to  a  pattern is a one-to-many mapping. 
 So, for the  Iowa  distribution  only  one  pattern  is  formally
 possible  (all  representations  must  be  conformal)  while  the
 mapping  of  the  model  is  multi-valued.  Consequently, while a
 single map describes  the Iowa pattern, there is no  cartographic
 summary of the pattern contained within the theoretical model.

 While  we  reduce  a  map  to a set of numbers we do not return a
 corresponding  set  of  numbers  to  the  map  form.  The cost of 
 reducing  the  Iowa  map  pattern  to  a  system   of   equations
 describing  an  imperfection  model  is  the  loss  of  the   map
 description of that pattern.  Whether this loss is compensated by
 the  substantially  greater  analytical utility of a mathematical
 construction  is  a  question  that each student must resolve for

 In  evaluating  these  questions the role of simulation should be
 correctly  interpreted.   Simulation  was  used  only  after  all
 parameters of the model  were  estimated.  This is not general in
 social science investigations  of large, complex systems by means
 of  simulation.   Often,  the model is simulated many times, each
 run  using a  different set of parameter values.  The model being
 simulated  is then  adjudged successful if some set of parameters
 provides a good fit to the data at hand.  This iterative approach
 is  based  upon  an  a  priori  acceptance of the model.  In this
 application  the simulation is used primarily to study properties 
 of a complex model, but it does not provide any independent means
 of verifying  the model itself.  Simulation was not used for this 
 purpose; for  the  imperfection  concept simulation serves as the
 poor  man's  (mathematically poor, that is) numerical integration
 of a  completely  specified  probabilistic model which can not be
 evaluated by analytic methods.
 \centerline{\bf Table 1}
 \noindent Frequency Distributions of Observed and Calculated
 Standardized Distances, $c_1$, from Geographic Center of
 Interior Counties Containing an Urban Place to Nearest Urban Place
   &Over 2.430 \quad &61 \quad &11.72 \quad &{\bf -1.85}\quad &
      ${{(f_0-f_c)^2}\over {f_0}}$&\cr %sample line
 \+&Distance&Freq. Dist.&&Error&${{(f_0-f_c)^2}\over {f_0}}$ \cr
 \+&$c_1/\sigma $&$f_0$&$f_c$&$f_0-f_c$& \cr
 \+&0-\phantom{1}.243&11          &11.72          
                     &-\phantom{1}.72      &0.471       \cr
 \+&-\phantom{1}.486 &11          &11.04          
                     &-\phantom{1}.04      &0.000       \cr
 \+&-\phantom{1}.729 &11          &\phantom{1}9.82
                     &\phantom{-}1.18      &0.127       \cr
 \+&-\phantom{1}.972 &\phantom{1}8&\phantom{1}8.23
                     &-\phantom{1}.23      &0.005       \cr
 \+&-1.215           &\phantom{1}6&\phantom{1}6.51
                     &-\phantom{1}.51      &0.237       \cr
 \+&-1.458           &\phantom{1}3&\phantom{1}4.85
                     &{\sl -1.85}          &{\sl 0.052} \cr
 \+&-1.701           &\phantom{1}5&\phantom{1}3.39
                     &{\sl \phantom{-}1.61}&            \cr
 \+&-1.944           &\phantom{1}2&\phantom{1}2.28
                     &{\bf -\phantom{1}.28}&{\bf 0.265} \cr
 \+&-2.187           &\phantom{1}2&\phantom{1}1.41
                     &{\bf \phantom{-1}.59}&            \cr
 \+&-2.430           &\phantom{1}2&\phantom{11}.83
                     &{\bf \phantom{-}1.17}&            \cr
 \+&Over 2.430       &\phantom{1}0&\phantom{11}.92
                     &{\bf -\phantom{1}.92}&            \cr
 \+&Total&61&61&&1.157 $(\equiv {\chi}^2)$ \cr
 \+&&&&&df=4 \cr
 \+&&&&$.90>Pr({\chi}^2 = 1.157)>.75$& \cr

 \noindent Iowa data, $f_0$ from Dacey (1963a).  The standard
 deviation is $\sigma =0.2286$.  The calculated frequency,
 $f_c$, is from the unit half-normal distribution.

 \centerline{\bf Table 2}
 \noindent Comparison of Observed Distribution of Urban Places
 per County in Iowa, 1950, with Expected Distribution
 of Points per Primitive cell of $M_2$
   &Number of \quad &Frequency \quad &Distributions\quad &\cr %sample line
 \+&Number of&Frequency Distributions& \cr
 \+&$x$&$g(x)$&$E(x)$ \cr
 \+&0       &21          &21.1           \cr
 \+&1       &64          &64.2           \cr
 \+&2       &13          &12.4           \cr
 \+&3       &\phantom{1}1&\phantom{1}1.2 \cr
 \+&$\geq 4$&\phantom{1}0&\phantom{11}.1 \cr

 \noindent Observed values are from Dacey (1963a).  Expected
 values are computed from (20) with $\lambda = .74$ and
 $\gamma = .2$.

 \centerline{\bf Table 3}
 \noindent Comparison of $j$ Order Distances for $M_2$ and Iowa Maps
   &Order \quad &65 \quad &1.74 \quad &${d_0}^{1/2}{\bar R}_j$\quad &
       Mi. \quad &${\tilde r}_j-{d_0}^{1/2}{\bar R}_j$\quad 
      &As \% of Iowa &\cr %sample line
 \+&Order&\quad $M_2$&&\quad Iowa&&\quad Error&&\cr
 \+&$j$&$n_j$&${\tilde r}_j$&${d_0}^{1/2}{\bar R}_j$
 &Mi.&${\tilde r}_j-{d_0}^{1/2}{\bar R}_j$&As \% of Iowa&\cr
 \+&\phantom{1}1&65&0.63&0.66&16&          -.03&4.7 \cr
 \+&\phantom{1}2&58&0.84&0.84&21&\phantom{-}.00&    \cr
 \+&\phantom{1}3&56&0.98&0.99&25&          -.01&1.4 \cr
 \+&\phantom{1}4&55&1.12&1.12&28&\phantom{-}.00&    \cr
 \+&\phantom{1}5&53&1.24&1.24&31&\phantom{-}.00&    \cr
 \+&\phantom{1}6&46&1.35&1.36&34&          -.01&1.0 \cr
 \+&\phantom{1}7&44&1.46&1.49&37&          -.03&2.1 \cr
 \+&\phantom{1}8&41&1.54&1.60&40&          -.06&4.0 \cr
 \+&\phantom{1}9&37&1.65&1.68&42&          -.03&2.0 \cr
 \+&10          &36&1.74&1.78&44&          -.04&2.0 \cr

 \noindent Iowa data are from Dacey (1963a).  

 \centerline{\bf Table 4}
 \noindent Comparison of $h$ Order Distances for $M_2$ and Iowa Maps
   &Order \quad &65 \quad &1.74 \quad &${d_0}^{1/2}{\bar R}_j$\quad &
       Mi.\quad &${\tilde r}_j-{d_0}^{1/2}{\bar R}_j$\quad 
      &As \% of Iowa &\cr %sample line
 \+&Order&\quad $M_2$&&\quad Iowa&&\quad Error&&\cr
 \+&$h$&$n_j$&${\tilde r}_h$&${d_0}^{1/2}{\bar R}_h$
 &Mi.&${\tilde r}_h-{d_0}^{1/2}{\bar R}_h$&As \% of Iowa&\cr
 \+&\phantom{1}1&40&0.42&0.41&10&\phantom{-}.01&4.7 \cr
 \+&\phantom{1}2&36&0.72&0.72&18&\phantom{-}.00&    \cr
 \+&\phantom{1}3&32&0.97&0.93&23&\phantom{-}.04&4.2 \cr
 \+&\phantom{1}4&31&1.07&1.13&28&          -.06&4.8 \cr    
 \+&\phantom{1}5&29&1.21&1.26&31&          -.05&4.0 \cr
 \+&\phantom{1}6&28&1.32&1.39&35&          -.07&4.8 \cr
 \+&\phantom{1}7&28&1.43&1.45&36&          -.02&1.8 \cr
 \+&\phantom{1}8&27&1.55&1.56&39&          -.01&0.8 \cr
 \+&\phantom{1}9&22&1.62&1.65&41&          -.03&1.6 \cr
 \+&10          &20&1.71&1.74&43&          -.03&1.9 \cr

 \noindent Iowa data are from Dacey (1963a).  

 October 14, 1963
 Philadelphia, Pennsylvania

 This original paper by Dacey, when printed  in  the {\sl Papers\/}
 of the Michigan In\-ter - Uni\-ver\-sity Community of Mathematical
 Geographers,  was  supplemented  with  an  `Addendum'   reflecting
 computer programs current at the time by Professor Duane F. Marble
 and Mr. Marvin Tener, and a second examination of the Iowa data by
 Dacey (December 13, 1963).  A Glossary by Nystuen offered expanded
 explanations of  complicated  material  for  readers uncomfortable
 with notation.  The added materials are not reprinted here.

 \noindent * 
     The support of the Regional Science Research Institute and
 of the National Science Foundation is gratefully acknowledged.

 \noindent **  Current address:  
       Department of Geography
       Northwestern University
       Evanston, IL

 \noindent {\bf References}

 \ref  Christaller, W.  1933.  {\sl Die zentralen Orte in
 S\"uddeutschland\/}.  Jena:  Fischer.

 \ref  Dacey, M. F.  1963a.  {\sl Iowa:  The Classic Plane or
 Croupier's Table\/}.  Mimeographed.

 \ref Dacey, M. F.  1963b.  {\sl A Poisson-Type Distribution for
 Dispersed Population\/}.  Mimeo.

 \ref Dacey, M. F.  1963c.  ``The Status of Pattern Analysis:
 Identification of Problems in the Statistical Analysis of
 Spatial Arrangement,"  paper presented at the Regional Science
 Association meetings, Chicago, 1963.

 \ref  Dalenius, T., J. Hajeck, and S. Zubrzycky.  1961.  On Plane
 Sampling and Related Geometrical Problems.  {\sl Proceedings
 of the Fourth Berkeley Symposium on Mathematical Statistics
 and Probability\/}, 1.  Berkeley:  University of California,

 \ref Feller, W.  1957.  {\sl An Intorduction to Probability
 Theory and its Applications, Vol. I\/}.  New York:  Wiley.
 Second edition.

 \ref Germond, H. H. 1950.  {\sl The Circular Coverage Function\/}.
 Santa Monica:  RAND, Memorandum 330.

 \ref L\"osch, A.  1939.  {\sl Die Raumliche Ordnung der
 Wirtschaft\/}.  Jena:  Fischer.  (Translated by W. H. Woglom
 and W. F. Stolper as the {\sl Economics of Location\/},
 New Haven:  Yale University Press, 1954.)

 \ref Seitz, F.  1952.  ``Imperfections in Nearly Perfect
 Crystals:  A Synthesis," {\sl Imperfections in Nearly Perfect
 Crystals\/} (W. Schockley, J. H. Hollomon, R. Maurer, F. Seitz,
 eds.) New York:  Wiley.

 \ref  Van Bueren, H. G.  1961.  {\sl Imperfections in Crystals\/}.
 New York:  Interscience.

 \ref Zubrzycki, S.  1961.  ``Concerning Plane Sampling."
 {\sl Second Hungarian Mathematical Cong\/}.  Budapest:
 Akademiai Kiado.


 Readers of {\sl Solstice\/} might also be interested to note the
 following additional references to Dacey's work, not noted in the
 MICMOG publication.

 \ref Dacey, M. F.  1960.  A note on the derivation of nearest
 neighbor distances, {\sl Journal of Regional Science\/}, 2, 81-87.

 \ref Dacey, M. F.  1960.  The spacing of river towns, {\sl Annals\/}
 Association of American Geographers, 50, 59-61.

 \ref Dacey, M. F. 1962.  Analysis of central place and point pattern
 by a nearest neighbor method, {\sl Lund Studies in Geography\/}
 24, 55-75.

 \ref Dacey, M. F.  1963.  Order neighbor statistics for a class of
 random patterns in multidimensional space, {\sl Annals\/}
 Association of American Geographers, 53, 505-515.

 \ref Dacey, M. F.  1963.  Certain properties of edges on a 
 polygon in a two dimensional aggregate of polygons having 
 randomly distributed nuclei.  Mimeo.

 \ref Dacey, M. F. 1964.  Two-dimensional random point patterns:
 A review and an interpretation, {\sl Papers\/}, Regional
 Science Association, 13, 41-55.

 \ref Dacey, M. F.  1964.  Modified Poisson probability law for 
 point pattern more regular than random, {\sl Annals\/}
 Association of American Geographers, 54, 559-565.

 \ref Dacey, M. F.  1965.  Order distance in an unhomogeneour
 random point pattern, {\sl The Canadian Geographer\/}, 9, 144-153.

 \ref Dacey, M. F.  1966.  A compound probability law for a 
 pattern more dispersed than random and with areal inhomogeneity,
 {\sl Economic Geography\/}, 42, 172-179.

 \ref Dacey, M. F.  1966.  A county seat model for the areal
 pattern of an urban system, {\sl Geographical Review\/},
 56, 527-542.

 \ref Dacey, M. F.  1966.  A probability model for central
 place location, {\sl Annals\/}, Association of American
 Geographers, 56, 550-568.

 \ref Dacey, M. F.  1967.  Description of line patterns,
 {\sl Northwestern Studies in Geography\/}, 13, 277-287.

 \ref Dacey, M. F.  1968.  An empirical study of the areal
 distribution of houses in Puerto Rico, {\sl Transactions\/},
 Institute of British Geographers, 45, 15-30.

 \ref Dacey, M. F.  1969.  Proportion of reflexive n-th order
 neighbors in spatial distributions, {\sl Geographical Analysis\/},
 1, 385-388.

 \ref Dacey, M. F.  1969.  A hypergeometric family of discrete
 probability distributions:  Properties and applications to
 location models, {\sl Geographical Analysis\/}, 1, 283-317.

 \ref Dacey, M. F. 1969.  Some properties of a cluster point process,
 {\sl Canadian Geographer\/}, 13, 128-140.

 \ref Dacey, M. F.  Similarities in the areal distributions of houses
 in Japan and Puerto Rico, {\sl Area\/}, 3, 35-37.

 \ref Dacey, M. F.  1973.  A central focus cluster process for
 urban dispersion, {\sl Journal of Regional Science\/}, 13, 77-90.
 \centerline{\bf 5. FEATURES}

 \centerline{\bf Construction Zone:
               The Braikenridge-MacLaurin Construction}
      The projective  plane  is  often thought of as the Euclidean
 plane with a line of infinity attached.   The line at infinity is
 composed of the infinity of points at infinity, each of which can
 be  viewed  as the intersection point for sets of parallel lines.
 Such generality can offer enlightenment. 

     The Braikenridge-MacLaurin construction (Coxeter 1974) offers
 a strategy  for constructing a conic through five given points in
 the   projective   plane.    Imaginary  lights  suggest  how  the
 construction  traces out  the locus of  a conic in the projective

      Given  five  points,  $A$,  $B$, $C$, $A'$, $B'$ (Figure 1). 
 Represent each  of  these by a relatively large white light bulb. 
 Join $A$ to  $B'$  and $A'$ to  $B$ by lighting, one at a time, a
 series of small  white  light bulbs from $A$ to $B'$ and from $B$
 to $A'$.   Designate  the  intersection point of these two lines,
 $N$, by a  white  bulb larger than those along the lines, but not
 quite  as  large  as  those  representing  the five given points.
 Choose  an  arbitrary  line,  $z_1$,  through  $N$; draw it using
 a sequence  of  small  red lights.  Join $A'$ to $C$ by a line of
 small red lights.  Label the intersection $M$, of $A'C$ and $z_1$,
 with a medium-sized red light.  Join $B'$ to $C$ by a line of red
 lights.   Label  the  intersection $L$  of $B'C$ and $z_1$ with a
 medium-sized  red  light.  Join $A$ to $M$ by a line of small red
 lights and  join $B$ to $L$ by a line of small red lights.  Label
 the intersection $C_1'$ of $AM$ and $BL$ with a medium red light.
 The point $C_1'$ lies on the conic. 

 \midinsert\vskip 4.5in
 Figure 1.  Braikenridge-MacLaurin Construction of a conic through
 five given points, $A$, $B$, $C$, $A'$, and $B'$ in the projective

 Now turn off all red lights except the  one  representing $C_1'$. 
 Draw,  using  a  sequence  of  small  green  lights, a line $z_2$
 (different  from  $z_1$), through $N$.  Repeat this construction,
 using green lights, producing in the end  another  point, $C_2'$,
 on the conic.  Leave the green light representing  $C_2'$  on and
 turn the others  (green ones)  off.  Repeat  this  process  using
 enough (three)  different  colors (a  ``Four-color Theorem"  type  
 of idea) to  trace out the locus of the conic in lights!
 \centerline{\bf Happy Holidays!}
 \noindent {\bf Reference}
 \ref Coxeter, H. S. M.  1974. {\sl Projective Geometry\/}, 2nd Ed.
 University of Toronto Press, Toronto.

 \centerline{\bf Population Environment Dynamics Course and Monograph}

 Once again, {\sl Solstice\/} board member William D. Drake invited
 S. Arlinghaus  to  co-teach  a  course  in  Population Environment
 dynamics based  on  Drake's  ideas  of transition theory.  For the
 third  consecutive  year their efforts, together with those of the
 many fine students,  have  resulted  in  an interesting monograph,
 authored almost totally by  the students.  The student authors and
 content of {\sl Population - Environment Dynamics:  Towards Public
 Policy Strategies\/} are as listed below:

 \noindent Deborah Carr, 
 Stability in Rural Communities:  Myth or Reality?

 \noindent Cheri DeLaRosia,
 Population-Environment Trends in the Modernization of Thailand;

 \noindent Rohinton Emmanuele,
 A City in Transition:  Urban Demographic Changes in Detroit
 and Their Impact on Urban Greenness and Climate;

 \noindent Noah Hall,
 Coastal Protection and the Coastal Population-Environment

 \noindent Timothy Macdonald,
 NAFTA and the Human Element, A Region in Transition;

 \noindent Soonae Park,
 Demographic Transition and Economic Growth in Korea:
 Comparison between Asian Countries;

 \noindent Carlos de la Parra,
 Analysis of Transitions in the U.S.-Mexico Border;

 \noindent Brent Plater,
 Population Policy and Environmental Quality;

 \noindent Shelley Price,
 A Framework of Pollution Prevention and Life-Cycle Design:
 Aiding Developing Nations through Transition to Industrialization;

 \noindent Richard Wallace,
 Motor Vehicle Transport and Global Climate Change:
 Policy Scenarios;

 \noindent Tracy Yoder,
 An Inquiry into Determinates of Fertility.



 \centerline{\bf  BACK ISSUES OF {\sl SOLSTICE\/} ON A GOPHER}

 \noindent {\sl Solstice\/} is available on a GOPHER from the
 Department of Mathematics at Arizona State University:
      PI.LA.ASU.EDU port 70 

 \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  IEVG;  type
 cd IEVG 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 

 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.  They should run, possibly
 with a few harmless ``vboxes" over or under.
  \# 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 7.  SOLSTICE--INDEX, VOLUMES I, II, III, IV}
 \noindent{\bf Volume V, Number 1, Summer, 1994}
 \noindent {\bf 1.}  Welcome to New Readers and Thank You.
 \noindent {\bf 2.} Press clippings, summary.
 \noindent {\bf 3.}  Reprints
 Getting Infrastructure Built
 Virginia Ainslie and Jack Licate
 Cleveland Infrastructure Team Shares Secrets of Success;
 What Difference Has the Partnership Approach Made?
 How Process Affects Products --- Moving Projects Faster
 Means Getting More Public Investment;  How Can Local
 Communities Translate These Successes to Their Own Settings?
 Center Here; Center There; Center, Center Everywhere
 Frank E. Barmore
 Abstract; Introduction; Definition of Geographic Center;
 Geographic Center of a Curved Surface; Geographic Center of 
 Wisconsin; Geographic Centern of the Conterminous United States;
 Geographic center of the United States; Summary and Recommendations;
 Appendix A:  Calculation of Wisconsin's Geographic Center;
 Appendix B:  Calculation of the Geographical Center of the
 Conterminous United States; References.
 \noindent {\bf 4.}  Articles.
 Equal-Area Venn Diagrams of Two Circles:  Their Use with Real-World
 Barton R. Burkhalter
 General Problem; Definition of the Two-Circle Problem; Analytic
 Strategy; Derivation of $B\%$ and $AB\%$ as a Function of
 $r_{B}$ and $d_{AB}$.
 Los Angeles, 1994 --- A Spatial Scientific Study
 Sandra L. Arlinghaus, William C. Arlinghaus, Frank Harary, and
 John D. Nystuen.
 Los Angeles, 1994; Policy Implications; References.
 Tables and Complicated Figures.
 \noindent {\bf 5.}  Downloading of Solstice
 \noindent {\bf 6.}  Index to Volumes I (1990), II (1991), III (1992),
 and IV (1993) of Solstice.
 \noindent {\bf 7.}  Other Publications of IMaGe
 \noindent{\bf Volume IV, Number 2, Winter, 1993}
 \noindent {\bf 1.}  Welcome to New Readers and Thank You Notes.
 \noindent {\bf 2.} Press clippings, summary.
 \noindent {\bf 3.}  Article
 Villages in Transition:  Elevated Risk of Micronutrient Deficiency.
 William D. Drake, S. Pak, I. Tarwotjo, Muhilal, J. Gorstein,
 R. Tilden.
 Abstract; Moving from Traditional to Modern Village Life:  Risks
 during Transition; Testing for Elevated Risks in Transition Villages;
 Testing for Risk Overlap within the Health Sector; Conclusions and
 Policy Implications.
 \noindent {\bf 4.}  Downloading of Solstice
 \noindent {\bf 5.}  Index to Volumes I (1990), II (1991), III (1992),
 and IV.1 (1993) of Solstice.
 \noindent {\bf 6.}  Other Publications of IMaGe
 \noindent {\bf 7.}  Selected recent publications of interest
 involving Solstice Board members, and some goings on about Ann Arbor.
 \noindent{\bf Volume IV, Number 1, Summer, 1993}
 \noindent {\bf 1.}  Welcome to New Readers.
 \noindent {\bf 2.} Press clippings, summary.
 \noindent {\bf 3.}  Goings on about Ann Arbor--ESRI and IMaGe Gift
 \noindent {\bf 4.}  Articles

 Electronic Journals:  Observations Based on Actual Trials,
 1987-Present, by Sandra L. Arlinghaus and Richard H. Zander.

     Abstract; Content issues; Production issues; Archival issues;

 Wilderness As Place, by John D. Nystuen.

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

 The Earth Isn't Flat.  And It Isn't Round Either:  Some Significant
     and Little Known Effects of the Earth's Ellipsoidal Shape,
     by Frank E. Barmore.
 reprinted from the {\sl Wisconsin Geographer\/}.

     Abstract; Introduction; The Qibla problem; The geographic
     center; The center of population; Appendix; References.

 Microcell Hex-nets? by Sandra L. Arlinghaus

     Introduction; Lattices; Microcell hex-nets; References.

 Sum Graphs and Geographic Information, by Sandra L. Arlinghaus,
     William C. Arlinghaus, Frank Harary.

     Abstract; Sum graphs; Sum graph unification:  construction;
     Cartographic application of sum graph unification; Sum graph
     unification:  theory; Logarithmic sum graphs; Reversed sum
     graphs; Augmented reversed logarithmic sum graphs; Cartographic
     application of ARL sum graphs; Summary

 \noindent{\bf 5.}  Downloading of {\sl Solstice\/}. 

 \noindent{\bf 6.} Index.

 \noindent{\bf 7.}  Other publications of IMaGe.
 \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 - Praximetrie\/}. 
 \smallskip \noindent {\sl
 1.  What Are They?  2.  Two Worlds:  Abstract and Empirical
 3.  Two Worlds:  Two Levels  4.  Two Levels:  Derivation and
 Selection  5.  Research Schema  6.  Sketches of Discovery
 7.  What Should They Be?

 \noindent {\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 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 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 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 entries 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.
 \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}

 \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.



 \centerline{\bf 8.  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 Gryphon Publishing

 1.  Sandra L. Arlinghaus and John D. Nystuen.  Mathematical
 Geography and Global Art:  the Mathematics of  David Barr's
 ``Four Corners Project,'' 1986. 
 2.  Sandra L. Arlinghaus.  Down the Mail Tubes:  the Pressured
 Postal Era, 1853-1984, 1986. 
 3.  Sandra L. Arlinghaus.   Essays on Mathematical Geography,

 4.  Robert F. Austin, A Historical Gazetteer of Southeast Asia,
 5.  Sandra L. Arlinghaus, Essays on Mathematical Geography--II,
 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.) 
 7.  Keith J. Tinkler, Editor, Nystuen---Dacey Nodal Analysis,

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

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

 10.  Daniel A. Griffith, Simulating $K=3$ Christaller Central
 Place Structures:  An Algorithm Using A Constant Elasticity of
 Substitution Consumption Function, 1989. 
 11.  Sandra L. Arlinghaus and John D. Nystuen,
      Environmental Effects on Bus Durability, 1990.  

 12.  Daniel A. Griffith, Editor.
 Spatial Statistics:  Past, Present, and Future,  1990. 
 13.  Sandra L. Arlinghaus, Editor.  Solstice --- I,  1990. 
 14.  Sandra L. Arlinghaus, Essays on Mathematical Geography
 --- III, 1991.
 15.  Sandra L. Arlinghaus, Editor, Solstice --- II, 1991.
 16.  Sandra L. Arlinghaus, Editor, Solstice --- III, 1992.

 17.  Sandra L. Arlinghaus, Editor, Solstice --- IV, 1993.
 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.
 1.  Allen K. Philbrick.  This Human World.
 2.  John F. Kolars and John D. Nystuen.  Human Geography.