Sandra Lach Arlinghaus

Geometry of Boundary Exchanges

The geometric concept of self-similarity, from fractal geometry, may be used to compress portions of long, linear boundaries separating contrasting landuse types, like lake and shore or forest and meadow, into small, compact regions. This process is applied to maps of an actual location on the northern shore of Lake St. Clair and to a hypothetical map of a national forest. Using fractal geometry to determine boundaries can offer visitors to such regions the opportunity to enjoy exchange across them and can simultaneously minimize the potential for overall damage to broad expanses of shore or forest. V ISITORS to forests and to shorelines share a desire to experience the variety that the landscape has to offer, to do so in relative privacy, and to have direct access to the sites they come to visit. Of concern are finding a way to offer a full measure of public access and minimizing the potential for environmental damage. The fractal concept of self-similarity is such an approach to this issue of maximum exposure in minimum space. An analogous problem arises in telecommunications with the introduction of facsimile transmissions and other data-intensive media. The problem is to increase data transmission without exceeding capacity. Fractal methods are employed to achieve data compression, so that additional information may be transmitted within a fixed capacity (Barnsley 1988). We seek ways to increase exchange with the environment without exceeding its capacity for exposure to people. In this article we apply the fractal compression of boundaries used in electronic systems to the terrestrial environment. Previous research has shown such geometric boundary compression to be an effective means of reducing the entire geometry of central-place theory to single fractal transformations; in such cases, the compression comes about by finding a fractallike form to fit a landscape form (Arlinghaus 1985; Arlinghaus and Arlinghaus 1989). We begin our study with a brief explanation of self-similarity. We then illustrate how self-similarity might be applied in seashore and forest settings to con- struct interdigitated patterns of land and water or of forest and meadow, by compressing the boundaries separating them within a small parcel of land. This approach, we believe, can help resolve the dilemma of maximizing contact between humans and natural landscapes in a minimum of terrestrial space and with a minimum of environmental hazard. SELF-SIMILARITY

An Application of Graphical Analysis to Semidesert Soils

Geometric models often contain elements of complex geographic processes. The shape of the curve selected to model a specific process can be crucial in determining interpretations. To illustrate this relationship we use the graphical analysis of chaos theory as a geometric model and apply it to the complex process of semidesert-soil production. This application suggests strategies for choosing when human intervention to improve soil production might be appropriate. A SYMMETRIC curves are useful as models of various real-world settings. All too often, however, the detail of the exact curve shape is of little concern in constructing these models, independent of what they might represent. In this article, we use graphical analysis (Feigenbaum 1980) to demonstrate that exact curve shape is important and that small variations in curve shape can make big differences in the geometric dynamics and the consequent interpretation of a geometric model. The case study is the application of graphical analysis to the study of semidesert soils. Curve shape can determine how matter and energy spread within a system. The different surface shapes of the tiger and the leopard may determine, for example, that the tiger will have stripes and that the leopard will have spots. When reactions cause diffusion rates of pigmentation in animal coats to vary, different spatial patterns, such as stripes or spots, emerge as standing waves of translation in the underlying surface (Xu, Vest, and Murray 1983; Murray 1988). Seiches, which are standing waves of oscillation rather than of translation, might form stripes on lakes and bays (Mosetti 1982). The position and periods of seiches depend on lake depth and coastline shape. Coastline shape might even cause reactions that influence the diffusion pattern of nearby urban traffic. The position of standing waves of all sorts is controlled by boundary shape as well as by the spread of matter or energy. In sum, shape matters. Graphical analysis, a tool from mathematical chaos theory, offers a way to understand how small geometric changes can force large geometric differences. Such analysis rests on an ordering of events, in which the output of one stage serves as the input for the next stage. Because graphical analysis cleverly uses the line y = x as a surrogate axis, it produces an easy-to* DR. ARLINGHAUS is the director of the Institute of Mathematical Geography, 2790 Briarcliff, Ann Arbor, Michigan 48105. DR. NYSTUEN is a professor of geography and urban planning at the University of Michigan, Ann Arbor, Michigan 48109. DR. WOLDENBERG is an associate professor of geography at SUNY Buffalo, Amherst, New York 14261. This content downloaded from 157.55.39.112 on Wed, 07 Sep 2016 04:30:35 UTC All use subject to http://about.jstor.org/terms

Geography of City Terrain Based on Bus Routes

A geography of city terrain, based on bus routes, supplies information about steepness that is vital to maintenance of equipment. This article presents a technique to determine vertical profiles of bus routes to evaluate topography and service demands in terrain stress on vehicles. A N arbitrary abstract bus-route network, superimposed on the undulating surface of a city, would logically follow lines of lowest topographic gradient when minimal terrain-imposed stress on equipment was a factor. The resultant routing strategy would fail to provide effective service to the population, because bus-route networks should also follow service gradients. The key issue centers on how to thread routes through an urban area so that they touch a set of high-demand locations like places of employment and then connect with areally spread residential zones. Another way to pose the question is how can the linear form of a bus route get as close as possible to an area, or how can a one-dimensional linear form be routed along a winding path to fill a portion of a two-dimensional market area? The latter question appears from general context to be one that is likely to respond to an abstract approach based on fractal geometry, and we conducted an empirical study of one set of bus routes. Bus-route vertical profiles were viewed as wiggly lines, attempting to fill some sector of the market area. To understand how these routes might fill space, we developed a procedure to measure the displacement of a bus-route vertical profile from a topographic baselevel. This base was established as a slope between adjacent points like river crossings on the profile. These points force a change in elevation of a route as the bus travels from origin to terminus. Routes along paths that would maintain baselevel are analogous to the arbitrary net dropped on a city. Displacement of all others from this minimal routing response is a function of the need to serve distinct points with high demands for service as well as areally spread residential markets with different demands for service. With this broad idea expressed in a terrain context, the test case begins by displaying briefly a general procedure to classify city terrain at a 1:250,000 scale; the results of applying this procedure to Ann Arbor, Michigan, as well * This research was partially supported by a grant from the Urban Mass Transportation Administration, U.S. Department of Transportation, to the University of Michigan Transportation Research Institute. * DR. ARLINGHAUS is director of the Institute of Mathematical Geography, 1441 Wisteria Drive, Ann Arbor, Michigan 48104. DR. NYSTUEN is a professor of geography and urban planning at the University of Michigan, Ann Arbor, Michigan 48109. This content downloaded from 157.55.39.225 on Mon, 12 Jun 2017 17:54:20 UTC All use subject to http://about.jstor.org/terms THE GEOGRAPHICAL REVIEW as San Francisco, Washington, D.C., and Detroit illustrate extreme positions in the taxonomy.1 Although this classification is useful to make broad terrain comparisons among cities, it does not permit identification of variations in elevation that result from residential-service needs to be made at the city level. Therefore the report proceeds with a comprehensive analysis of the terrain in Ann Arbor at a scale of 1:24,000, based on vertical profiles of local bus routes. These profiles are partitioned by a discrete set of critical values that force variation in elevation into continuous intervals over which routeelevation displacement from a baselevel is measured. The study concludes by mapping the results of the displacement and discussing the implications of a geography of terrain based on bus routes for transit managers. TERRAIN CLASSIFICATION Terrain with a gradient in excess of 8 percent causes problems for virtually any type of vehicle, and most railroad tracks follow a grade of less than 2 percent.2 Thus a city with a large percentage of 8 percent grade might be classified as steep, one with terrain largely less than 2 percent as flat, and all others as intermediate. To determine these percentages for a specific city, we employed the following technique. The city boundary was represented as a circle to facilitate classificatory comparisons among cities; allometry was used to represent the city as a circle with a radius proportional to population. Then evenly spaced lines were used to sample the unevenly spaced contour lines within the allometric circle and to classify the underlying terrain as steep, intermediate, or flat.3 The mechanics of analyzing the terrain within a circle required sampling of the spacing between the pattern of contour lines. Generally contour lines are wiggly; however, locally all are topologically equivalent to short straightline segments. Thus a sequence of parallel short straight-line segments became a contour comb to disentangle contour lines. When the segments were spaced to represent 2 percent and 8 percent grades on a 1:250,000 topographical map with a 50-foot contour interval, they were in a form suitable for use with a topographic map of the same scale (Fig. 1). When the contour combs were applied to the pattern of contour lines representing topography on U.S. Geological Survey maps, the spacing in a set of lines appeared to fall mostly between the 2 percent and 8 percent pattern, but much of it was closer to the 2 percent end (Fig. 2). Ann Arbor was found to have terrain of Sandra Arlinghaus and John Nystuen, Terrain Effects on Bus Durability, report prepared for University of Michigan Transportation Research Institute in cooperation with Urban Mass Transportation Administration, U.S. Department of Transportation, National Technical Information Service, Springfield, Va., 1986; Sandra Arlinghaus and John Nystuen, Terrain Effects on Bus Maintenance Performance, Transportation Research Record, forthcoming. 2 Edward H. Hammond, Analysis of Properties in Land Form Geography: An Application to BroadScale Land Form Mapping, Annals of the Association of American Geographers 54 (1964): 11-19; Edward L. Ullman, The Railroad Pattern of the United States, Geographical Review 39 (1949), 242-256. 3 Waldo R. Tobler, The Spectrum of U.S. 40, Papers of the Regional Science Association 23 (1969): 4552; Hammond, footnote 2 above. 184 This content downloaded from 157.55.39.225 on Mon, 12 Jun 2017 17:54:20 UTC All use subject to http://about.jstor.org/terms CITY TERRAIN AND BUS ROUTES

Category theory in geography

Abstract Is mathematical category theory a unifying tool for geography? Here we look at a few basic category theoretical ideas and interpret them in geographic example. We also offer links to indicate how category theory has been used as such in other disciplines. Finally, we announce the direction of our research program on this topic as a way to facilitate the learning, and maintenance of learning, of GIS software – and in the spirit of Quaestiones Geographicae, invite debate, comment, and contribution to this program in spatial mathematics.

Environmental Modeling with GIS

Keywords: Environnement ; modele ; modelisation ; GIS Reference Record created on 2005-06-20, modified on 2016-08-08

Structural Models in the Subterranean World

FOR many years geographers have employed various indexes based on the ideas and concepts of graph theory to measure connectivity and other facets of network analysis. They have noted the limits of these indexes and have sought other tools, some, such as those drawing on combinatorial topology, intimately related to graph theory but others farther afield (Garrison 1960; Nystuen and Dacey 1961; Tinkler 1988). One difficulty with applying mathematics to real-world settings is that strategies which serve in laboratory sciences often do not function equally well in the real world. Thus indexes that can be used over and over in carefully controlled laboratories may not be well suited to complexities in applications that lack such controls. Modifications of theoretical tools that yield good results for one project might not be suited when further modified for another project. Thus it is often preferable to return to the original theoretical underpinnings and adjust the mathematics to fit the real-world situation at hand. In such situations tailor-made mathematical suits are superior to those taken from the racks of traditional models. There are geographical examples of this art, and there is room for more (Harary 1969; Arlinghaus 1994; Arlinghaus, Arlinghaus, and Harary forthcoming). In the latter spirit I offer two simple examples that display both the power and the elegance that carefully constructed structural models can bring to geographical analysis. PROBLEM OF LAYERS Any system, such as a transportation system, that is forced to have different physical levels for entry and exit becomes a target for policy difficulties of various kinds. Elevated and subterranean trains are a response to providing efficient commuting in a densely populated environment: to prevent collisions these networks generally have a number of different horizontal layers. Clearly it is inconvenient for passengers to move vertically as well as horizontally during transfers from one route to another. Although the extra dimension removes a collision hazard between trains, it increases the potential for collisions among passengers. Different layers add a host of security problems for security personnel on the lookout for muggers and thieves who prey on a population closely confined underground. Unless elevators or ramps are installed, multiple-level stations exclude transfer possibilities for individuals confined to wheelchairs. In some locales funding is directly tied to the extent of handicap access to be provided in a public project. Different access layers offer a way to overcome congestion and collisions; however, their mere presence can keep an otherwise worthwhile project from being funded, if they are not constructed in a manner that permits barrier-free access. To begin, therefore, I propose a theorem that determines when a set of layers is arranged to achieve the goal of barrier-free access. ELEVATOR THEOREM A persistent issue in any logical approach to a problem is knowing whether or not a solution exists. It is counterproductive to search for solutions that are known not to exist. One standard theorem from the realm of continuous mathematics is the intermediate-value theorem. Stated informally, this theorem ensures that a continuous function on [a, b] assumes all values between f(a) and f(b). That is, choose a value m on the y axis between f(a) and f(b)--there exists a value c in the interval on the x axis between a and b such that f(c) = m. In this context m is the problem, and the value c is its solution. Stated more formally, the intermediate-value theorem is often cast in the following manner: suppose that a function f is continuous throughout a closed interval [a, b] and that m is any number between f(a) and f(b); then there is at least one number c in [a, b] such that f(c) = m. Geometrically it is not difficult to imagine that instead of a continuous function f over a continuous closed interval [a, b], there might be a discrete function f with separated values. …