Fractals Take a Central Place Author(s): Sandra Lach Arlinghaus Source: Geografiska Annaler. Series B, Human Geography, Vol. 67, No. 2 (1985), pp. 83-88 Published by: Blackwell Publishing on behalf of the Swedish Society for Anthropology and Geography Stable URL: http://www.jstor.org/stable/490419 Accessed: 17/06/2009 19:43 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=black. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact [email protected]. Swedish Society for Anthropology and Geography and Blackwell Publishing are collaborating with JSTOR to digitize, preserve and extend access to Geografiska Annaler. Series B, Human Geography. http://www.jstor.org FRACTALS TAKE A CENTRAL PLACE BY SANDRA LACH ARLINGHAUS* ABSTRACT. The geometry of central place theory is shown to nal space (Hahn 1956, pp. 1965-66), suggesting in- be a (small) proper'subset of the geometry of fractal curves: cur- the of differen- ves of fractional dimension which have been dis- tegration, analytic companion only recently tiation. played in a graphically provocative manner as computer-gene- rated images (Mandelbrot 1977; 1983). The exact procedure for The use of space-filling curves to disprove topo- making this correspondence between a theory from economics logical conjectures continues to the present (Steen and geography with one from pure mathematics is displayed in and Seebach and the the text. It lends itself to hand or machine. As 1970, pp. 137-38), by early replication by by twentieth mathematicians Fatou and Ju- is usual with alignments of this sort, a wide variety of related century, projects follows naturally; some of these are indicated at ap- lia focused on a systematic theoretical organiza- propriate points in the text. tion of these sets extending beyond the calculus (Sullivan, 1982). Today, mathematicians speciali- zing in topological dynamics create theorems Introduction about these sets and about sets that fill only part "Fractal" curves are curves that "fill" a fractional of a space; that the broader mathematical commu- part of a space; such curves have long been a sour- nity sees the constructive potential for this activity ce of pathological counterexample in mathema- is suggested by the series of four invited addresses tics, and more recently, have been applied to fun- on dynamical systems given in the "Colloquim" damental problems in a variety of disciplines. Ele- Lectures to the American Mathematical Society mentary calculus exploits the use of the absolute (Sullivan, 1982). value function, y = Ix , to disprove that continuity Moreover, prior to the presence of high-speed implies differentiability; this V-shaped, conti- computing machinery, we could visualize these nuous, absolute value function has only one point, complicated curves only in the mathematical at the "corner," at which the function is non-diffe- twilight of our minds. Thus the emergence of rentiable. Karl Weierstrass, a nineteenth century Benoit Mandelbrot's works (Mandelbrot 1977; mathematician, sought a continuous curve that 1983), displaying an elegant array of computer- was nowhere differentiable and found one using a generated "fractal"curves, suggests various appli- sequence of alternations of the absolute value cations for these curves in situations where shift function of the following sort; transform the letter in scale is fundamental. Mandelbrot's computer- "V," replacing each segment by two suitably generated three-dimensional landscapes are remi- placed copies of the letter "N," and repeat this niscent of Erwin Raisz's block diagrams (Raisz procedure through n steps (Hahn 1956, pp. 1962- 1948, pp. 120-121); his description of Minkows- 63). The limit as n approaches infinity yields a ki's "sausage" for smoothing curves is similar to curve that is continuous but is composed only of John Nystuen's use of epsilon discs to identify the corner points and so is nowhere differentiable. domain of the boundary dweller (Nystuen 1967); Koch superimposed the endpoints of Weier- and his concern for "How long is the coast of strass's curve, producing a continuous curve that Britain?" echoes the persistence of the geographic is nowhere differentiable surrounding a bounded scale problem (Mandelbrot 1983, pp. 264-65, 32, zone of the plane (Mandelbrot 1983, p. 41). Filling 25). Cartographers Waldo Tobler and Harold a bounded zone, using Weierstrass's procedure Moellering observed the potential for fractals to with lines, led to the notion of "space-filling" contribute to their research in shape theory and curves. Peano created curves formed from one- the transformation of shape; Michael Goodchild dimensional line segments, which, when twisted noted Mandelbrot's "model for the Pareto distri- and transformed infinitely, filled a two-dimensio- butions observed for certain geographic areas" in an article dealing with various aspects of the lo- cation-allocation problem; and John Nystuen saw the significance of applying Mandelbrot's notion * of to the of urban facilities Dr. Sandra Lach Director, in Mathe- "self-similarity" design Arlinghaus, Colloquium on networks for and matical Geography, 1441 Wisteria Drive, Ann Arbor, MI 48104, dependent dendritic entry USA. exit (cf. Mandelbrot, 1977; Tobler, 1984; Moelle- GEOGRAFISKA ANNALER *67 B (1985) ?2 83 ARLINGHAUS ring, 1978; Goodchild 1979, p. 247; Nystuen, I Ctnlral Plarp ?`ra(tal Iteration Sepquences: K = 3, 4 , 7 1 1978; 1984). This paper exhibits procedure to generate frac- -- uv tal sets and then uses it to show that the entire geo- u u u~V- metry of central place theory is but a small subset of the theory of fractal geometry; for, one style of fractal iteration sequence alone, using various a I~~~~~ (but related) "generators," will produce all possi- 7v ble central place nets. Consequently, the align- B .0, ment of central place theory with fractal geometry :: I does not merely produce one, two, or three (i.e., K = 3,4,7) cases of central place nets; rather, be- cause fractal iteration sequences deal with infinite processes, they yield all cases. Nor does this align- ment present mere technique for verification of a geometry of central places; the geometry of the central place model is well-known and has been clearly, and comprehensively, discussed in Mi- chael Dacey's masterful 1965 article (Dacey 1965). The material presented below does show exact procedure for the merging of two separate theories - one from pure mathematics and one 1. from economics and geography. In doing so, it Figure suggests, in general, the power of one to enrich the other through the lodging of one discipline in the house of the other, and in particular, a theory above the arrow, generating the closed curve on in its own right derived from associating mathema- the right. Thus in Figure 1.a, side UV of the tically-specialized central place concepts with hexagon in Figure 1.a.i is replaced (outside the mathematically-broader fractal concepts. Recent- hexagonal boundary) by a 'bent' or 'broken' shape ly, R. H. Atkin has demonstrated the richness of with the included angle equal to 120?. This same this sort of approach in (among other things) his shape replaces the side adjacent to UV at V (with- analysis of the internal dynamics of urban structu- in the hexagonal boundary), and so forth around re using material from combinatorial and alge- the hexagon, until U is reached from the left-hand braic topology (Atkin, 1974; 1981). In a parallel side. Since the application of this shape alternated vein, the broader conceptual base offered here back and forth between 'inside' and 'outside' the could present means for assessing the dynamic hexagon, the area in Figure 1.a.ii is the same as structure of' shared space between cities by eva- that in Figure 1.a.i. Similarily, this pattern could luating the changing dimensions of an entire urban be applied, at a scale made to match the length of landscape across the continuum of fractional va- a side, to Figure 1.a.ii creating Figure 1.a.iii; in lues that reflects the infinity of variation in real- this case the broken shape UV' is used to replace world constraints. each side of Figure 1.a.ii. Iteration of this proce- dure produces increasingly complicated curves; the shape at the left end of this sequence is the for Procedure generating simple fractal sets 'initiator,' the pattern that is applied to the initia- As with Weierstrass's creation of a nowhere diffe- tor is the 'generator,' and the shapes that appear rentiable curve, the strategy that underlies the at various stages in the iteration sequence are 'te- physical development of fractal curves involves re- ragons' (Mandelbrot 1983, pp. 50, 48). placing, successively, the edges of a given regular The 'broken' character of the teragons derives polygon with a pre-determined pattern. To repre- from the application of a 'broken' generator; in- sent this replacement, the notation of Figure 1 will deed, Mandelbrot comments that "I coined fractal prove convenient; in that Figure, the shape above from the Latin adjective fractus.