An Electronic Journal of
Geography and Mathematics.
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Final version of IMaGe logo created by Allen K. Philbrick from original artwork from the Founder.

Educational Research Collaboration

Sandra L. Arlinghaus, Ph.D. and Joseph J. Kerski, Ph.D.

     As exciting new vistas open doors in both geographical and mathematical visualization, it may be easy to get caught up in the focus within one discipline or the other.  The richness that comes from interplay between disciplines can get cast aside, only to be rediscovered much later when single disciplines have matured and are looking for the fresh and new, beyond original disciplinary boundaries.
     When such loss occurs at the research level, it is disappointing because much creative activity can slow down while more conventional activity plays out its course within established, comfortable, and conventional limits.  When such loss occurs in the education of children it can be far more dramatic, indeed tragic.  Generations of future citizens, voters, municpal authorities, students, researchers, and teachers may become trapped in curricular conventions of a particular decade.
     A number of scholars, in wide-ranging fields, have long seen this difficulty.  It becomes perhaps more pronounced now with the technological revolution within which so many projects, scholarly and educational alike, are set.
     We have been interested, for at least a part of our careers, in helping to bridge this interdisciplinary gap between geography and mathematics (particularly geometry).  To set this collaboration in motion, we offer a taste of our interests with materials and links to selections from our previous works.   These items illustrate some of our educational and scholarly interests and suggest directions for future work.  Some are simple but not necessarily "easy"; indeed, "simple" is often the hallmark of elegance that piques curiosity and stimulates  both the motivation to learn eagerly and the imagination to pursue new directions.

Conceptual Framework
    A number of years ago, the American Mathematical Monthly introduced a novel feature:  "Proofs without Words" (David and Tomei, 1989).   Others picked up on that theme and this interesting feature has appeared in a variety of contexts over the decades (Nelsen, 1997, 2001; Polster, 2004).   The goal is to visualize mathematics without using notation.  Naturally, some worried that  such an approach is not "good" mathematics.  Most, however, appreciate being able to grasp concepts without notation; some of us are more visually, and less notationally, oriented than are others.  The world of paper and print publication is limited in what can be offered.  Indeed, the world of black and white print is even more limited.  There are some Internet applications of this approach dating from the 1990s (try a search engine using "maps math" as a pair of keywords).  Our goal in this project is to explore a broad range of visual connection between maps and math, especially as this connection might be inserted into the mathematics curriculum, and to present visual results employing current technology.  The level of advancement of the mathematics is not what matters; what matters is clarity of concept.
     The abstract view in Figure 1 captures this conceptual framework  (Arlinghaus and Arlinghaus, 2005).  Each large hexagon represents a single scholarly discipline:  mathematics on the left and geography on the right (as an example).  A single field (geometry) is identified (deep red trapezoid) from the broad discipline of mathematics, and is aligned with a counterpart (deep red trapezoid) field (cartography) from the other broad discipline of geography.  Despite the alignment of classically appropriate entities, a gap remains.  The goal, from both a research and an educational standpoint, is to seal the gap and to unify aligned, related structures.

Figure 1.  Conceptual framework of aligning and unifying interdisciplinary studies (Arlinghaus and Arlinghaus, 2005).

Finding the Size...

          One well known classical approach to this style of integrative scholarship appears in the work of Eratosthenes of Alexandria (c. 276-194 B.C.).  Eratosthenes made critical discoveries in both mathematics and geography that might seem unrelated, at first.  His prime number sieve permitted the complete characterization of all whole numbers and offered, therefore, an understanding of the whole number system.  The idea of understanding a whole, that one could never see, also applies to his measurement of the circumference of the Earth.  In his role as Librarian of the great library of Alexandria, he had access to works of his many outstanding predecessors.  He worked with spherical geometry, latitude and longitudeearth-sun relations, and a variety of geometric and trigonometric ideas.  Eratosthenes sealed the gap between number theory and the real world with the idea of using abstract tools to understand more than what one could see (material from Spatial Synthesis (cited below) used with permission of the authors).
     Two millennia later, we know how to "see" not only with theoretical tools based in mathematics, but also with field tools that drawn on connections with the sky and man-made satellites!  As the lesson summary to the linked, downloadable lesson, notes (Kerski, 2007):

With a GPS receiver, find out how you and your students can accurately measure the circumference of the Earth to within 1% of the accepted value. In this lesson, students start by learning about Eratosthenes, who calculated the circumference 2,500 years ago. They then go into the field to calculate the circumference in a variety of different ways, and also consider mass and volume. This lesson incorporates scale, measurement, field work, coordinate systems, and brings together mathematics, geography, and physics.

     Return to this classical experiment in mathematical geography not only to avoid re-inventing the wheel but also to gain, and foster in students, a deeper appreciation of both the endurance and power of this approach.

Finding the Center...
     Once we know the size of something, often we begin to ask questions about its internal structure.  Where are the edges of the object; where is its center?  Frank Barmore offers some interesting work using physics reference integrated with geographic questions (1992; 1994).  More elemetary approaches, integrating contemporary software with concepts of "center" and "scale" appear in lessons available online.     As the lesson summary to the linked, downloadable lesson notes (Kerski, 2010):

Use spatial analysis and GIS to determine and analyze the population centers of the USA and individual states over space and time. Objectives: 1) Understand the definition of a mean center and weighted mean center; 2) Understand the definition of a population weighted mean center; 3) Learn how to calculate mean population centers for the USA and for individual states using GIS tools; 4) Understand how and key reasons why the US population center moved from 1800 to the present; 5) Analyze how and key reasons why states’ population centers moved from 1900 to the present.

     The case study below illustrates another, more advanced (mathematically), approach to using maps to display theorem proof--directly in the spirit of "proofs without words"  (based on material from (Arlinghaus, Arlinghaus, and Harary, 2002).  The illustrated result is a theorem in graph theory.  The mapping employed ArcView 3.1 (ESRI) (and its predecessor software, Atlas GIS), done in and prior to 2002.   In this study, an animated map of the Berlin Rohrpost (pneumatic postal network from the early twentieth century) illustrates the method of proof of the Jordan-Sylvester Center Theorem.  Conventional proof of that Theorem deconstructs an arbitrary graph by removing edges until no others can be removed, thus finding a graphical "center."  The conventional notation is often cumbersome and difficult to understand.  An animated map (Figure 2) makes the proof technique easy to grasp:  it illustrates the proof of the Theorem through seven stages of edge removal until the Theorem's predicted center is reached.  The idea is simple:  peel off the perimeter nodes and edges, in stages, until no more can be done thus leaving one (or possibly two) nodes as the "center."

Figure 2.  Animated Berlin Rohrpost illustrates proof technique of the Jordan-Sylvester Center Theorem of graph theory.

Finding a Path...

     Once one understands the nature of one object, it becomes natural how to ask how to get to different or similar objects elsewhere in space.  In terms of capturing this concept for education purposes,
the lesson summary to the linked, downloadable lesson notes (Kerski, 2010):

The travel problem is an important one in mathematics, because several concepts can be taught and understood through it, such as algebra, measurement, calculations, units of measure, conversions, percentages, time, and distance. Within a GIS environment, the travel problem can be analyzed in the context of real places and distances on maps.

Current modes of travel, that are "fun" as well, can serve to stimulate the curiosity and imagination although the lessons that they teach transcend transport mode as the "current" becomes the "past."   The sport of paragliding offers one such example.  See all three of the links below for lesson utilizing this remarkable form of transport (Kerski,  2008 and later). 
A summary from these links appears below:

Many activities are inherently spatial in nature. Paragliding is an activity in which a person flies in a harness suspended by lines below a large fabric wing. The lack of a motor and complete dependence on the environment makes paragliders extremely sensitive to all spatial aspects of their equipment, terrain, and atmospheric conditions. The wing’s shape is formed by the pressure of air entering its front vents, and the pilot has some control over height and speed by pulling in certain ways on the lines. Air rises as the sun differentially heats ground features, while landforms force wind in different directions. The shape and elevation of the terrain over which the pilot flies, and the way in which the pilot leans affect flight speed, height, and distance. The Earth is a dynamic planet, so the unexpected can happen, but fortunately, paragliders carry an emergency parachute!

     Another case study (Figure 3) illustrates the foundation of graph theory--by showing the Königsberg Bridge Problem set on an historical map of Königsberg of the time (based on material from (Arlinghaus, Arlinghaus, and Harary, 2002).  Follow the challenge originally posed of traversing the seven bridges on one afternoon stroll without ever crossing the same bridge twice--it cannot be done and the animation on the map in Figure 3 illustrates why that is the case as one attempt at solution is traced out in dark red "footprints". In this case, five bridges have been crossed and the pedestrian is back at home. If a sixth is to be crossed (grayed  tracks) then the stroller does not return home. All seven bridges cannot be traversed once each (the reader might wish to make a few other attempts).

Figure 3.  The Königsberg Bridge problem.  Source of base map: Baedeker, Atlas of Northern Germany, 1890. Electronic image supplied by Generations Press,

Both authors of this work (in progress) have years of experience employing spatial tools to visualize mathematics (links to selected works appear below).  Thus, they are embarking on a series of books, designed to implement mapping in the mathematics curriculum at all levels, from the kindergarten through college levels.  (The broad title of this project is:  MatheMaPics, with the "P" as an uppercase letter to emphasize its role in both the word "MaP" as well as in the word "Pics" ("Pictures").)  Because the possibilities are without end, the authors invite others from around the world to participate with them:  as secondary authors who present an actual case study for inclusion, or as extra input folks who offer a link or two of interest.  The Internet makes sharing of information easy to achieve; however, finding all that is out there is not easy.  If you would like to work with us (we reserve editorial rights of inclusion/exclusion) please e-mail one of the two principal authors and indicate your possible interest.  We look forward to producing a wealth of materials using maps to create enthusiasm for mathematics; we also look forward to using the Internet as an inclusive research collaboration tool.*
Related Materials
  • Arlinghaus, Sandra L., Arlinghaus, William C., and Harary, Frank. Graph Theory and Geography: An Interactive View E-Book. Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley and Sons (New York), April 2002. 400 pages depending on screen resolution. This book was a finalist, 2002, in the international Pirelli INTERNETional Award competition, (shows current information only).  It was installed as part of a permanent display in the "Library Covers" exhibit of the University of Michigan Shapiro Science Library display, February 24, 2010.
  • Arlinghaus, Sandra L., and Arlinghaus, William C.  Spatial Synthesis: Volume I, Centrality and Hierarchy. Book 1.  Ann Arbor:  Institute of Mathematical Geography, 2005.
  • Barmore, Frank.
    • Where Are We? Comments on the Concept of Center of Population.  Solstice:  An Electronic Journal of Geography and Mathematics, Vol. III, No. 2, 1992.  Ann Arbor:  Institute of Mathematical Geography.
    • Center Here; Center There; Center, Center Everywhere.  Solstice:  An Electronic Journal of Geography and Mathematics, Vol. V, No. 1, 1994.  Ann Arbor:  Institute of Mathematical Geography.
    • Other
  • Links Arlinghaus:
    • Link to books of the Institute of Mathematical Geography (Part of book on Spatial Synthesis, Vol. I, Book 1--a Pirelli INTERNETional award Semi-Finalist)
    • Link to current issue of Solstice:  An Electronic Journal of Geography and Mathematics
    • Link to Archive of Solstice:  An Electronic Journal of Geography and Mathematics
    • Link to home page of Institute of Mathematical Geography; persistent archive in Deep Blue.
In this lesson, you will understand (1) how to access and format data from the USA County database from the US Census Bureau within a GIS environment; (2) how to analyze these data, specifically about water and by extension, other variables—using GIS and spatial statistics techniques, including regression analysis, hot spot, scatterplot, and others. With updates and improvements December 2009!

*  Kerski notes that on May 24 ESRI rolled out ArcGIS Explorer in a web browser, with some measurement capability in it; see screen below.  So, an expanding number of tools are available!

Author affiliations
  • Arlinghaus:  Adjunct Professor of Mathematical Geography and Population-Environment Dynamics, School of Natural Resources and Environment, The University of Michigan, Ann Arbor MI 48109; President, Arlinghaus Enterprises; Founding Director, Institute of Mathematical Geography; Member Board of Trustees and Executive Committee, Community Systems Foundation.  sarhaus@umich.edu
  • Kerski:  Education Manager, Environmental Systems Research Institute, Inc.- ESRI, 1 International Court, Broomfield CO 80021-3200  USA  E:;  V: 303-449-7779 x 8237 ; F: 303-449-8830


Solstice:  An Electronic Journal of Geography and Mathematics
Volume XXI, Number 1
Institute of Mathematical Geography (IMaGe).
All rights reserved worldwide, by IMaGe and by the authors.
Please contact an appropriate party concerning citation of this article:

Solstice was a Pirelli INTERNETional Award Semi-Finalist, 2001 (top 80 out of over 1000 entries worldwide)

One article in Solstice was a Pirelli INTERNETional Award Semi-Finalist, 2003 (Spatial Synthesis Sampler).

Solstice is listed in the Directory of Open Access Journals maintained by the University of Lund where it is maintained as a "searchable" journal.

Solstice is listed on the journals section of the website of the American Mathematical Society,
Solstice is listed in Geoscience e-Journals
IMaGe is listed on the website of the Numerical Cartography Lab of The Ohio State University:

Congratulations to all Solstice contributors.
Remembering those who are gone now but who contributed in various ways to Solstice or to IMaGe projects, directly or indirectly, during the first 25 years of IMaGe:

Allen K. Philbrick | Donald F. Lach | Frank Harary | H. S. M. Coxeter | Saunders Mac Lane | Chauncy D. Harris | Norton S. Ginsburg | Sylvia L. Thrupp | Arthur L. Loeb | George Kish
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Ann Arbor, MI 48104