SOLSTICE:  AN ELECTRONIC JOURNAL OF GEOGRAPHY AND MATHEMATICS.
Volume V, Number 2.  Winter, 1994.
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\centerline{\big SOLSTICE:}
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\centerline{\bf AN ELECTRONIC JOURNAL OF GEOGRAPHY AND MATHEMATICS}
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\centerline{\bf WINTER, 1994}
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\centerline{\bf Volume V, Number 2}
\centerline{\bf Institute of Mathematical Geography}
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\centerline{\bf Ann Arbor, Michigan}
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\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 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
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Concern for manuscript security.  Prospective authors  sometimes
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are introduced at the whim  of the  Editor.  Any  re-transmitted
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This  document is produced using the typesetting  program,
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Notation  in  the electronic file is in accordance with that  of
hard copy for on The University of Michigan's Xerox 9700 laser--
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that University.

Unless otherwise noted, all regular features"  are  written by
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\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
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 $$R_{i1} 0$$ $$f(\theta) = (2\pi)^{-1} \qquad 0 < \theta < 2\pi .$$ 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.
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