Technical Terms:
adjacency, connected, developable surfaces, dodecahedron, edge, eulerian, gnomonic projection, graph, hamiltonian, latitude, longitude, multigraph, node, non-Euclidean, stereographic projection, trail, walk.
Link to Chapter 1 Complement
The first discovery of this beautiful and most useful branch of combinatorial mathematics is due to the Swiss mathematician, Leonhard Euler. He published (Euler, 1736) the solution to the problem of trying to traverse each of the seven bridges of the Prussian city, Königsberg, exactly once and return to the starting place. His solution was a negative one. Not only did he prove that the problem cannot be solved for this case, but also he characterized (in his theorem) precisely those configurations, now called graphs, for which the problem can be solved. The first publication in graph theory resolved a problem in geography. Since that early beginning, many other applications of graph theory to various areas of geography have been derived. It is our purpose to present several of these in this book.
The physicist, Gustavus Kirchhoff, made the second discovery of graph theory 111 years after Euler (Kirchhoff, 1847). He needed to determine the number of spanning trees of a connected graph in order to calculate the amount of current in each "branch" of an electric network consisting of the so-called passive LRC "elements". Here L stands for inductance, R for resistance, and C for capacitance. The network also has a voltage input on one of its branches. His mathematical model had a graph to represent the "topology" of the network. This again can be regarded as a geographic model. The edges can be considered as representing roads and the nodes as places where three or more roads meet. The currents flowing along the edges can be viewed as traffic along the roads.
One of the many independent discoveries of graphs as representations of the structure of a network, in this case the "topology" of a molecule, was made first by Kekule who modeled a benzene molecule by a regular hexagon and benzenoid organic compounds by polyhexes. Later the mathematician Cayley (1874) succeeded in enumerating the alkanes which are the full hydrocarbons of the form C(n)H(2n+2) as a favor for his friend, the chemistry professor at Cambridge. The graph captures the structure of the molecule. Its nodes are chemical elements and its edges chemical bonds. Thus, a molecule, too, can be interpreted as a road network. A closer look at a few examples sets the stage for the remaining essays in the book: as attempts to capture, using the theory of graphs, the structural essence of geographical situations.
Long ago (as early as 1736), citizens of Königsberg (contemporary Kaliningrad) challenged themselves to take a walk around the city so that they crossed each of the seven bridges over the River Pregel exactly once and returned to their point of departure. (In Figure 1.1, the position of the bridge nearest the bottom edge of the map has been altered to fit inside the map). Soon, they discovered that their attempts were in vain; however, the problem is sufficiently complicated that it is not clear from empirical trials why the suggested stroll is impossible.
One attempt at solution is traced out in red "footprints" (Figure 1.1). In this case, five bridges have been crossed and the pedestrian is back at home. If a sixth is to be crossed (green tracks) then the stroller does not return home. All seven bridges cannot be traversed once each (we invite the reader to make a few other attempts); one cannot cross the seventh bridge without crossing an already-crossed bridge.
Figure 1.1. Map of a portion of Königsberg from 1890 showing the seven bridges; one attempt at crossing each bridge exactly once is shown in this animated map. (Source of base map: Baedeker, Atlas of Northern Germany, 1890. Electronic image from Generations Press, www.GenerationsPress.com, website, used with their permission).
Figure 1.2. The Seven Bridges of Königsberg (location of bridge 7 has been modified to fit well on the map) highlighted and labeled on the map. (Source of base map: Baedeker, Atlas of Northern Germany, 1890. Electronic image supplied by Generations Press, www.GenerationsPress.com).
When the apparently complicated connection patterns were partitioned into the two categories, "regions" and "bridges" (Figures 1.2 and 1.3), and then the regions represented as nodes and the bridges as edges (Figure 1.4), a graphical model that captured only the structural elements of the problem emerged. Thus, it became possible for Leonhard Euler to show why such a walk was impossible.
Figure 1.3. Regions of the city are represented as A, B, C, and D. (Source of base map: Baedeker, Atlas of Northern Germany, 1890. Electronic image supplied by Generations Press, www.GenerationsPress.com).
Figure 1.4. The seven bridges of Königsberg appear as a part of a graph. Structural model (multigraph) based on geographic map. Only the nodes (representing regions on the map) are labeled, A, B, C, and D. Edges represent associated bridges 1 through 7.
A trip of the sort desired by the townspeople is a sequence of nodes where each adjoining pair of nodes is joined by a different edge, and all seven edges are included. For example, A, D, B, C, A is a sequence starting and ending at A (returning to the point of departure), but only four edges, AD, DB, one of BC, and one of CA, of the seven are included. In any such sequence of nodes and edges, at each node, an edge comes in and an edge goes out. Thus, the total number of edges at each node must be even (this number is called the degree of that node). In the multigraph of the Königsberg bridges, not just one but also all of the nodes have odd degree (Figure 1.4). Therefore, a traversal is impossible. In fact, Euler proved that a multigraph that comes in one piece (a connected multigraph) has a traversal of the type mentioned (eulerian trail) precisely when the degree of every node is even.
Another question that might be posed about a graph such as the Königsberg bridge graph (Figure 1.4) is the following:
Obviously, in the case of Königsberg, some bridge(s) must be crossed more than once to ensure that each of the seven bridges is crossed at least once. In graph-theoretic terms, how long must a closed trail be to include each edge at least once? This general problem is known as the Chinese Postman Problem (Kwan, 1962) and is named after its proposer, Mei-Ko Kwan (Guan Mei-gu or Meigu, under current transliteration), who has revealed to one of the authors (Harary) that he actually served as a postman during the cultural revolution in China. In his formulation, of course, a postman wants to find the smallest trail he must walk to deliver letters to every house on his route. For a fuller discussion of this problem, see Michaels.
It is important to have replicable, systematic characterizations of real-world environments. When the "real" situation changes, as generally it does, consistent analytic strategy can capture these changes. Indeed, the situation in Königsberg has changed radically, due to political confrontations of various sorts, since the days of Euler. Harary notes (Harary, 2000) from recent field evidence that in 1950 there may have been eight bridges and that modern-day Königsberg (Kaliningrad) has a mere five bridges (bridges 2 and 4 from the above map are missing) while World War II Königsberg had no bridges. Euler's Theorem applies equally to all, independent of real-world positions. "It was the best of times. It was the worst of times." in this Tale of One City (Harary, 2000).
When a graph is used to represent the structural elements of a real-world situation, as it was above, we refer to it as a structural model. Often, it is useful to look at some of the deeper issues represented in the process of creating models, for it is there that one can find parallels to guide applications in various directions. In the case of Königsberg, the representation of the geographic situation as a graphical model enabled Euler to solve the immediate superficial problem of a walk through a particular town. As suggested above, in the comments about Eulerian trails, this representation also enabled him to extend the initial idea to solve an entire class of mathematical problems.
Often, with a bit of imagination, one can use mathematics to capture geographical structure and use geographical analysis to further mathematical inquiry. Such was the case for Königsberg; another classical case involved Hamilton's interest in traversability.
Hamilton was interested in a different type of graphical traversal (Harary, 1969). Instead of having every edge occur exactly once, he wanted each node represented exactly once along the traversal. For instance, in the graphical representation of the cube below (Figure 1.5), the sequence A, B, C, D, E, F, G, H, A, would produce a hamiltonian circuit.
Figure 1.5. Cube represented as a graph. The sequence A, B, C, D, E, F, G, H, A, produces a hamiltonian circuit.
Hamilton himself presented this problem in the form of a puzzle called The Traveller's Dodecahedron: A Voyage Round the World (Biggs, Lloyd, and Wilson, 1976; originally called The Icosian Game), a graphical representation of the dodecahedron with city names (according to Biggs, Lloyd, and Wilson these are: Brussels, Canton, Delhi,...,Zanzibar) attached to each node. See if you can solve Hamilton’s puzzle, using the graph of the dodecahedron in Figure 1.6.
Figure 1.6. Dodecahedron represented as a graph. Each node represents a city. Is this graph hamiltonian? It is, as the pattern of node labeling suggests.
It can be quite difficult to determine whether a given graph is or is not hamiltonian. Here are a few easy examples of graphs that are not hamiltonian (Figure 1.7). To traverse all nodes, and return to the origin, requires passing through a node twice.
Figure 1.7. Graphs that are not hamiltonian; it is not possible to traverse each node exactly once and return to the origin. In the example on the left, it is not possible to traverse each node exactly once; in the example on the right, it is possible to traverse each node exactly once but not possible to do so and return to the origin.
Hamilton shows us that there is more going on behind a simple global issue, such as traversing a set of cities, than meets the eye. Hamilton and Euler saw "geo-graphs" as an initial stimulus for far-reaching research. Research that is more contemporary involves problems of finding tours of nodes and edges, in the Travelling Salesman problem (Dantzig, Fulkerson, and Johnson, 1954) and the Steiner problem (Cockayne, 1967). Careful initial alignment of the geographical and mathematical elements can penetrate in many different directions (Arlinghaus, 1977). It is such alignments and their consequences that are the thrust of this book.
The most famous problem in graph theory, and one of the most enduring in all of mathematics, is the Four Color Conjecture, which only in the past few decades has been proven as the Four Color Theorem (Appel and Haken, 1976). According to May (1965), the conjecture states that
any map on a plane or the surface of a sphere can be colored with only four colors so that no two adjacent countries have the same color. Each country must consist of a single connected region, and adjacent countries are those having a boundary line (not merely a single point) in common. The conjecture has acted as a catalyst in the branch of mathematics known as combinatorial topology and is closely related to the currently fashionable field of graph theory. More than half a century of work by many (some say all) mathematicians has yielded proofs for special cases...The consensus is that the conjecture is correct but unlikely to be proved in general. It seems destined to retain for some time the distinction of being both the simplest and most fascinating unsolved problem of mathematics.
The Four Color problem is expressed in graph theoretic terms by representing each country as a node and the boundary lines between adjacent countries as edges linking adjacent nodes. Appel and Haken's solution depended on the use of modern computing machinery; hence the timeless, yet timely, character of this geo-graphic problem. Other timeless mathematical concerns lurk behind the geographic. We present an intuitive view here of a few of these. Readers wishing more are referred to the literature.
A simple closed curve is one that is topologically equivalent to a circle: it is like a circular rubber band. There is a theorem about simple closed curves, critical in both classical and contemporary approaches to geography, that is due to Camille Jordan (1838–1922). It was first presented in his Cours d'Analyse, a work that was critical in fostering contemporary rigor in mathematics. Jordan's original proof was not valid, and, as was the case with the Four Color Problem, proof of this "obvious" theorem took time. One reason for the difficulty was the slippery character of providing rigorous definition for the intuitively obvious.
Theorem 1.1. Jordan Curve Theorem. A simple closed curve J in the plane separates the plane into two distinct domains, each with boundary J.
The domains into which J partitions the plane are often called the inside and the outside of J. To determine which is the inside, imagine walking along J in a counterclockwise direction with your left arm extended to the side. Your left hand points to the inside of the curve; you are serving as a continuously turning line tangent to the curve. An example of a curve that is not a simple closed curve is the numeral 8. On the counterclockwise traverse, first your left hand points to the bounded "inside" and then when the crossing point is traversed, your left hand points to the unbounded plane as the inside. It is important for mapmakers to know what kinds of curves they are working with. If they do not, they might assign an address to the wrong side of a street. If, for example, all even-numbered addresses are assigned to the "inside" and all odd-numbered ones to the outside of a street that is topologically equivalent (homeomorphic) to a figure 8, then the even-numbered addresses might appear within the bounded lower half of the figure 8 and on the unbounded side of the upper half of the figure 8: contrary to the real-world situation.
This problem is easily overcome. Simply split any complex curve, such as the figure 8, into simple closed curves: view the figure 8 not as a single curve with a crossing point but as two simple closed curves with a common point. When all complex curves are split apart so that all curves on the map are simple closed curves, then no errors of this sort arise. Software that manipulates images is easy to test to see that coloring algorithms acknowledge the Jordan Curve Theorem. Draw a figure 8 in Adobe PhotoShop or Windows PaintBrush. Make sure there are no gaps in the boundary. Use the "fill" tool on the lower, bounded, portion of the figure 8. If the Jordan Curve Theorem has been considered, only the lower half of the figure 8 will fill; if it has not, then the lower half and the entire plane beyond the figure 8 will fill. Figure 1.8 shows the result of this experiment in Adobe PhotoShop 5.5. The software treats the curve as two simple closed curves even though it was drawn as a single closed curve. From the graph-theoretic viewpoint, the software inserts a new node at the geometric intersection point of two lines.
Figure 1.8. A figure 8 decomposed by a "fill" function into two simple closed curves.
To underscore the importance of this theorem, simple closed curves are often referred to as Jordan curves. As suggested above, when complex curves arise in mapping they must be disaggregated into simple closed curves in order to have confidence that paper or virtual maps will match the real-world situation being mapped. The U.S. Bureau of the Census records are topologically encoded to maintain direction and connectivity of objects (without regard for distance). The so-called TIGER (Topologically Integrated Geographic Encoding and Referencing) files developed (Census/USGS) for the 1990 U.S. Decennial Census of the population improved earlier Geographic Base File/Dual Independent Map Encoding (GBF/DIME) files to distinguish inside from outside.
The following geometric analysis illustrates why the Jordan Curve Theorem is true.
Given a simple closed curve J and any two points a and b (not on J) in the plane. The following strategy determines whether a and b lie on the same side, or on opposite sides, of J.
Figure 1.9. Simple closed curve (black) and set of intersecting segments illustrate why the Jordan Curve Theorem is true.
The sphere is the simplest and most useful model of the Earth. Known properties of the sphere then serve as the base on which to build mapping of the sphere. A few of these simple properties are enumerated below. Suppose the center of the sphere is denoted as O. Choose any point P on the surface of the sphere. Join P to O. The line PO will pierce the sphere in another point, P'. The points P and P' are said to be antipodal (from "anti" "pedes": "opposite" "feet"): they are points at either end of the diameter of the earth sphere.
A plane may cut the earth sphere; it may be tangent to the sphere; or, it may not touch the sphere. If the plane cuts the sphere, and passes through O, then the trace the plane makes on the sphere surface is called a great circle. A great circle is the largest circle that can be drawn on the sphere. If the plane does not pass through O then the resulting circle on the sphere surface is called a small circle (smaller than a great circle). An infinite number of great circles and small circles are possible. The shortest distance (geodesic) between two points on a sphere is measured along an arc of a great circle. For antipodal points, geodesics are not unique; for all other pairs of points on the sphere, they are.
Coordinate systems are often useful (Figure 1.10). Great and small circles are conventionally used as coordinates on the earth sphere. Select the great circle that bisects the distance between designated north and south rotational poles of the Earth. Call that great circle the Equator. Introduce a set of small circles north and south of the Equator whose planes are parallel to the equatorial plane. These small circles are called parallels. The Equator serves as a horizontal axis used to reference positions of small circles (2nd parallel south, and so forth). Find a vertical axis. A pencil of planes orthogonal to the equatorial plane contains the north-south equatorial axis as a common line. Each of these orthogonal planes traces out a great circle passing through north and south poles. Halves of great circles going from pole to pole are called meridians. Contemporary use is to designate a half of a great circle that passes through Greenwich, England as the vertical reference line, or Prime Meridian. Positions of other meridians are thought of as east or west of the Prime Meridian.
Figure 1.10. The Earth sphere: coordinates using great and small circles.
Using the scheme described above, one might locate a point on the earth sphere as being at the intersection of the second parallel north and the fifth meridian east. While this description will work, it is relative to the globe in use. An infinite number of parallels and meridians are available. What is one person’s second parallel may in fact not be another person’s second parallel. Hence, this sort of relative coordinate scheme is not portable from one globe to another. Reference to standard spherical measure overcomes this problem, independent of the number of lines actually drawn on the globe. To calculate the displacement of a point P from the horizontal axis of the Equator, measure the number of degrees in a central angle of the sphere one of whose sides lies in the equatorial plane and the other of whose sides goes from O to P (Figure 1.11). This measure of north–south displacement is called latitude. To calculate the displacement of P from the Prime Meridian, measure the number of degrees in a central angle of the sphere one of whose sides lies in the plane of the Prime Meridian and the other of whose sides goes from O to P (Figure 1.11). This measure of east–west displacement is called longitude. These measures are standard from globe to globe.
Figure 1.11. Latitude and longitude displayed in the Earth sphere. (Base figure electronically modified from GeoSystems Global Corporation, http://www.geosys.com/cgi-bin/genobject/mapskills_latlong/tigdd27).
Even though the Earth is not a sphere, the globe is a fine model of the Earth in many regards. It is not easily portable and one cannot see all of it at once. Thus, an enduring problem in both mathematics and geography is to find ways to represent the spherical globe on a plane surface. The transformations that map the globe to the plane are called projections.
Cartographer Arthur Robinson notes, (Robinson, p. 55):
No matter how the spherical surface may be transformed to the plane surface the relationships on the spherical surface cannot be entirely duplicated on the plane. Because of the necessary scale alterations, a number of kinds of deformation involving angles, areas, distances, and directions must or may take place; any system of projection will involve some or all of the following deformations:
If the sphere were unpeeled and flattened perfectly in the plane, that flat image would be an exact representation of the sphere in the plane. There would be no need for more than that; however, the sphere cannot be spread out perfectly on the surface of the plane. The following illustration shows that to be the case.
Stereographic projection of the sphere to the plane employs a center of projection at one point, N, on the surface of the sphere, or generating globe, and a plane of projection tangent to the sphere at a point S, antipodal to N. The transformation is carried by lines of projection: a ray emanating from N, pierces the sphere at point P and continues until it pierces the tangent plane at point P' (Figure 1.12). In this manner, the point P projects stereographically to the point P'. Every point on the sphere, except N, projects to the plane. The line of projection from N is parallel to the plane of projection and therefore does not intersect it. If Euclid's Parallel Postulate were not in force behind this projection, then we might project the entire sphere, with N mapping to an ideal point at infinity. We live, however, in the world of Euclid; the fertile non-Euclidean geometric environment has not seen widespread application in the cartographic or geographic realm.
Figure 1.12. Stereographic projection: generating globe and projection surface.
Stereographic projection is the best we can do; that one extra point cannot be mapped into the plane along with the others because of parallelism. Distortion is severe in this projection farther from the point of tangency. Spacing between successive parallels increases gradually farther from the point of tangency. Numerous compromises may be made in transferring the surface of the sphere (itself a compromise on earth-shape) to the plane. One is to preserve area, but sacrifice shape; this compromise creates "equal-area" projections (such as the linked Lambert Equal Area, Azimuthal). Another is to preserve local shape; this compromise creates "conformal" projections. These might be well suited to navigation (as is the commonly available Mercator projection; normally, one sees the image cropped in one way or another). There are projections that are so-called "compromise" projections; they are neither equal-area nor conformal (the Robinson projection is one example in common use in contemporary mapping software). There are infinitely many ways available to classify projections (Snyder, 1993).
Another style of compromise involves the manner of projection. One is to move the center of projection to any of the infinite number of positions available throughout three-dimensional space. Figure 1.13 shows the difference in the resulting grid in the plane when the center of projection is moved from N (stereographic projection) to the center of the sphere (gnomonic projection). The distortion away from the point of tangency is even more severe in the gnomonic projection than it is in the stereographic projection as is the increase in spacing between successive parallels. Moving the center of projection, to create an infinite number of maps, is but one way to create an infinite set of maps. Another way is to move the position of the tangent plane relative to the projection center. Still another way is to move the plane of the projection to have more than one, or no, intersections with the earth-sphere. Indeed, there are an infinite number of ways of creating these infinite sets. Cartography is a rich and vibrant field, crossing the boundaries of mathematics and geography (see references for a variety of supplementary readings).
Figure 1.13. Generating globe sits on two grids: stereographic projection grid (three heavy parallels in the plane grid) and gnomonic projection grid (two heavy parallels in the plane grid).
The manner in which a projection is made, and the choice of projection, can have profound implications for meaning of output. Indeed, the notion of projection need not be confined to geography. D'Arcy Thompson (On Growth and Form, 1917), transformed one species of fish into another by projecting one grid to another in a manner highly reminiscent of map projection (background fish in image based on cover of reference, reprinted with the permission of Cambridge University Press). Thompson's work in biology in the earlier part of the 20th century is yet another example of interdisciplinary activity that helped to stimulate further geographic work (Tobler, 1961 and other references in the linked list).
Because there is no perfect mapping of the globe to the plane (by the one-point compactification of the plane), there is an infinite number of possible projections of the sphere to the plane. As with Thompson's fish, the pattern of distortion of the underlying grid gives strong clues as to the pattern of distortion in the surfaces draped over that grid: be they fish-flesh masses or land masses.
Another way to create more projections that are new is to bend the plane surface onto which the sphere is projected. "Developable" surfaces are those surfaces that can be cut to unroll perfectly into the plane. Conversely, they are surfaces that can be created by rolling up a segment of the plane. A sheet of paper may be rolled up into a cylinder. One might then consider projecting the surface of the generating globe onto the surface of the cylinder, tangent at a great circle, and then unrolling it. There are many so-called cylindrical projections and classes of projections based on this idea.
There are other developable surfaces. A cylinder may be made into a torus (doughnut) by joining the circular top and bottom ends of the cylinder. Both of these surfaces may be unrolled into bounded portions of the plane and either might have a map projected upon it from a generating sphere. The linked table shows maps in the plane and the torus and indicates the number of colors required to color each in order to distinguish regions that have a common line-segment as boundary. The issue of how many colors are required to color any map was proved on the torus in advance of being proved in the plane.
Another developable surface is formed as follows: take a rectangle and give it a half-twist. Then, join the ends as if to make a cylinder: this action transforms a rectangle to a Möbius strip. The Möbius strip is a developable surface; it is also a one-sided surface. Waldo Tobler noted that if one maps the world on a Möbius strip, that a pin poked through the paper will go in at point P and emerge at the point antipodal to P (Tobler, 1961). A map, Terrae Antipodum, takes the information from the world map on the Möbius strip and puts it back on a flat map (using an equal area projection) (Arlinghaus, S., 1987, reprinted here with permission of the Institute of Mathematical Geography, and earlier). This map found actual application in the world of art (Arlinghaus, S. and Nystuen, J., 1986).
A fourth surface can be developed from a rectangle using the Möbius strip as the base. To make a torus, a cylinder was used as the base, and the ends were glued together (abstractly) to form the torus. Instead, join the ends of the Möbius strip (as if to make a doughnut from it); what is formed is called a Klein bottle (Newman, 1956). Why a map on a Klein bottle might be interesting in a real-world setting is difficult to imagine. That theoretical issue is open. Developable surfaces from rectangles, as mapping surfaces, are of interest in both theory and practice.
Other developable surfaces can be created by distorting the rectangle and then by rolling up the distorted rectangle to use as a projection surface. A cone is an example. Place a cone on the earth-sphere, with the apex of the cone lying on the polar axis of the generating globe (simple conic projection). Use the center of the generating globe as a projection center from which to project part of the surface of the globe onto the surface of the cone. The cone is tangent at a small circle; the projection becomes increasingly distorted as one moves away from the circle of tangency in the map surface. One way to improve this situation is to allow the cone to intersect the surface of the earth—then distortion increases as one moves away from each of these small circles in this "secant" projection. These small circles are referred to as "standard parallels" and a brief description of any conic projection should tell you the location of the standard parallels. When the cone is positioned as above, distortion is reduced in east/west direction. Thus, this sort of projection is better suited to a landmass or a nation with greater east/west extent than north/south extent, such as the United States. This observation suggests that what is critical in making good choices for projections is to let the projection fit the underlying area as well as is possible.
What "as well as" means will of course vary from project to project, depending on emphasis. The infinities of projections available suggest that at least one is available to suit the needs of any project; it may not, however, be easy to find. To learn more, we refer the reader to the cartographic literature.
Adjacency is a concept that is both graphical and geographical. In a geographical setting, adjacency often leads one to ask about clustering of information, or its lack. One tool that is important for making base maps that show clusters is the dot or dot density map. Typically, these maps might show 1 dot representing 200 people in some spatial unit or 1 dot representing 1 percent of the population of some spatial unit. While these can be powerful tools, they are also tools that are often misunderstood. To be effective, the dot scatter should be randomized at a relatively local scale and then viewed for pattern and clustering at a scale more global than the scale of randomization. In Figure 1.14, dot scatter has been randomized at the U.S. Census Block Group level; it is then viewed successively at the U.S. Census Tract level (tracts are larger than block groups) and then at the state level. Any pattern in the scatter of dots within individual block groups is meaningless: the scatter is random. Within tracts, broad clusters are evident; and, even more obviously, within the state clusters of dots delineate urban areas (Figure 1.14).
Figure 1.14. Southeastern Michigan. Dot scatter is random at the block group level (black polygons). Clustering is sensible at the tract (red polygons) and state level.
The dots in the animated map in Figure 1.14 are simply counts of population: 1 dot represents 200 people according to the 1990 U.S. Census of the Population. When other data sets are considered, thematic maps that reveal various aspects of demographic pattern may be the result. Maps motivate (Arlinghaus S. and Arlinghaus W., 1997)!
Figure 1.15. Animated map racial polarization in Wayne County, Michigan.
Source of data and selected polygons: United States Bureau of the Census, 1990. Source of base maps: Environmental Systems Research Institute (ESRI). Maps reflect accuracy standards imposed by ESRI and the Census.
The above maps of Wayne County, Michigan, Figures 1.15, show considerable clustering of block groups according to dominant racial type. Census block groups that are predominantly populated by white population (purple) tend to be directly adjacent to block groups of similar character; census block groups that are predominantly populated by non-white (green) population tend to be directly adjacent to block groups of similar character (Figure 1.15). The racial polarization between white and non-white is as evident on the map as it is to county residents. Indeed, a similar situation exists for second-order adjacency patterns (neighbors) as well (Figure 1.15). The buffer zones surrounding Detroit appear almost as topographic ridges in the adjacency pattern of racial relations.
Graphical and Geo-graphical intersections, such as those suggested by these classical and modern examples, will form the backbone of this book. Some examples will aim at careful alignment of basic material; others will extend the alignment into more complex directions. All will make ample use of graphic materials, from historical to contemporary maps.