WINTER, 1995
Founding Editor-in-Chief:
Sandra Lach Arlinghaus, University of Michigan;
Institute of Mathematical Geography (indep)
Editorial Advisory Board:
Michael F. Goodchild, University of California, Santa Barbara
Daniel A. Griffith, Syracuse University
Jonathan D. Mayer, University of Washington
John D. Nystuen, University of Michigan
William C. Arlinghaus, Lawrence Technological University
Neal Brand, University of North Texas
Kenneth H. Rosen, A. T. & T. Bell Laboratories
Engineering Applications.
William D. Drake, University of Michigan
Frederick L. Goodman, University of Michigan
Robert F. Austin, Austin Communications Education Services.
Technical Editor.
Richard Wallace, University of Michigan.








The purpose of 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. Individuals wishing to submit articles or other material should contact an editor, or send e-mail directly to ------------------------------------------------------------------------

Volumes I-V (1990 to 1994) are typeset using TeX. Later volumes are formatted in a variety of ways. File names of all volumes, the first five and later ones, carry the journal name, tagged with the number and volume year in the file extension--thus, solstice.190, is number 1 of the 1990 volume of Solstice.

WORLD WIDE WEB: Back issues are available on a site under construction. Universal Resource Locator (URL):
Thanks to William E. Arlinghaus for his WebSite work.

GOPHER: Solstice may be found under Arizona State University Department of Mathematics, Electronic Mathematics Journals and Newsletters, Solstice - An Electronic Journal of Geography & Mathematics on GOPHER: PI.LA.ASU.EDU port 70. Thanks to Bruce Long for his continuing advice and support.

FTP--until June 1, 1996. Open anonymous FTP host account IEVG. Once in the system, type cd IEVG and then type on the next line, ls. Then type get filename (substitute a name from the directory).
The electronic files are issued yearly as copyrighted hardcopy in the Monograph Series of the Institute of Mathematical Geography. This material will appear in Volume 20 in that series, ISBN 1-877751-58-8. To order hardcopy, and to obtain current price lists, write to the Editor-in-Chief of Solstice at 2790 Briarcliff, Ann Arbor, MI 48105, or call 313-761-1231.

Suggested form for citation: cite the hardcopy. To cite the electronic copy, note the exact time of transmission from Ann Arbor, and cite all the transmission matter as facts of publication. Any copy that does not superimpose precisely upon the original as transmitted from Ann Arbor should be presumed to be an altered, bogus copy of Solstice.

One reason that planning of any sort is a difficult process is that it involves altering natural boundaries to fit human needs and desires. While it may not be "nice to fool Mother Nature" the act of planning may be predicated on such an attempt, especially when the balance between human and environmental needs is tipped strongly toward the human side. At a very general level, planning how to use the Earth's surface involves what space to use and when to use it. The "what" issues are those that involve spatial planning; they typically involve the concept of scale. The "when" issues involve temporal planning; they typically involve the concept of sequence.

Particular spatial issues might address whether or not boundaries of a parcel of land are clearly designated with respect to one's neighbors; whether or not a proposed land use is consistent with the general character of a larger region; or whether or not a developer's site plans give sufficient attention to natural features. Temporal issues might address the long range and the short range view of a traffic circulation pattern; the sequence, in years, in which lands are to be annexed to a city; or the length of time trees need to have lived in order to be designated landmark trees. When one considers that budget concerns often function as an underlying factor that can help to sway this balance, the fragility of the art of planning becomes apparent.

One way to view complicated issues is to consider them at an abstract level in order to understand the logic that links them. The two-valued system of logic on which much of mathematics is based offers one structure that exposes logical connections. When using this structure in conjunction with real-world settings, which often defy the Law of the Excluded Middle, one generally has a number of difficult decisions to make; it is in the act of making these decisions that thoughts can become clearer.

The preservation of natural features is an issue that can be a developer's nightmare, just as development can be the bete noir of the environmentalist. When man-made boundaries are superimposed on the natural environment, there is often little correspondence between the two partitions of space. Abstractly it is not surprising, therefore, that individuals using one way to partition space will be at loggerheads with those using a different partition of space.

When the topography of a region is altered, it is necessarily the case that the natural features on that surface are also altered. Considering the contrapositive of this statement, a logical equivalent, leads to the idea that the preservation of natural features is dependent on the preservation of topography. When this idea is coupled with the notion that the fundamental topographic unit is the drainage basin or watershed (Leopold, Wolman, and Miller, Fluvial Processes in Geomorphology), the following principle emerges.

Watershed Principle.
If the preservation of natural features depends upon the preservation of topography and if the fundamental topographic unit is the watershed, then the preservation of natural features depends upon the watershed.

If one accepts this Principle, then it may well be a small step to the following Corollaries.

Corollary 1.
When environmental concerns are involved, the drainage basin should be the fundamental planning unit.

Corollary 2.
When the drainage basin is the fundamental planning unit, the partition of wetlands and other elements of the drainage network, by man-made planning unit boundaries, is not possible.

Decisions as to the impact a proposed development project will have on a wetland are facilitated by having the entire wetland contained within the legal boundaries of the parcel; using the drainage basin as the fundamental planning unit ensures that such set-theoretic containment will be the case. Issues involving the welfare of the entire watershed also become tractable under such an alignment: neighbors become neighbors with respect to the drainage pattern rather than with respect to superimposed human boundaries. Indeed, what my neighbor does three miles upstream from me may have far more impact on my land that does the action of a neighbor 100 feet away who is in a different drainage basin. Current technology (Geographic Information Systems, for example) might make it possible to alter the inventory of lands to create suitable, substantial changes, along these or along other lines, in legal definitions. The use of technological capability to make legal definitions correspond more closely to natural definitions can lead to the resolution of conflicts: the closer the fit between natural and man-made boundaries the fewer the disagreements.

The basic idea behind the Watershed Principle might be captured as one that minimizes damage to the environment and maximizes satisfaction of human needs and desires. Viewed more broadly, the Watershed Principle might be recast as a MiniMax Principle which can then be recast downstream abstractly, in a number of other more specific forms (such as the Watershed Principle).

MiniMax Principle
An optimal plan is one which minimizes alteration of existing entities and maximizes the common good.

Highly general principles, such as this one, demand attention to definitional matters: what is meant by "common good" or how might one measure "alteration." These are difficult problems: one advantage to an abstract view is to bring important and difficult issues into focus.

One case in which the fit between natural and man-made boundaries is done in a style consistent with the Minimax Principle is the spatial layout of reckoning time (thus, time becomes transformed in a "meta" fashion into space). Much of the developed world measures the passage of time by the position of Earth relative to our Sun. One unit of time, the year, corresponds roughly to one revolution of the Earth around the Sun. Another, smaller, unit of time, the day, corresponds roughly to the rotation of the Earth on its axis--the man-made boundaries in both cases are set by the natural planetary motions in space.

When planetary motions do not permit any further refinement of the day into even smaller units, we subdivide the day into hours. When the partition of the day into 24 hours is put into correspondence with the grid system based on latitude and longitude, one hour corresponds to fifteen degrees of longitude. Fifteen degrees of longitude corresponds to a central angle of fifteen degrees intercepted along the Equatorial great circle. Thus, 24 man-made time zones of 15 degrees of longitude each envelop the Earth--man-made boundaries again follow (although a bit indirectly) from natural boundaries. The Earth becomes a "clockwork orange" of 24 sections, each 1 hour wide, with boundaries along meridians spaced 15 degrees apart. Across oceans, this alignment of time-zone and longitude may reasonably have boundaries along meridians; interior to a continent, however, human needs and desires may reasonably prevail, making it prudent to bend the natural alignment for the common good.

** The author wishes to thank her colleagues on the City of Ann Arbor Planning Commission and in the Planning Department of the City of Ann Arbor. The challenge and stimulation fostered by this lively Commission helped to generate this viewpoint.
The attached file contains a Mollweide equal area projection of the world made from Geographic Information System (GIS) technology. A map from a vector GIS (MapInfo for Windows) was copied onto the Windows clipboard and then pasted into Windows Paintbrush. It was then zipped using PKZIP 204G. Readers of Solstice can download this raster file from the attachments to Solstice. Other (free) maps are available from the MapBank on the WebSite

When these downloaded maps are put into Windows Paintbrush (or other software) and are projected from the computer screen onto the wall (using a data-show and overhead projector, or some such) their resolution is of about the same quality as that of the original on-screen display in the GIS. Thus, wall-maps can be carried around on diskette. This strategy offers an easy way for university professors and pre-collegiate teachers alike to give lectures with maps tailored to their needs--from base maps for simple place-name recognition, to maps showing voting patterns by party in presidential elections, to maps showing global vehicle registration patterns, to detailed topographic maps. Naturally, the first in the MapBank series of maps offered for this style of communication are base maps.


July 18, 1995, the International Society of Spatial Sciences (ISSS--I-triple-S) was founded as a division of the non-profit Community Systems Foundation of Ann Arbor, MI. This primarily electronic society has as board members:

Sandra L. Arlinghaus (founder), W. C. Arlinghaus, M. L. Bird, B. R. Burkhalter, W. D. Drake, F. L. Goodman, F. Harary, J. A. Licate, A. L. Loeb, K. E. Longstreth, J. D. Nystuen, W. R. Tobler. To follow its activities, browse the under-construction WebSite with direct links to material of the American Geographical Society and the Th&uumlnen Society. internet:

The focus of this new society is to place in a core position those sciences, of spatial character, that are often relegated to the periphery within academic institutional structure. Such sciences are, to name only a few, geology, geography, and astronomy. In moving along this continuum from the center of the Earth to the outer reaches of the universe, one might imagine a whole host of sciences that could also be included (from oceanography to atmospheric science to regional science). Thus, ISSS offers a platform from which individuals, institutions, and professional societies devoted to some aspect of spatial science might further their interests.

The previous article announces one of the projects of ISSS. For the past two years, during the developmental stages of ISSS, a continuing project has been the development of a MapBank. This is a bank composed of maps made by students; most of the maps are thematic maps made to supplement student term papers or as maps to be used in the classroom by students of Education. For teachers to use the MapBank, free of charge, they must make a deposit of an electronic map. Currently the MapBank numbers more than 100 electronic maps. Look for thematic maps to appear in future issues of Solstice and on the WebSite of ISSS. There are currently ten base maps of the world on the ISSS MapBank WebSite: http//




Volume VI, Number 1, June, 1995.

Fifth Anniversary of Solstice

New format for Solstice and new Technical Editor

Richard Wallace. Motor Vehicle Transport and Global Climate Change: Policy Scenarios.

Expository Article. Discrete Mathematics and Counting Derangements in Blind Wine Tastings. Sandra L. Arlinghaus, William C. Arlinghaus, John D. Nystuen


Volume V, No. 2, Winter, 1994.

Sandra L. Arlinghaus, William C. Arlinghaus, Frank Harary: The Paris Metro: Is its Graph Planar?
Planar graphs; The Paris Metro; Planarity and the Metro; Significance of lack of planarity.

Sandra Lach Arlinghaus: Interruption!
Classical interruption in mapping; Abstract variants on interruption and mapping; The utility of considering various mapping surfaces--GIS; Future directions.

Reprint of Michael F. Dacey: Imperfections in the Uniform Plane. Forewords by John D. Nystuen.
Original (1964) Nystuen Foreword; Current (1994) Nystuen Foreword; The Christaller spatial model; A model of the imperfect plane; The disturbance effect; Uniform random disturbance; Definition of the basic model; Point to point order distances; Locus to point order distances; Summary description of pattern; Comparison of map pattern; Theoretical 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.

Sandra L. Arlinghaus: Construction Zone: The Brakenridge-MacLaurin Construction.

William D. Drake: Population Environment Dynamics: Course and Monograph--descriptive material.


Volume V, No. 1, Summer, 1994.

Virginia Ainslie and Jack Licate: Getting Infrastructure Built. Cleveland infrastructure team shares secrets of sucess;
What difference has the partnership approach made; How process affects products--moving projects faster means getting more public investment; 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?

Frank E. Barmore: Center Here; Center There; Center, Center Everywhere.
Abstract; Introduction; Definition of geographic center; Geographic center of a curved surface; Geographic center of Wisconsin; Geographic center of the conterminous U.S.; Geographic center of the U.S.; Summary and recommendations; Appendix A: Calculation of Wisconsin's geographic center; Appendix B: Calculation of the geographical center of the conterminous U.S.; References.

Barton R. Burkhalter: Equal-Area Venn Diagrams of Two Circles: Their Use with Real-World Data
General problem; Definition of the two-circle problem; Analytic strategy; Derivation of B% and AB% as a function of r(B) and d(AB).

Sandra L. Arlinghaus, William C. Arlinghaus, Frank Harary, John D. Nystuen. Los Angeles, 1994 -- A Spatial Scientific Study.
Los Angeles, 1994; Policy implications; References; Tables and complicated figures.


Volume IV, No. 2, Winter, 1993.

William D. Drake, S. Pak, I. Tarwotjo, Muhilal, J. Gorstein, R. Tilden. Villages in Transition: Elevated Risk of Micronutrient Deficiency.
Abstract; Moving from traditional to modern village life: risks during transtion; Testing for elevated risks in transition villages; Testing for risk overlap within the health sector; Conclusions and policy implications

Volume IV, No. 1, Summer, 1993.

Sandra L. Arlinghaus and Richard H. Zander: Electronic Journals: Observations Based on Actual Trials, 1987-Present.
Abstract; Content issues; Production issues; Archival issues; References

John D. Nystuen: Wilderness As Place.
Visual paradoxes; Wilderness defined; Conflict or synthesis; Wilderness as place; Suggested readings; Sources; Visual illusion authors.

Frank E. Barmore: The Earth Isn't Flat. And It Isn't Round Either: Some Significant and Little Known Effects of the Earth's Ellipsoidal Shape.
Abstract; Introduction; The Qibla problem; The geographic center; The center of population; Appendix; References.

Sandra L. Arlinghaus: Micro-cell Hex-nets?
Introduction; Lattices: Microcell hex-nets; References

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


Volume III, No. 2, Winter, 1992.

Frank Harary: What Are Mathematical Models and What Should They Be? What are they?
Two worlds: abstract and empirical; Two worlds: two levels; Two levels: derivation and selection; Research schema; Sketches of discovery; What should they be?

Frank E. Barmore: Where Are We? Comments on the Concept of Center of Population.
Introduction; Preliminary remarks; Census Bureau center of population formulae; Census Bureau center of population description; Agreement between description and formulae; Proposed definition of the center of population; Summary; Appendix A; Appendix B; References.

Sandra L. Arlinghaus and John D. Nystuen: The Pelt of the Earth: An Essay on Reactive Diffusion.
Pattern formation: global views; Pattern formation: local views; References cited; Literature of apparent related interest.

Volume III, No. 1, Summer, 1992.

Harry L. Stern: Computing Areas of Regions with Discretely Defined Boundaries.
Introduction; General formulation; The plane; The sphere; Numerical examples and remarks; Appendix--Fortran program.

Sandra L. Arlinghaus, John D. Nystuen, Michael J. Woldenberg: The Quadratic World of Kinematic Waves.


Volume II, No. 2, Winter, 1991.

Reprint of Saunders Mac Lane: Proof, Truth, and Confusion, The Nora and Edward Ryerson Lecture at The University of Chicago in 1982.
The fit of ideas; Truth and proof; Ideas and theorems; Sets and functions; Confusion via surveys; Cost-benefit and regression; Projection, extrapolation, and risk; Fuzzy sets and fuzzy thoughts; Compromise is confusing.

Robert F. Austin: Digital Maps and Data Bases: Aesthetics versus accuracy.
Introduction; Basic issues; Map production; Digital maps; Computerized data bases; User community.

Volume II, No. 1, Summer, 1991.

Sandra L. Arlinghaus, David Barr, John D. Nystuen: 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 Solstice, Vol. I, No. 1.).

Sandra L. Arlinghaus: Construction Zone--The Logistic Curve.

Educational feature--Lectures on Spatial Theory.


Volume I, No. 2, Winter, 1990.

John D. Nystuen: 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, Geographical Review, April, 1974) in the Detroit metropolitan regions of 1974. As with Clifford's Postulates, reprinted in the last issue of Solstice, note the timely quality of many of the observations.

Sandra Lach Arlinghaus: 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.

Sandra Lach Arlinghaus: Parallels Between Parallels.
The earth's sun introduces a symmetry in the perception of its trajectory in the sky that naturally partitions the earth's surface into zones of affine and hyperbolic geometry. The affine zones, with single geometric parallels, are located north and south of the geographic parallels. The hyperbolic zone, with multiple geometric parallels, is located between the geographic tropical parallels. Evidence of this geometric partition is suggested in the geographic environment--in the design of houses and of gameboards.

Sandra L. Arlinghaus, William C. Arlinghaus, and John D. Nystuen: 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, Geographical Analysis, 1990). In that previous research, we applied the algorithm to the generalized road network graph surrounding San Francisco Bay. Here, we examine consequent changes in matrix entries when the underlying adjacency pattern of the road network was altered by the 1989 earthquake that closed the San Francisco--Oakland Bay Bridge.

Sandra Lach Arlinghaus: Fractal Geometry of Infinite Pixel Sequences: "Super-definition" Resolution?
Comparison of space-filling qualities of square and hexagonal pixels.

Sandra Lach Arlinghaus: Construction Zone--Feigenbaum's number; a triangular coordinatiztion of the Euclidean plane; A three-axis coordinatization of the plane.

Volume I, No. 1, Summer, 1990.

Reprint of William Kingdon Clifford: Postulates of the Science of Space.
This reprint of a portion of Clifford's lectures to the Royal Institution in the 1870s 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?

Sandra Lach Arlinghaus: Beyond the Fractal.
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.

William C. Arlinghaus: Groups, Graphs, and God.

Sandra L. Arlinghaus: Theorem Museum--Desargues's Two Triangle Theorem from projective geometry.

Construction Zone--centrally symmetric hexagons.