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%Arizona State University Department of Mathematics
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\centerline{\big SOLSTICE:}
\vskip.5cm
\centerline{\bf AN ELECTRONIC JOURNAL OF GEOGRAPHY AND MATHEMATICS}
\vskip5cm
\centerline{\bf WINTER, 1994}
\vskip12cm
\centerline{\bf Volume V, Number 2}
\centerline{\bf Institute of Mathematical Geography}
\vskip.1cm
\centerline{\bf Ann Arbor, Michigan}
\vfill\eject
%----------------------------------------------------------------
%----------------------------------------------------------------
\hrule
\smallskip
\centerline{\bf SOLSTICE}
\line{Founding Editor--in--Chief:
{\bf Sandra Lach Arlinghaus} \hfil}
\line{Institute of Mathematical Geography and University of Michigan \hfil}
\smallskip
{\bf GOPHER: on Arizona State University Department
of Mathematics gopher}
\smallskip
\centerline{\bf EDITORIAL BOARD}
\smallskip
\line{{\bf Geography} \hfil}
\line{{\bf Michael Goodchild},
University of California, Santa Barbara. \hfil}
\line{{\bf Daniel A. Griffith},
Syracuse University. \hfil}
\line{{\bf Jonathan D. Mayer},
University of Washington;
joint appointment in School of Medicine.\hfil}
\line{{\bf John D. Nystuen},
University of Michigan.\hfil}
\smallskip
\line{{\bf Mathematics} \hfil}
\line{{\bf William C. Arlinghaus},
Lawrence Technological University. \hfil}
\line{{\bf Neal Brand},
University of North Texas. \hfil}
\line{{\bf Kenneth H. Rosen},
A. T. \& T. Bell Laboratories. \hfil}
\smallskip
\line{{\bf Engineering Applications} \hfil}
\line{{\bf William D. Drake},
University of Michigan, \hfil}
\smallskip
\line{{\bf Education} \hfil}
\line{{\bf Frederick L. Goodman},
University of Michigan, \hfil}
\smallskip
\line{{\bf Business} \hfil}
\line{{\bf Robert F. Austin, Ph.D.} \hfil}
\line{President, Austin Communications Education Services \hfil}
\smallskip
\hrule
\smallskip
The purpose of {\sl Solstice\/} is to promote interaction
between geography and mathematics. Articles in which elements
of one discipline are used to shed light on the other are
particularly sought. Also welcome, are original contributions
that are purely geographical or purely mathematical. These may
be prefaced (by editor or author) with commentary suggesting
directions that might lead toward the desired interaction.
Individuals wishing to submit articles, either short or full--
length, as well as contributions for regular features, should
send them, in triplicate, directly to the Editor--in--Chief.
Contributed articles will be refereed by geographers and/or
mathematicians. Invited articles will be screened by suitable
members of the editorial board. IMaGe is open to having authors
suggest, and furnish material for, new regular features.
The opinions expressed are those of the authors, alone, and the
authors alone are responsible for the accuracy of the facts in
the articles.
\smallskip
\noindent {\bf Send all correspondence to:}
Sandra Arlinghaus, Institute of Mathematical Geography,
2790 Briarcliff, Ann Arbor MI 48105.
sarhaus@umich.edu
\smallskip
Suggested form for citation. If standard referencing to the
hardcopy in the IMaGe Monograph Series is not used (although we
suggest that reference to that hardcopy be included along with
reference to the e-mailed copy from which the hard copy is
produced), then we suggest the following format for citation of
the electronic copy. Article, author, publisher (IMaGe) -- all
the usual--plus a notation as to the time marked electronically,
by the process of transmission, at the top of the recipients
copy. Note when it was sent from Ann Arbor (date and time to
the second) and when you received it (date and time to the
second) and the field characters covered by the article (for
example FC=21345 to FC=37462).
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.
\smallskip
{\nn Upon final acceptance, authors will work with IMaGe
to get manuscripts into a format well--suited to the
requirements of {\sl Solstice\/}. Typically, this would mean
that authors would submit a clean ASCII file of the
manuscript, as well as hard copy, figures, and so forth (in
camera--ready form). Depending on the nature of the document
and on the changing technology used to produce {\sl
Solstice\/}, there may be other requirements as well.
Currently, the text is typeset using {\TeX}; in that way,
mathematical formul{\ae} can be transmitted as ASCII files and
downloaded faithfully and printed out. The reader
inexperienced in the use of {\TeX} should note that this is
not a ``what--you--see--is--what--you--get" display; however,
we hope that such readers find {\TeX} easier to learn after
exposure to {\sl Solstice\/}'s e-files written using {\TeX}!}
{\nn Copyright will be taken out in the name of the
Institute of Mathematical Geography, and authors are required to
transfer copyright to IMaGe as a condition of publication.
There are no page charges; authors will be given permission to
make reprints from the electronic file, or to have IMaGe make a
single master reprint for a nominal fee dependent on manuscript
length. Hard copy of {\sl Solstice\/} is available at a cost
of \$15.95 per year (plus shipping and handling; hard copy is
issued once yearly, in the Monograph series of the Institute of
Mathematical Geography. Order directly from IMaGe. It is the
desire of IMaGe to offer electronic copies to interested parties
for free. Whether or not it will be feasible to continue
distributing complimentary electronic files remains to be seen.
Presently {\sl Solstice\/} is funded by IMaGe and by a generous
donation of computer time from a member of the Editorial Board.
Thank you for participating in this project focusing on
environmentally-sensitive publishing.}
\vskip.5cm
\copyright Copyright, December, 1994 by the
Institute of Mathematical Geography.
All rights reserved.
\vskip1cm
{\bf ISBN: 1-877751-56-1}
{\bf ISSN: 1059-5325}
\vfill\eject
%----------------------------------------------------------------
%----------------------------------------------------------------
\centerline{\bf TABLE OF CONTENT}
\smallskip
\noindent{\bf 1. WELCOME TO NEW READERS AND THANK YOU}
\smallskip
\noindent{\bf 2. PRESS CLIPPINGS---SUMMARY}
\smallskip
\noindent{\bf 3. ARTICLES}
\smallskip
\noindent{\bf The Paris Metro: Is its Graph Planar? }
\smallskip
\noindent{\bf Sandra L. Arlinghaus,
William C. Arlinghaus,
Frank Harary}
\smallskip
Transmitted as part 2 of 9.
\smallskip
Planar graphs;
The Paris Metro;
Planarity and the Metro;
Significance of lack of planarity.
\smallskip
\noindent{\bf Interruption! }
\smallskip
\noindent{\bf Sandra Lach Arlinghaus}
\smallskip
Transmitted as part 3 of 9.
\smallskip
Classical interruption in mapping;
Abstracts variants on interruption and mapping;
The utility of considering various mapping surfaces--GIS;
Future directions.
\smallskip
\noindent{\bf 4. REPRINT}
\smallskip
\noindent {\bf Imperfections in the Uniform Plane}.
\smallskip
\noindent{\bf Michael F. Dacey}
\smallskip
\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.
\smallskip
Reprinted from {\sl Michigan Inter-university Community of
Mathematical Geographers\/}, Papers, John D. Nystuen, Editor.
Reprinted here with permission.
\smallskip
Original (1964) Nystuen Foreword;
Current (1994) Nystuen Foreword;
Article:
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;
Evaluation;
Tables.
\smallskip
\noindent{\bf 5. FEATURES}
\smallskip
\noindent{\bf Construction Zone:
The Braikenridge-MacLaurin Construction}
\smallskip
Transmitted as part 8 of 9.
\smallskip
\noindent{\bf Population Environment Dynamics: Course and Monograph}
\smallskip
\noindent{\bf William D. Drake}
\smallskip
Transmitted as part 8 of 9.
\smallskip
\noindent{\bf 6. DOWNLOADING OF SOLSTICE}
\smallskip
\noindent{\bf 7. INDEX to Volumes I (1990), II (1991),
III (1992), IV (1993) and V (1994, part 1) of {\sl Solstice}.}
\smallskip
\noindent{\bf 8. OTHER PUBLICATIONS OF IMaGe }
All transmitted as part 9 of 9.
\vfill\eject
%----------------------------------------------------------------
%----------------------------------------------------------------
\centerline{\bf 1. WELCOME TO NEW READERS AND THANK YOU}
Welcome to new subscribers! We hope you enjoy participating
in this means of journal distribution. Instructions for
downloading the typesetting have been repeated in this issue,
near the end. They are specific to the {\TeX} installation at
The University of Michigan, but apparently they have been helpful
in suggesting to others the sorts of commands that might be used
on their own particular mainframe installation of {\TeX}. New
subscribers might wish to note that the electronic files are
typeset files---the mathematical notation will print out as
typeset notation. For example,
$$
\Sigma_{i=1}^n
$$
when properly downloaded, will print out a typeset summation as
$i$ goes from one to $n$, as a centered display on the page.
Complex notation is no barrier to this form of journal
production.
\vskip.5cm
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\/}.
\smallskip
%---------------------------------------------------------------
%---------------------------------------------------------------
\centerline{\bf 2. PRESS CLIPPINGS---SUMMARY}
\noindent
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.
\smallskip
\noindent 2. {\bf Science News}, ``Math for all seasons"
by Ivars Peterson, January 25, 1992, Vol. 141, No. 4.
\smallskip
\noindent 3. {\bf Newsletter of the Association of American
Geographers}, June, 1992.
\smallskip
\noindent 4. {\bf American Mathematical Monthly},
``Telegraphic Reviews" --- mentioned as
``one of the World's first electronic journals using {\TeX},"
September, 1992.
\smallskip
\noindent 5. {\bf Harvard Technology Window}, 1993.
\smallskip
\noindent 6. {\bf Graduating Engineering Magazine}, 1993.
\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.
\vfill\eject
Publications of the Institute of Mathematical Geography have,
in addition, been reviewed or noted in
\smallskip
1. {\sl The Professional Geographer\/} published
by the Association of American Geographers;
\smallskip
2. The {\sl Urban Specialty Group Newsletter\/}
of the Association of American Geographers;
\smallskip
3. {\sl Mathematical Reviews\/} published by the
American Mathematical Society;
\smallskip
4. {\sl The American Mathematical Monthly\/} published
by the Mathematical Association of America;
\smallskip
5. {\sl Zentralblatt\/} fur Mathematik, Springer-Verlag, Berlin
\smallskip
6. {\sl Mathematics Magazine\/}, published by the Mathematical
Association of America.
\smallskip
7. {\sl Newsletter\/} of the Association of American Geographer.
\smallskip
8. {\sl Journal of The Regional Science Association\/}.
\smallskip
9. {\sl Journal of the American Statistical Association\/}.
\smallskip
\vfill\eject
\centerline{\bf 3. ARTICLES}
\smallskip
\centerline{\bf The Paris Metro: Is Its Graph Planar?}
\smallskip
\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.}
\smallskip
\smallskip
\smallskip
\smallskip
\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
\smallskip
\smallskip
\smallskip
\noindent{\bf To appear in
{\sl Structural Models in Geography\/}
by this set of authors.}
\smallskip
\smallskip
The reader should read this article with a map of the Paris Metro
in hand.
\smallskip
\smallskip
\smallskip
\smallskip
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 $Z$.
\endinsert
\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.
\endinsert
\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
plane.
\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 $Y'$.
\endinsert
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.
\endinsert
\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
$X_1X_3$, $X_1X_4$, $X_2X_4$, $X_2X_5$, $X_3X_1$, $X_3X_5$.
\endinsert
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.
\endinsert
\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.
\endinsert
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.
\vfill\eject
\noindent{\bf References}
\ref Harary, F. 1969. {\sl Graph Theory\/}. Reading, Mass.,
Addison-Wesley.
\ref Kuratowski, K. 1930. Sur le probl\`eme des courbes
gauches en topologie. {\sl Fund. Math.\/}, {\bf 15}, 271-283.
\vfill\eject
\centerline{\bf Interruption!}
\smallskip
\centerline{\bf Sandra Lach Arlinghaus}
\centerline{The University of Michigan}
\smallskip
\smallskip
{\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,
1987).
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.
\endinsert
\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
practice.
\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.
\vfill\eject
\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,
Baltimore.
\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.
\vfill\eject
\centerline{\bf 4. REPRINT}
\smallskip
\centerline{\bf Imperfections in the Uniform Plane}
\smallskip
\centerline{\bf Michael F. Dacey}
\smallskip
\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}
\smallskip
\centerline{\bf John D. Nystuen}
\smallskip
\smallskip
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
generation.
\smallskip
\smallskip
\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}
\smallskip
\centerline{\bf John D. Nystuen}
\smallskip
\smallskip
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
parts.
\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 ${\chi}^2$
statistic.
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.
\vfill\eject
\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.
\vfill\eject
\centerline{\bf Imperfections in the Uniform Plane}
\smallskip
\centerline{\bf Michael F. Dacey}
\smallskip
\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
specification.
\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
identifiable.
These imperfections are not given precise definitions. In
constructing the imperfection model more precise definitions are
given.
\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.
\endinsert
\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 $(at_1 + bt_2)$
is
$$
s_{ab} = \beta_{ab}(at_1 + bt_2 + \xi_{ab}) \eqno(6)
$$
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 $R$,
put
$$
p=\pi \gamma r^2 \eqno(8)
$$
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 $M$
and
$$
M=S \cup R. \eqno(9)
$$
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
pattern.
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
parameters.
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
$$
R_{i1}< \cdots < R_{ij} < \cdots < R_{iJ} \eqno(10)
$$
and the $R_{ij}$ is called the $j$ order distance. For
description of a bounded map pattern the $j$ order distance is
recorded only if $R_{ij}$ is less than the distance from $i$ to
the nearest map boundary. The chance of bias due to the influence
of boundaries is reduced by this constraint, but there is loss of
information to the pattern description because all distance
relations are not utilized.
The $R_{ij}$ measurements reflect the arbitrary map metric. The
dimensional constant which eliminates effect of scale is
$d^{1/2}$, where $d$ is the density of points in $Q$.
Measurements in $Q$ are reduced to standardized distance by the
transformation
$$
r_{ij}=d^{1/2}R_{ij}. \eqno(11)
$$
Standard distances are used in this report to describe all
patterns.
Let $I$ denote a collection of points in $Q$, and $i\in I$. One
description of $Q$ uses standard distances from each origin point
$i\in I$ to the $J$ nearest points.
\noindent Locus to Point Order Distances
A second description of pattern uses distance measurements from
coordinate locations to points. Let $L$ define a set of locations
in $Q$ and in general a locus $\ell \in L$ is not a point symbol
of $Q$. The measured distance in $Q$ from locus $\ell $ to the
$h$ nearest point is denoted by $R_{\ell h}$. The measurements
from $\ell $ are ordered by distance and put in standard form; in
symbols
$$
r_{\ell 1}< \cdots < r_{\ell h} < \cdots < r_{\ell H}
\eqno(12)
$$
$$
r_{\ell h}=d^{1/2}R_{\ell h}. \eqno(13)
$$
The second description of $Q$ uses standard distances from each
locus $\ell \in L$ to the $H$ nearest points. The boundary
constraint pertains to these distances also.
\noindent Sampling Methods
The elements of $I$ may consist of all or a sample of points in
$Q$. For this study a census was taken, largely because of small
pattern size.
The loci in $L$ necessarily constitute a sample, and these
locations may be designated by random, stratified or uniform
sampling methods. The most efficient mesh for plane sampling has
been studied by a number of writers, as Zubrzycki (1961) and
Dalenius, Hajek, and Zubrzycki (1961), but there are no general
conclusions. This study used random sampling, largely because the
patterns of interest contain high degree of uniformity in spacing
and random sampling is probably less sensitive to this type of
spatial bias. However, this topic requires study.
\noindent Summary Description of Pattern
A point pattern may be summarized by (i) the lower moments of the
$j$ and $h$ order distances or (ii) the frequency distributions
of these order distances. The $j$ order point to point distances
provide a quantitative summary of the arrangement of points with
respect to other points of the pattern, but these distances do
not explicitly reflect the arrangement of points with respect to
the map space. The complementary $h$ order locus to point
distances provide a quantitative summary of the arrangement of
points with respect to the loci in $L$. To the degree the sample
mesh of $L$ is a measure of the map space, $h$ order distances
also summarize the arrangement of points with respect to the map
space. Because these two classes of distances reflect two
different aspects of pattern, this type of summary statement
captures many of the subtle characteristics composing a point
pattern.
\noindent{\bf Comparison of Map Patterns}
The descriptive measures provide a basis for evaluating the
degree of similarity between two or more patterns. patterns are
called similar if the order distances summarizing each of the
patterns have the same statistical parameters. The standardized
distances allow direct comparison of any two point patterns, for
the distances represented by the variable $r$ (either $r_{ij}$
or $r_{\ell h}$ are normalized to account for differences in
scale, unit measurement and density of points. Using either
means or frequency distributions of order distances, the
hypothesis that two or more sets of measurements belong to the
same statistical population may be tested by standard procedures.
\noindent{\bf Theoretical Order Distances}
This paragraph considers the basic derivation of order distances
for imperfection models. The derivations are simplified by
studying (i) lattices for which $t_1=t_2$, (ii) nearest neighbor
situations only, and (iii) the stochastic variable $\xi $
defined by the normal law.
Two nearest neighbor lattice sites are separated by the distance
$t$ ($=t_1=t_2$). Let the random variable $X$ denote the distance
between two disturbed points associated with any two nearest
neighbor lattice sites. It requires only elementary geometry to
show that the distance between points $(\rho_1, \theta_1)$ and
$(\rho_2, \theta_2)$ is
$$
x=
((\rho_1 \hbox{cos}\,\theta_1 - \rho_2 \hbox{cos}\,\theta_2 + t)^2
+
((\rho_1 \hbox{sin}\,\theta_1 - \rho_2 \hbox{sin}\,\theta_2)^2)^{1/2}.
\eqno(14)
$$
The simplest derivation of order distances is for the complete
periodic disturbance model ($\lambda = 1$ and $\gamma = 0$) on
the hexagonal lattice. Let $m$ ($=6$) denote the number of
nearest neighbors to each lattice site. We consider the distances
from an arbitrary point $i$ at $(at_1 + bt_2 +\xi_{ab})$. It is
assumed that the $m$ nearest points to $i$ are disturbed from
nearest neighbor lattice sites only. The $x_k$ is the distance
from point $i$ to the $k$ ($= 1, 2, \ldots , m$) nearest point.
If the disturbance term is identical and independent for each
lattice site, the $m$ distances from $i$ may be interpreted
as $m$ independent observations in a sample of size $m$ from
the population defined by the random variable $X$. Because the
observations are ordered from shortest to longest, $x_k$ is the
kth order statistic. It is well known that the distribution
function of the kth order statistic is given by
$$
\Psi (x_k) =
{{m!}\over{(k-1)!(m-k)!}} F^{k-1} (\omega )
F^{k-1}(\omega) (1-F(\omega))^{m-k} f(\omega ) \eqno(15)
$$
where $f(\omega ) = dF(\omega )$ and the variable $X$, after
making the probability transformation for a specified $f(\rho )$
and $f(\theta )$, is substituted for $\omega $. The $z$ crude
moment of the $k$ order statistic for the complete periodic
disturbance model is
$$
{\mu_z}'(x_k)
=
{{m!}\over{(k-1)!(m-k)!}} F^{k-1} (\omega )
\int_0^\infty \omega^z F^{k-1}(\omega )
(1-F(\omega))^{m-k} f(\omega ) d\omega .\eqno(16)
$$
The derivation is far more complex if the lattice is not hexagonal
and undoubtedly requires more advanced concepts than provided by
elementary probability methods. Moreover, even in this simplified
case, numerical evaluation of (16) is not necessarily possible by
elementary procedures.
In the statement of disturbance models the normal law was
interpreted in polar coordinates by the folded half-normal
distribution; that is, the distribution function for location
about a lattice site is
$$
F(\xi ) = F(\rho, \theta) =
\int_0^\rho \,\, \int_0^\theta f(\rho)\,f(\theta)\, d\rho\, d\theta
\eqno(17)
$$
where
$$
f(\rho ) = {\sqrt 2} \hbox{exp}(-\rho^2/2\sigma^2)/
(\sigma {\sqrt \pi}) \qquad \rho > 0
$$
$$
f(\theta) = (2\pi)^{-1} \qquad 0 < \theta < 2\pi .
$$
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
intractable.
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
published.
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
counties,
\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
himself.
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.
\vfill\eject
\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
\smallskip
\vskip.5cm
\hrule
\vskip.5cm
\settabs\+\qquad\qquad\qquad\qquad
&Over 2.430 \quad &61 \quad &11.72 \quad &{\bf -1.85}\quad &
${{(f_0-f_c)^2}\over {f_0}}$&\cr %sample line
\smallskip
\+&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
\smallskip
\+&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
\smallskip
\+&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.
\vfill\eject
\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 $M_2$
\smallskip
\vskip.5cm
\hrule
\vskip.5cm
\settabs\+\qquad\qquad\qquad\qquad
&Number of \quad &Frequency \quad &Distributions\quad &\cr %sample line
\smallskip
\+&Number of&Frequency Distributions& \cr
\+&Places&Observed&Expected\cr
\+&$x$&$g(x)$&$E(x)$ \cr
\smallskip
\+&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
$\gamma = .2$.
\vfill\eject
\centerline{\bf Table 3}
\noindent Comparison of $j$ Order Distances for $M_2$ and Iowa Maps
\smallskip
\vskip.5cm
\hrule
\vskip.5cm
\settabs\+
&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
\smallskip
\+&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
\smallskip
\+&\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).
\vfill\eject
\centerline{\bf Table 4}
\noindent Comparison of $h$ Order Distances for $M_2$ and Iowa Maps
\smallskip
\vskip.5cm
\hrule
\vskip.5cm
\settabs\+
&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
\smallskip
\+&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
\smallskip
\+&\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).
\vfill\eject
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
\vfill\eject
\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,
125-150.
\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.
\vfill\eject
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.
\vfill\eject
\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
plane.
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
plane.
\endinsert
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!
\smallskip
\centerline{\bf Happy Holidays!}
\smallskip
\smallskip
\noindent {\bf Reference}
\smallskip
\ref Coxeter, H. S. M. 1974. {\sl Projective Geometry\/}, 2nd Ed.
University of Toronto Press, Toronto.
\vfill\eject
\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
Dynamic;
\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.
\vfill\eject
\centerline{\bf 6. SAMPLE OF HOW TO DOWNLOAD THE ELECTRONIC FILE}
\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
\centerline{\bf BACK ISSUES OF {\sl SOLSTICE\/} AVAILABLE ON FTP}
\noindent This section shows the exact set of commands that work
to download {\sl Solstice\/} on The University of Michigan's
Xerox 9700. Because different universities will have different
installations of {\TeX}, this is only a rough guideline which
{\sl might\/} be of use to the reader. (BACK ISSUES AVAILABLE
using anonymous ftp to open um.cc.umich.edu, account 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 sarhaus@umich.edu.
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.
\noindent
ASSUME YOU HAVE SIGNED ON AND ARE AT THE SYSTEM PROMPT, \#.
\smallskip
\# create -t.tex
\# percent-sign t from pc c:backslash words backslash
solstice.tex to mts -t.tex char notab
(this command sends my file, solstice.tex, which I did as
a WordStar (subdirectory, ``words") ASCII file to the
mainframe)
\# run *tex par=-t.tex
(there may be some underfull (or certain over) boxes that
generally cause no problem; there should be no other
``error" messages in the typesetting--the files you
receive were already tested.)
\# run *dvixer par=-t.dvi
\# control *print* onesided
\# run *pagepr scards=-t.xer, par=paper=plain
\vfill\eject
\centerline{\bf 7. SOLSTICE--INDEX, VOLUMES I, II, III, IV}
\smallskip
\noindent{\bf Volume V, Number 1, Summer, 1994}
\smallskip
\noindent {\bf 1.} Welcome to New Readers and Thank You.
\smallskip
\noindent {\bf 2.} Press clippings, summary.
\smallskip
\noindent {\bf 3.} Reprints
\smallskip
Getting Infrastructure Built
\smallskip
Virginia Ainslie and Jack Licate
\smallskip
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?
\smallskip
Center Here; Center There; Center, Center Everywhere
\smallskip
Frank E. Barmore
\smallskip
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.
\smallskip
\noindent {\bf 4.} Articles.
\smallskip
Equal-Area Venn Diagrams of Two Circles: Their Use with Real-World
Data
\smallskip
Barton R. Burkhalter
\smallskip
General Problem; Definition of the Two-Circle Problem; Analytic
Strategy; Derivation of $B\%$ and $AB\%$ as a Function of
$r_{B}$ and $d_{AB}$.
\smallskip
\smallskip
Los Angeles, 1994 --- A Spatial Scientific Study
\smallskip
Sandra L. Arlinghaus, William C. Arlinghaus, Frank Harary, and
John D. Nystuen.
\smallskip
Los Angeles, 1994; Policy Implications; References.
Tables and Complicated Figures.
\smallskip
\noindent {\bf 5.} Downloading of Solstice
\smallskip
\noindent {\bf 6.} Index to Volumes I (1990), II (1991), III (1992),
and IV (1993) of Solstice.
\smallskip
\noindent {\bf 7.} Other Publications of IMaGe
\smallskip
\smallskip
\noindent{\bf Volume IV, Number 2, Winter, 1993}
\smallskip
\noindent {\bf 1.} Welcome to New Readers and Thank You Notes.
\smallskip
\noindent {\bf 2.} Press clippings, summary.
\smallskip
\noindent {\bf 3.} Article
\smallskip
Villages in Transition: Elevated Risk of Micronutrient Deficiency.
\smallskip
William D. Drake, S. Pak, I. Tarwotjo, Muhilal, J. Gorstein,
R. Tilden.
\smallskip
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
\smallskip
\noindent {\bf 5.} Index to Volumes I (1990), II (1991), III (1992),
and IV.1 (1993) of Solstice.
\smallskip
\noindent {\bf 6.} Other Publications of IMaGe
\smallskip
\noindent {\bf 7.} Selected recent publications of interest
involving Solstice Board members, and some goings on about Ann Arbor.
%___________________________________________________________________
%___________________________________________________________________
\smallskip
\noindent{\bf Volume IV, Number 1, Summer, 1993}
\smallskip
\noindent {\bf 1.} Welcome to New Readers.
\smallskip
\noindent {\bf 2.} Press clippings, summary.
\smallskip
\noindent {\bf 3.} Goings on about Ann Arbor--ESRI and IMaGe Gift
\smallskip
\noindent {\bf 4.} Articles
\smallskip
Electronic Journals: Observations Based on Actual Trials,
1987-Present, by Sandra L. Arlinghaus and Richard H. Zander.
Headings:
Abstract; Content issues; Production issues; Archival issues;
References.
\smallskip
Wilderness As Place, by John D. Nystuen.
Headings:
Visual paradoxes; Wilderness defined; Conflict or synthesis;
Wilderness as place; Suggested readings; Sources; Visual
illusion authors
\smallskip
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\/}.
Headings:
Abstract; Introduction; The Qibla problem; The geographic
center; The center of population; Appendix; References.
\smallskip
Microcell Hex-nets? by Sandra L. Arlinghaus
Headings:
Introduction; Lattices; Microcell hex-nets; References.
\smallskip
Sum Graphs and Geographic Information, by Sandra L. Arlinghaus,
William C. Arlinghaus, Frank Harary.
Headings:
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
\smallskip
\noindent{\bf 5.} Downloading of {\sl Solstice\/}.
\smallskip
\noindent{\bf 6.} Index.
\smallskip
\noindent{\bf 7.} Other publications of IMaGe.
%----------------------------------------------------------------
%----------------------------------------------------------------
\smallskip
\noindent {\bf Volume III, Number 2, Winter, 1992}
\smallskip
\noindent {\bf 1.} A Word of Welcome from A to U.
\smallskip
\noindent {\bf 2.} Press clippings--summary.
\smallskip
\noindent {\bf 3.} Reprints:
\smallskip
\noindent {\bf A.}
What Are Mathematical Models and What Should They Be?
by Frank Harary, reprinted from {\sl Biometrie - Praximetrie\/}.
\smallskip \noindent {\sl
1. What Are They? 2. Two Worlds: Abstract and Empirical
3. Two Worlds: Two Levels 4. Two Levels: Derivation and
Selection 5. Research Schema 6. Sketches of Discovery
7. What Should They Be?
\/}
\smallskip
\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
\/}
\smallskip
\noindent {\bf 4.} Article:
\smallskip
The Pelt of the Earth: An Essay on Reactive Diffusion,
by Sandra L. Arlinghaus and John D. Nystuen.
\smallskip \noindent {\sl
1. Pattern Formation: Global Views 2. Pattern Formation:
Local Views 3. References Cited 4. Literature of Apparent
Related Interest.
\/}
\smallskip
\noindent {\bf 5.} Feature
Meet new{\sl Solstice\/} Board Member William D. Drake;
comments on course in Transition Theory and listing of
student-produced monograph.
\smallskip
\noindent {\bf 6.} Downloading of Solstice.
\smallskip
\noindent {\bf 7.} Index to Solstice.
\smallskip
\noindent {\bf 8.} Other Publications of IMaGe.
\smallskip
%----------------------------------------------------------------
%----------------------------------------------------------------
\noindent {\bf Volume III, Number 1, Summer, 1992}
\smallskip
\noindent{\bf 1. ARTICLES.}
\smallskip\noindent
{\bf Harry L. Stern}.
\smallskip\noindent
{\bf Computing Areas of Regions With Discretely Defined Boundaries}.
\smallskip\noindent
1. Introduction 2. General Formulation 3. The Plane 4. The Sphere
5. Numerical Example and Remarks. Appendix--Fortran Program.
\smallskip
\noindent{\bf 2. NOTE }
\smallskip\noindent
{\bf Sandra L. Arlinghaus, John D. Nystuen, Michael J. Woldenberg}.
\smallskip\noindent
{\bf The Quadratic World of Kinematic Waves}
\smallskip
\noindent{\bf 3. SOFTWARE REVIEW}
\smallskip
RangeMapper$^{\hbox{TM}}$ --- version 1.4.
Created by {\bf Kenelm W. Philip}, Tundra Vole Software,
Fairbanks, Alaska. Program and Manual by {\bf Kenelm W. Philip}.
\smallskip
Reviewed by {\bf Yung-Jaan Lee}, University of Michigan.
\smallskip
\noindent{\bf 4. PRESS CLIPPINGS}
\smallskip
\noindent{\bf 5. INDEX to Volumes I (1990) and II (1991) of
{\sl Solstice}.}
\smallskip
%----------------------------------------------------------------
%----------------------------------------------------------------
\noindent {\bf Volume II, Number 2, Winter, 1991}
\smallskip
\noindent 1. REPRINT
Saunders Mac Lane, ``Proof, Truth, and Confusion." Given as the
Nora and Edward Ryerson Lecture at The University of Chicago in
1982. Republished with permission of The University of Chicago
and of the author.
I. The Fit of Ideas. II. Truth and Proof. III. Ideas and Theorems.
IV. Sets and Functions. V. Confusion via Surveys.
VI. Cost-benefit and Regression. VII. Projection, Extrapolation,
and Risk. VIII. Fuzzy Sets and Fuzzy Thoughts. IX. Compromise
is Confusing.
\noindent 2. ARTICLE
Robert F. Austin. ``Digital Maps and Data Bases:
Aesthetics versus Accuracy."
I. Introduction. II. Basic Issues. III. Map Production.
IV. Digital Maps. V. Computerized Data Bases. VI. User
Community.
\noindent 3. FEATURES
Press clipping; Word Search Puzzle; Software Briefs.
\smallskip
%----------------------------------------------------------------
%----------------------------------------------------------------
\noindent {\bf Volume II, Number 1, Summer, 1991}
\smallskip
\noindent 1. ARTICLE
Sandra L. Arlinghaus, David Barr, John D. Nystuen.
{\sl The Spatial Shadow: Light and Dark --- Whole and Part\/}
This account of some of the projects of sculptor David Barr
attempts to place them in a formal, systematic, spatial setting
based on the postulates of the science of space of William
Kingdon Clifford (reprinted in {\sl Solstice\/}, Vol. I, No. 1.).
\smallskip
\noindent 2. FEATURES
\item{i} Construction Zone --- The logistic curve.
\item{ii.} Educational feature --- Lectures on ``Spatial Theory"
\smallskip
%----------------------------------------------------------------
%----------------------------------------------------------------
\noindent{\bf Volume I, Number 2, Winter, 1990}
\smallskip
\noindent 1. REPRINT
John D. Nystuen (1974), {\sl A City of Strangers: Spatial Aspects
of Alienation in the Detroit Metropolitan Region\/}.
This paper examines the urban shift from ``people space" to
``machine space" (see R. Horvath, {\sl Geographical Review\/},
April, 1974) in the Detroit metropolitan region of 1974. As
with Clifford's {\sl Postulates\/}, reprinted in the last issue
of {\sl Solstice\/}, note the timely quality of many of the
observations.
\noindent 2. ARTICLES
Sandra Lach Arlinghaus, {\sl Scale and Dimension: Their Logical
Harmony\/}.
Linkage between scale and dimension is made using the
Fallacy of Division and the Fallacy of Composition in a fractal
setting.
\smallskip
Sandra Lach Arlinghaus, {\sl Parallels between Parallels\/}.
The earth's sun introduces a symmetry in the perception of
its trajectory in the sky that naturally partitions the earth's
surface into zones of affine and hyperbolic geometry. The
affine zones, with single geometric parallels, are located
north and south of the geographic parallels. The hyperbolic
zone, with multiple geometric parallels, is located between the
geographic tropical parallels. Evidence of this geometric
partition is suggested in the geographic environment --- in the
design of houses and of gameboards.
\smallskip
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.
\smallskip
Sandra Lach Arlinghaus, {\sl Fractal Geometry of Infinite Pixel
Sequences: ``Su\-per\--def\-in\-i\-tion" Resolution\/}?
Comparison of space-filling qualities of square and hexagonal
pixels.
\smallskip
\noindent 3. FEATURES
\item{i.} Construction Zone --- Feigenbaum's number; a
triangular coordinatization of the Euclidean plane.
\item{ii.} A three-axis coordinatization of the plane.
\smallskip
%----------------------------------------------------------------
%----------------------------------------------------------------
\noindent{\bf Volume I, Number 1, Summer, 1990}
\noindent 1. REPRINT
William Kingdon Clifford, {\sl Postulates of the Science of Space\/}
This reprint of a portion of Clifford's lectures to the
Royal Institution in the 1870's suggests many geographic topics
of concern in the last half of the twentieth century. Look for
connections to boundary issues, to scale problems, to self-
similarity and fractals, and to non-Euclidean geometries (from
those based on denial of Euclid's parallel postulate to those
based on a sort of mechanical ``polishing"). What else did, or
might, this classic essay foreshadow?
\noindent 2. ARTICLES.
Sandra L. Arlinghaus, {\sl Beyond the Fractal.}
An original article. The fractal notion of self-similarity
is useful for characterizing change in scale; the reason
fractals are effective in the geometry of central place theory
is because that geometry is hierarchical in nature. Thus, a
natural place to look for other connections of this sort is to
other geographical concepts that are also hierarchical. Within
this fractal context, this article examines the case of spatial
diffusion.
When the idea of diffusion is extended to see ``adopters" of
an innovation as ``attractors" of new adopters, a Julia set is
introduced as a possible axis against which to measure one class
of geographic phenomena. Beyond the fractal context, fractal
concepts, such as ``compression" and ``space-filling" are
considered in a broader graph-theoretic setting.
\smallskip
William C. Arlinghaus, {\sl Groups, Graphs, and God}
\smallskip
\noindent 3. FEATURES
\smallskip
\item{i.} Theorem Museum --- Desargues's Two Triangle Theorem
from projective geometry.
\item{ii.} Construction Zone --- a centrally symmetric hexagon is
derived from an arbitrary convex hexagon.
\item{iii.} Reference Corner --- Point set theory and topology.
\item{iv.} Educational Feature --- Crossword puzzle on spices.
\item{v.} Solution to crossword puzzle.
\smallskip
\noindent 4. SAMPLE OF HOW TO DOWNLOAD THE ELECTRONIC FILE
\smallskip
\vfill\eject
\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,
1986.
4. Robert F. Austin, A Historical Gazetteer of Southeast Asia,
1986.
5. Sandra L. Arlinghaus, Essays on Mathematical Geography--II,
1987.
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,
1988.
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.
%----------------------------------------------------------------
%----------------------------------------------------------------
\smallskip
DISCUSSION PAPERS--ORIGINAL
Editor, Daniel A. Griffith
Professor of Geography
Syracuse University
1. Spatial Regression Analysis on the PC:
Spatial Statistics Using Minitab. 1989.
Cost: \$12.95, hardcopy.
%----------------------------------------------------------------
%----------------------------------------------------------------
\smallskip
DISCUSSION PAPERS--REPRINTS
Editor of MICMG Series, John D. Nystuen
Professor of Geography and Urban Planning
The University of Michigan
1. Reprint of the Papers of the Michigan InterUniversity
Community of Mathematical Geographers.
Editor, John D. Nystuen.
Cost: \$39.95, hardcopy.
Contents--original editor: John D. Nystuen.
1. Arthur Getis, ``Temporal land use pattern analysis with the
use of nearest neighbor and quadrat methods." July, 1963
2. Marc Anderson, ``A working bibliography of mathematical
geography." September, 1963.
3. William Bunge, ``Patterns of location." February, 1964.
4. Michael F. Dacey, ``Imperfections in the uniform plane."
June, 1964.
5. Robert S. Yuill, A simulation study of barrier effects
in spatial diffusion problems." April, 1965.
6. William Warntz, ``A note on surfaces and paths and
applications to geographical problems." May, 1965.
7. Stig Nordbeck, ``The law of allometric growth."
June, 1965.
8. Waldo R. Tobler, ``Numerical map generalization;"
and Waldo R. Tobler, ``Notes on the analysis of geographical
distributions." January, 1966.
9. Peter R. Gould, ``On mental maps." September, 1966.
10. John D. Nystuen, ``Effects of boundary shape and the
concept of local convexity;" Julian Perkal, ``On the length
of empirical curves;" and Julian Perkal, ``An attempt at
objective generalization." December, 1966.
11. E. Casetti and R. K. Semple, ``A method for the
stepwise separation of spatial trends." April, 1968.
12. W. Bunge, R. Guyot, A. Karlin, R. Martin, W. Pattison,
W. Tobler, S. Toulmin, and W. Warntz, ``The philosophy of maps."
June, 1968.
%----------------------------------------------------------------
%----------------------------------------------------------------
Reprints of out-of-print textbooks.
1. Allen K. Philbrick. This Human World.
2. John F. Kolars and John D. Nystuen. Human Geography.
\bye