|Coordinates||Series A, No. 7|
|How to Map a Sandwich:
Surfaces, Topological Existence Theorems and the Changing
Nature of Modern Thematic Cartography, 1966-1972
Persistent URL for citation: http://purl.oclc.org/coordinates/a7.htm
of Publication: March 15, 2009
John Hessler (e-mail: firstname.lastname@example.org) is Senior Cartographic Librarian at the Library of Congress, Geography and Map Division, 101 Independence Ave. SE, Madison Bldg., Room LMB01, Washington D.C. 20540-4650.
Abstract: This paper is meant to be the beginning of a project that examines the use of abstract mathematics and the changing ontology of mapmaking in the early years of the development of computer cartography. The history of the conceptual developments that took place during this revolutionary period in the history of mapmaking is both controversial and incomplete. Much of the primary source material has yet to be examined by historians, residing as it does in obscure journals, government archives and in obsolete software. This study provides a look at one example of this conceptual development in the early years of computer cartography through a close reading of two papers on existence theorems published by the Harvard Laboratory for Computer Graphics and Spatial Analysis. It attempts to highlight the changing conceptual and mathematical foundations of mapmaking during this period and in doing so provides a case study for the difficulties that historians of modern cartography face in researching this critical period in its history.
Keywords: Existence theorems; computer mapping; surfaces; topological data structures; Harvard Laboratory for Computer Graphics and Spatial Analysis.
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Introduction: Abstraction in Early Computer Cartography
What does it mean to obtain a new concept of the surface of a
The science of cartography in the period after World War II saw revolutionary changes in its methods, in its data and in its conceptual foundations. New sources of data from satellites, the development of new numerical and mathematical techniques and the creation of computers and the graphical displays that accompanied them, all changed the science of cartography in ways that historians are only just beginning to come to terms with. In the 1960s and 1970s some of the most important work being accomplished in cartography from the mathematical standpoint had to do with the topological properties of surfaces, their relationship to geographical and spatial analysis, and the ontology of cartographic objects. The Harvard Laboratory for Computer Graphics and Spatial Analysis was a hotbed of such work and was led into new areas of research by the ideas of the theoretician William Warntz (1922-1988). During this critical period in the history of cartography Warntz’s group and others at the Harvard Lab took a research path that essentially rethought the meaning of what it meant to create a map. In these years researchers there, and in other venues, let their imaginations run wild and experimented with formerly untapped areas of mathematics and computation, planting one of the many seeds that grew into modern Geographic Information Systems (GIS). It was an era that saw the melding of mathematics with new geographical concepts and in which increasing levels of geometric and mathematical abstraction would become an integral part of the visual and pragmatic science of cartography.
The history of the conceptual developments in cartography during this era are especially difficult to research, as much of the primary source material is still to be examined by historians, residing as it does in obscure journals, in government archives, in old computer programs and on obsolete hardware. The following paper is meant to be an example of those difficulties and makes no claim to completeness in the subject matter it examines, as the early history of computer mapmaking and its mathematical foundations is both controversial and incomplete. Rather, it is meant as a case study, a beginning point, in the larger project of assessing the changing role of mathematical abstraction in the early years of computer cartography and role it played in how modern maps are created and perceived.
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Figure 1: Surface model of Warntz’s
1960 Population Map
During the 1960s and 1970s new levels of mathematical abstraction were especially evident in research that centered on thematic cartography. While many researchers in the field were looking at the numerical properties of thematic maps and at the statistical relationships in the data they displayed, William Warntz of Harvard looked to understanding the topology of the surfaces that the data formed. He recognized that the most important properties of surfaces from a mathematical point of view had nothing to do with numbers and specific values, but rather with the surface’s invariance under transformations. Warntz described the relationship of the topological properties of a surface to cartography in a number of papers that adopted a terminology and methodology built on the work of the mathematicians Arthur Cayley (1821-1895)  and James Clerk Maxwell (1831-1879). Warntz generalized Cayley’s vocabulary that described the contours on geographic and topographic maps for use in the description of surfaces. Cayley’s lexicon categorized particular features on maps that he called summits, pits, immits, and other terms that Warntz used in describing the geometry of the surfaces of thematic maps. In Cayley’s vocabulary Warntz found a natural and geometrically descriptive lexicon for pointing out particularly interesting features of surfaces that had analogues on topographic maps, resembling contours, singularities and extrema (maxima and minima).
In the preface to “Geography and an Existence Theorem” Warntz briefly wrote about the lack of attention given to surfaces in the fields of geography and cartography:
The topological nature of the surface has not received as much attention and the requirements (Begin Page 4) that this places upon the theories of spatial process in geography have not been recognized, to the disadvantage of those theories.
Warntz was particularly interested in mapping and graphically displaying thematic surfaces and adopted a macro-geographical theoretical perspective that led not only to fundamental mathematical breakthroughs but also yielded philosophical insight into the nature of the objects described by the ‘science’ of cartography.
Today geographers, regional scientists, and others are taking the geo in geometry literally, and the study of earth related surfaces and paths has now been expanded far beyond its original application to such things as land form, contour mapping, drainage patterns, temperatures, pressures, precipitation, and the like in physical geography alone. The modern scholar conceives of surfaces based also on social, economic, and cultural phenomena, portraying not only conventional densities but other things such as field quantity potentials and also probabilities, costs, times, and so on.
Warntz goes on to state that all of these quantities can be mapped:
Always however, these conceptual surfaces may be regarded as capable of overlying the surface of the real earth, and the geometric and topological characteristics of these surfaces, as transformed, could thus describe aspects of the geography of the real world.
As an example of this type of thematic cartography, Warntz includes in the paper his map of the potentials of population as a three-dimensional model (figure 1) and also reproduces his “Map of the Potentials of Population” that he published with the American Geographical Society (figure 2). Warntz describes the map as showing a true “macrogeographic” quantity, one that varies continuously over the surface of the map. The fact that the surface could be portrayed as a continuous function, and not just composed of discrete values, opened up new areas of geographical analysis and research quickly developed around finding efficient algorithmic ways of smoothing surfaces and calculating their properties.
Much of Warntz’s research on this type of thematic cartographic analysis was inspired by the work of John Q. Stewart (1894-1972,) who attempted to formalize the study of population distribution and its cartographic nature in a series of papers in the late 1940s and early 1950s, which drew upon physical science models and potential theory. Stewart, writing in the American Journal of Physics, described variables that could be mapped thematically as “demographic indices” that had attraction, interactance and influence on population at a distance. One article, aptly named “The Development of Social Physics,” had some influence on geographers and thematic cartographers at the time, and suggested that thematic variables could be treated mathematically in cartography with the same sort of equations that scientists used to describe Newton’s Law of Gravitation. Warntz thought Stewart’s contribution to thematic cartography (Begin Page 5) to be important enough that he included him with other more recognizable geographers, like Ptolemy, in his book Breakthroughs in Geography, which he wrote with Peter Wolff in 1971.
The use of geo-potentials is just one type of surface analysis that was pioneered by Warntz for use in thematic cartography. He and his students would exploit other mathematical and physical analogs as they rethought the types of surfaces and variables that could be thematically important for geographic analysis. These explorations would carry them into more and more abstract territory and, as we shall see, into more and more original cartographic applications.
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Figure 2: Potentials of Population
by William Warntz, 1960.
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II. How to Map a Sandwich: Existence Theorems and the Nature of Thematic Cartography
From what rests on the surface we are led into the depths.
Given any three sets in space, each of finite outer Lebesque measure, there exists a plane which bisects all three sets, in the sense that the part of each set which lies on one side of the plane has the same outer measure as the part of the same set which lies on the other side of the plane.
This statement of the Sandwich Theorem at the beginning section of Warntz’s introduction to “The Sandwich Theorem: A basic one for geography”  hardly seems at first reading to have import to the history of thematic cartography. Warntz however, thought it to be of “paramount importance.” In the introduction Warntz asks us to assume the earth to be a solid sphere and to picture the infinite number of planes that one could choose to bisect its volume. After presenting this rather straightforward set of bisecting planes, he then moves to discuss other types of partitioning of the earth’s surface that are more abstract and employ less physically imaginable variables. He says that:
The idea of partitioning geographic
and social variables and to mapping their distribution on the
surface of the earth has always been one of the mainstays of
thematic cartography and the above theorem, though highly abstract,
gives hope that any group of sets on a map could be partitioned
into “regions” of
equal size. The name of the Sandwich Theorem comes
from the most straightforward example of its application.
If one builds a sandwich of bread, cold cuts, cheese,
and then spreads it with butter, is there a way to
cut all of these elements that make up the sandwich precisely
in half with a single cut of a plane? The theorem implies
that the cut would produce two sections of the sandwich
each of which has precisely half of the bread, half
of the cold cuts and half of all of the other things at
the same time no matter how they were originally oriented
or distributed on the sandwich.
Even this relatively simple computer assistance in finding spatial solutions to the given existence theorem demonstrates the possibility of using computer mapping programs to find spatial solutions to that whole class of existence theorems having explicit or implicit spatial connotations. After initial problems related to adopting the particular existence theorem to a computer solution are resolved, computer mapping techniques are capable of quick and accurate spatial solutions with a minimum of human manipulation. This is another indication that the problem solving capabilities of computer mapping are at least as diverse and effective as are its proven capacities graphically to represent stores of spatially ordered information. These capabilities lay largely untapped, waiting to be exploited.
Begin Page 10Lindgren in his analysis decides to take a surface-oriented approach to the problem and begins by looking at the geometry of the theorem, “It just happens that geometry provides such possibility and, if one wished to put claim on discovery, perhaps one should recognize this priority to geometry as a whole. As it turned out, the sandwich theorem, under the light of geometric synthesis, is only a conclusion.” What Lindgren is describing here is an approach to problems that the Harvard Lab would often experiment with, namely, the development of algorithms and cartographic approaches to solving problems previously known only as results in pure mathematics.
Figure 3: Bisection of two spatial
distributions by a plane. Figure 3 from Sandwich Theorem
Lindgren, whose main research interest at the time centered on four-dimensional geometries, used a geometric analog of angle bisection to picture the partitioning of cartographic surfaces. In figure 3 Lindgren describes the layering of two surface distributions and the plane that bisects them. The top layer represents schematically the surface distribution of population. The top surface in the diagram is not flat, but is curved in places to show the value of the population much like the potential surfaces in Warntz’s population potential map. The bottom distribution is geographic area, say, for example, of the United States or any finite region. What follows in the paper is a complicated treatment of the notion of angle and the possible iterative solutions to the problem of bisecting all of the distributions simultaneously (figure 4). Lindgren’s solution and that of Eduardo Lozano, who also writes in the Sandwich Theorem paper, corresponds to a series of geometric constructions that divide the various distributions in half. The solution to the overall problem was found using an iterative process that calculated various partitioning schemes for each distribution until one is found that corresponds to the partition for all of them.
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Figure 4 Lindgren’s Schematic
of the distribution as angular bisector
The iterations produced
not an exact solution to the partitioning problem but
an approximate solution that could be made more accurate
depending on the amount of computer time one was willing
to spend on the calculation. The actual computer code
for the program to accomplish the partitioning was
authored by Katherine Kiernan and used a common numerical
programming technique of searching for a minimum by
constantly reducing the space that was being analyzed.
It is not known how much computer time was expended,
but multiple iterations must have been necessary since
there were 3,070
counties used in the calculation. The
approximate nature of the solution can be seen in figure
5, which is the computer printout from the Sandwich
Theorem paper, and shows the percentages of the three
thematic distributions that were partitioned. If this
were an exact solution all the values for area, income
and population would read 50%. Figure
6 shows the map of the line dividing the three thematic
variables from the Sandwich Theorem paper.
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Figure 5: Computer printout from Kiernan’s
program showing percentages of each of the partitions.
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Figure 6: Map from the Sandwich Theorem Paper showing
the line that simultaneous divides, area, population and
income for the continental United States.
Although much of what we have discussed in this paper so far may appear to the historian of cartography to have been a mere exercise in mathematics, it had much more effect on the future of cartography and the philosophical nature of what cartography would become than is generally realized. Warntz, in the preface to the Existence Theorem paper discussed in some detail the philosophical and conceptual shifts that would be brought about by this new ability to import abstract ideas from mathematics into cartographic analysis. After a description of the Borsuk-Ulam theorem , another existence theorem, he writes about this new tool in cartography:
Begin Page 14This definition of mapping is radically innovative and extremely interesting. Cartography and mapping are no longer seen as just printed planar representations of the real world, but are algebraically defined and treated as applications of the theory of sets. Maps have become more abstract and axiomatic, but because of their ability to be algebraically manipulated, they have also become more useful tools for complex geographical analysis. This way of looking at maps was to have important implications for the development of GIS and other forms of computerized mapping. Warntz goes on:
Cartography is the geographical example in the application of this concept. That which is ordinarily called a map by geographers and laymen is technically “a graphical image of a mapping.” Whatever useful roles graphics plays in science generally also can be claimed for cartography with relation to geographical science. This is especially true now that most geographers increasingly employ geometry as an appropriate vehicle to carry their discipline.
Thus, cartography was conceived of by Warntz in much the same way that the graph of an equation is conceived of in the mathematical sciences, as a tool for analysis. Warntz saw the map as an algebraically changeable image that is a visual representation of the abstract topology of the surfaces that made them up. For him this did not diminish the role of cartography but expanded it. Warntz described his conception of the nature of a map best when he said:
There is already, of course, an accumulated stock of knowledge and experience concerning maps as stores of spatially stored information. We hope, however, to examine the expanded roles that mapping seems well suited to play in the sciences viewed from the standpoint of theoretical cartography and in the disciplines employing its models for decision making purposes.
III. Conclusion: Just the Beginning
--If the intended application of mathematics is essential, how about parts of mathematics whose application
—or at least what mathematicians take for their application—is quite fantastic?
I would like to thank the
members of the Philosophy of Mathematics reading group
at Columbia University who, during our sessions last May,
read the papers on the Sandwich Theorem, and gave helpful
information on the ontology of set theory and spatial representation.
I would also thank the two anonymous reviewers who made
helpful comments on this article, along with David Allen,
Ralph Erhenberg, and Ronald Grim who did the same. And
finally, a mention of Catherine Delano-Smith, editor of Imago
Mundi, for discussions
on the nature and ontology of the type of material that
makes up the history of cartography.
2. The changes in cartography that occurred during this period fit in well with the philosophic model for the types of changes that occur during periods of rapid and radical epistemological change in the sciences discussed in Thomas Kuhn’s The Structure of Scientific Revolutions, (Chicago: University of Chicago Press, 1962). Kuhn’s philosophic models of paradigm shifts and lexical change are a good starting point for all those trying to discuss the nature of the changes that took place in cartography in the post-war period.
3. The research being accomplished currently for the 20th Century volume of The History of Cartography, edited by Mark Mommonier of Syracuse University and to be published by the University of Chicago Press, (Begin Page 16) should provide a starting point for future historians. The current author’s article in that volume “Mathematics and Cartography” attempts to broadly deal with the more specific points highlighted in this study.
4. For a more complete look at the history of early GIS see Timothy W. Foresman editor, The History of Geographic Information Systems ( Upper Saddle River, NJ: Prentice Hall, 1998). The contributors to this volume discuss the evolution of GIS at various University and Government agencies in Canada and the United States. One need only look at the diagram of the historical pathways and connections involved in the genesis of GIS on page 7 of Foresman’s Introduction to understand the complex problems faced by historians in writing its history.
5. For purposes of this paper a thematic map will be defined as one that attempts to map the characteristics of a geographic phenomenon to reveal its spatial order and organization. See Judith Tyner, Introduction to Thematic Cartography, Englewood Cliff, NJ: Prentice Hall (1992), 10-11.
8.William Warntz, preface to Stephen Selkowitz, Geography and an Existence Theorem: A Cartographic Solution to the Localization on a Sphere of Sets of Equal-Valued Antipodal Points for Two Continuous Distributions with Practical Applications to the Real Earth. Harvard Papers in Theoretical Geography 21 (Cambridge, Mass. : Laboratory for Computer Graphics and spatial analysis, Center for Environmental Design Studies, Harvard University, 1968), i
12. There are many examples of this type of research being conducted at the Harvard Lab, including work by Frank Rens from the Harvard School of Architecture. Rens worked to find methods for creating smooth surfaces from data sets that had only discrete point observations. See Frank Rens “The Smoothing of Topographic Surfaces”, in Selected Projects: Harvard Laboratory for Computer Graphics and Spatial Analysis, ed. Carl Steinitz, 1970.
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16. Nicholas R. Chrisman, “Concepts of Space as a Guide to Cartographic Data Structures,” in First International Advanced Study Symposium on Topological Data Structures for Geographic Information Systems, Volume 7, Spatial Semantics: Understanding and Interacting with Map Data. (Cambridge Mass.: Laboratory for Computer Graphics and Spatial Analysis, 1978) 1-19.
18. This would spur a debate in philosophical circles regarding the nature of geographical objects that continues today. See Achille Varzi, “Introduction: Philosophical Issues in Geography”, Topoi 20 (2001), 119-130, “Vagueness in Geography”, Philosophy & Geography 4 (2001), 49-65, and “Fiat and Bona Fide Boundaries” (with Barry Smith), Philosophy and Phenomenological Research 60 (2000), 401-420. For an more geographically oriented take on these debates see J.F. Raper, “Spatial Representation: A Scientists Perspective”, in Paul E. Longley et. al., Geographic Information Systems: Principles and Technical Issues (New York: John Wiley and Sons, 1999) 61-80; John Hessler, “From Topology to Mereology: Notes Toward a Philosophy of Cartography” Topos (forthcoming 2009).
22. William Warntz “Introduction” to Warntz et. al., The Sandwich Theorem: A Basic One for Geography. Harvard Papers in Theoretical Geography 44 (Cambridge, Mass.: Laboratory for Computer Graphics and Spatial Analysis, Harvard Univ.1971), i.
23. The Sandwich Theorem volume is divided into several parts. The first is Warntz’s Introduction followed by a full translation of Hugo Steinhaus’ original paper; “A Geometrical Analysis Concerning the Sandwich Theorem” by C. Ernesto Lindgren; “Solution to the Sandwich theorem with Two Distributions by Approximation” by Eduardo Lozano; “Euclidean Model for the Sandwich Theorem,” by Luis Bonfiglioli; and “Implementation: Computer Program and One Example of the Halving of Three Distribution,” by Ernesto Lindgren and Katherine Kiernan.
27. For historical background on this debate see Mathieu Marion , “Kronecker’s Safe Haven of Real Mathematics” Quebec Studies in the History of Science 1 (1995): 189-215; David Corfield, Toward a Philosophy of Real Mathematics (Cambridge: Cambridge University Press, 2006); and Jeremy Gray, Plato’s Ghost: the Modernist Transformation of Mathematics (Princeton, N.J.: Princeton University Press, 2008).
36. The importance of the role of the Harvard Lab itself in the development of GIS is a controversial issue but the work accomplished at the Lab was very much in keeping with developments going on elsewhere in computer cartography. Applications of pure mathematics of the type we are discussing in this paper were attempted by many researchers and the discussion here is merely an example. Those interested in a broader look at primary research from a number of different labs should consult the publications found in the Proceedings of the First International Advanced Study Symposium on Topological Data Structures for Geographic Information Systems, 8 volumes, Geoffrey Dutton editor, (Cambridge, Mass.: Fellows of Harvard University, 1979).
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42. John Hessler, “Archive Fever: Challenges in Preserving Conceptual and Foundational Cartographic Materials from the Twentieth Century and Beyond,” Journal of Map and Geography Libraries 3 (2007): 79-95.