Fractals Take a Central Place Author(S): Sandra Lach Arlinghaus Source: Geografiska Annaler

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

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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-generated 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 location-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 alignment 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 procedure 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. The correspond- each arrow indicates that each edge of the closed ing Latin verb frangere means 'to break' to create curve on the left is to be replaced by the pattern irregular fragments" (Mandelbrot 1983, p. 4). The

84 GEOGRAFISKA ANNALER 67 B (1985) ?2 FRACTALS TAKE A CENTRAL PLACE curve formed as the limit of an infinite iteration had remained mutually unrelated [and] . . . few sequence is a "fractal" curve if and only if its di- definitions of dimension were used more than mension, D, does not coincide with one of the once" (Mandelbrot 1983, p. 16). Indeed, a tradi- standard Euclidean dimensions 0, 1,2, or 3. Man- tional difficulty in identifying a basic set of spatial delurot's formula for determining the dimension assumptions from which to classify spatial pheno- D is D = (log N) / (log k) where N represents the mena (as noted for example by Nystuen (Nystuen number of segments of equal length into which the 1968)), involves problems associated with the generator is broken (e.g., in Figure 1.a, N = 2), placement of objects into more than one class as and k is derived from the concept of "self-similari- a result of changes in the scale of observation. ty," discussed below (Mandelbrot 1983, p. 44). Mandelbrot's continuum of fractional dimensions In Figure 1.a, application of the generator to the offers the potential to resolve this dichotomy in initiator produces the first teragon which contains spatial classification. three copies of the initiator at a reduced scale (suggested by the dashed lines inside the first teragon). of the to the first Application generator teragon Fractal generation of central place nets transforms it into the second which con- teragon The invariant of k2, in tains three of the first self-similarity, produced images teragon (again sug- the iteration of l.a to dashed lines within the second sequence Figure appears gested by teragon). serve the same function as the K-value of central of the Repeated application generator produces noted a of this with but in all place theory. (Dacey relationship teragons increasingly lacy edges, sort between central cases the contains three place theory, repetition (n+l)st-teragon copies, and iterative of various kinds; at a reduced of the This nume- theory, processes scale, nth-teragon. Mandelbrot's work was not available rical invariant that measures but regrettably, shape permits to him in 1965 1965, The remain- scale to will be called the invariant of self- (Dacey p. 115)). shift, der of this section determines the relation of the similarity. In the case of Figure 1, the smallest fractal k-value to the central place K-value and Euclidean dimension in which all teragons can be exhibits fractal iteration sequences that generate embedded is 2; thus, the invariant three, of this the standard central place nets associated with K will be represented as k2, where the ex- example, = 3, K = 4, and K = 7, as well as those derived ponent corresponds to this minimum embedding from other points of a triangular lattice, selected criterion. (Had three appeared as an invariant of to expose the reader to generator selection techni- self-similarity in a sequence of three-dimensional que appropriate to obtaining nets for higher K- (one-dimensional) teragons, we should write k3 = values associated with an arbitrary lattice point. 3 (k = 3)). Thus k2 = 3 in Figure 1.a, so that k = V/-, and since N = 2, D = (log 2)/(log \/T) = 1.2618595. Thus the curve that results from the in- Standard central hierarchies finite sequence associated with Figure l.a is a place "fractal"curve of fractional dimension 1.2618595. The generator in the iteration sequence in Figure The plausibility of assigning a "fractional" di- l.a transforms a hexagonal initiator (Figure l.a.i) mension to a curve comes from viewing it as a into a first teragon composed of three hexagons highly contorted one-dimensional Euclidean line (Figure l.a.ii) which is again transformed (by the that appears to "fill" more Euclidean space than same generator) into the shape in Figure l.a.iii does a single straight line, but less than does a which contains three copies of the first teragon. Euclidean plane region. Mandelbrot's formula for The following stacking procedure of the initiator D produces larger values for D as larger amounts and teragons yields the K = 3 central place hier- of space are "filled," reflecting this plausibility archy (Figure 1.a): from a notational standpoint (Polya, 1954). The a) stack the initiator on the first teragon so notion of fractional dimension goes beyond stan- that 00 is superimposed on 01 as a geo- dard Euclidean dimension since it includes it as metric translation in the direction of the proper subset; Mandelbrot observes that "In fact, arrow; having recognized the inadequacies of standard b) stack the first teragon on the second dimension, numerous scholars . . . had already teragon so that 01 is superimposed on been groping towards broken, anomalous, conti- 02 as a geometric translation in the di- nuous dimensions of all kind. These approaches rection of the arrow;

GEOGRAFISKA ANNALER ?67 B (1985) ?2 85 ARLINGHAUS

c) continue this sequence indefinitely, as Initiator First considerations of scale demand. teragon This generator produces teragons with cells of exactly the right size to use to form a central place net K = 3; the stacking procedure used to align Y Y the teragons to form the entire central place net is a V/ . \ X <;)) not as straightforward as it is in the cases that fol- low. Figure 1.b shows a fractal iteration sequence whose invariant of self-similarity is k2 = 4. When initiator and are stacked, with centers V V' V teragons U 00, 01, 02, ... superimposed in the obvious way A A Y Y (as a geometric translation in the direction of the arrow), a K = 4 central place net emerges. App- ~b ~~ ~ ~ ' ' .1 lication of the generator shown in Figure l.c, to 2' Y ' the hexagonal initiator, produces a fractal iteration sequence with an invariant of self-similarity k2 = the cells of dia- 7; teragons provide hexagonal VI meter suited to forming a K = 7 central place hier- U V A V archy. Again, the entire central place net appears easily when the teragons are stacked in such a way that centers O, 0, 02, ?. ., and vertices U, V, V',... line up in the natural 1.c.ii y r Y way. (Figure Acc is tilted to fit the teragons together neatly; other- wise, stacking follows the arrows, as above). In all cases of Figure 1, unit hexagons form the basis of the second teragons in order to ease visual compa- Figure 2. risons of the most complicated forms in that Figure. The key to using fractal geometry to obtain cen- and to construct the corresponding central place tral place nets rests in choosing the correct genera- hierarchy, from a fractal iteration sequence, to tor to apply to the hexagonal initiator. Once the any level of detail. shape of such a generator has been determined The teragons of Figure 1 show only the first two (not always an easy task), it remains to construct stages in fractal iteration sequences; passing ab- the generator. Variations in detail of the procedu- stractly to the limit as the number of stages appro- re used to construct the generator for K = 7 (Figu- aches infinity yields the dimension of each central re 1.c) will yield positions for precise generator place landscape and consequently, a measure of placement for any K-value. To produce the K = how completely that net "fills" the space that con- 7 central place net, it is required that

86 GEOGRAFISKA ANNALER ?67 B (1985) ?2 FRACTALS TAKE A CENTRAL PLACE

Central place lattice

K-value generating function:

f(x,y)= x2+xy+y2 ______

D= 1.63

T4^f_~ ~ ______J----(3-1-- 2) (4 /N=3;k2=4 N=5;k =13 N=7;k =28 D=1.59 D=1.25 D= 1.17 ) T7..T 0, (2o)______-(2,1 -- (3,2)e N= 3;k = 7 N=5;k2=19 D= .13 D=1 09

T / (21)3 6d2) 3 N=2; k 3 N=4;k2=1 2 D=1.26 D= 1.12

Figure 3.

ne central place K-values are identical to the frac- associated central place hierarchies (K = 12, K = tal invariant of self-similarity, k2, in which the ex- 19, and K = 13, or f(2,2), f(3,2), and f(3,1) derived ponent represents an upper bound for the dimen- from the generating function f(x,y) = x2 + xy + sion of any fractal curve in the plane. Thus K = y2 for lattice points (x,y) (Dacey 1965, p. 113)), k2 is the relation that serves to formulate central emerge by superimposing initiator and teragon place nets from fractal iteration sequences in the centers in the natural way. plane (this paper deals with existence criteria, uniqueness is not addressed although it appears that such investigation might prove fruitful). Classification of generators The generator in Figure 2.a is composed of two copies of the generator in Figure l.a; it may be Higher order central place nets shown using algebraic technique that fractal gene- To gain added insight into methods for generator rators for central place nets associated with the lat- selection, Figure 2 displays the beginnings of frac- tice point (n,n) (n an integer) are composed of tal iteration sequences required to produce central In copies of the generator in Figure l.a. This place hierarchies associated with lattice points observation suggests grouping all such lattice (2,2), (3,2), and (3,1); here again, the most comp- points into one class, called T3 from the use of the licated teragon boundaries contain the unit hex- K = 3 generator. These points all fall on one hori- agons. All exhibit only the first step in the fractal zontal line in Figure 3, and this collinearity condi- iteration sequence; the hard part in creating these tion suggests looking at the other horizontal lines is to find the generator. In Figure 2.a, a generator to see if they determine sets of lattice points asso- of four sides, applied alternately within and out- ciated with other generator types. Indeed, it ap- side the initiator, yields a first teragon containing pears that higher central place K-value fractal se- twelve copies of the initiator. Thus k2 = 12 and, quences sort naturally into one of three mutually if this sequence were carried out indefinitely, the exclusive generator types grouped by lattice dimension of the limiting position would be D = points on these horizontal lines: (log 4)/(log V/-2) = 1.1157718. In Figure 2.b a a) Type T3: the images of the initiator in- five-sided generator, which crosses the path of UV side the first (and subsequent) teragon. when applied to the initiator produces a first tera- Figueres l.a and 2.a exhibit this sort of gon with mineteen images of the initiator. The configuration. limiting position for this sequence would have di- b) Type T4: the characteristic of T3 does mension D =(log 5)/(log VT9) = 1.0932051. not hold, and in addition, the generator Finally, Figure 2.c shows a five-sided generator, does not cross UV or any other initiator applied outside UV and then rotated about V to or teragon side. Figures l.b and 2.b lie within the initiator, that leads to an ultimate show this style of net. partition of the plane with dimension D = (log 5) / c) Type T7: the characteristic of T3 does (log /-3) = 1.2549471. In these three cases, the not hold, and in addition, the generator

GEOGRAFISKA ANNALER *67 B (1985) ?2 87 ARLINGHAUS

does cross UV and all other initiator and = k2, also issues a challenge. That challenge, re- teragon sides. Figures 1.c and 2.c de- sulting from the merging of two disparate bodies monstrate this quality. of literature, is to explore the power of this geometric alignment in uncovering distortions to cen- Evidence from additional fractal constructions, tral place nets caused by barriers, to understand with results coded the horizontal lines in Fi- along the implications of this merger for partitioning the 3 that show the these gure relationships among ty- landscape according to underlying networks, and in a that pes triangular lattice, suggests to demonstrate how related concepts, such as the horizontal line a) along any single exactly fractional dimension D, might explain the poten- one of or T3, T4, T7 holds; tial of an areally spread market to communicate, as one moves from left to a b) right along along teragon links, with point sources of central horizontal the dimension of the line, goods and services. central place fractal decreases; c) there is a cyclic pattern in the order of T3, T7, T4; d) values of N, the number of sides in the References generator, increase by an increment of Atkin, R. H., 1974: Mathematical Structure in Human Affairs. 2, as one moves from left to right along Crane, Russak, New York. 212 p. - 1981: Multidimensional Man. York. a single horizontal line. Penguin, New 196 p. Dacey, M.F., 1965: The geometry of central place theory. Geogr. Annaler, 47, pp. 111-124. Goodchild, M. F., 1979: The aggregation problem in location- Conclusions allocation. Geogr. Analysis, 11, pp.240-255. This for Hahn, H.: The crisis in intuition, lecture reprinted in Newman, paper presents explicit technique generat- J. 1956: The world mathematics. Simon and central nets from fractal iteration se- R., of Shuster, ing place New York, pp. 1956-1976. quences. It begins with the lowest level central Mandelbrot, B. F., 1977: Fractals: Form, Chance, and Dimen- place nets (Figure 1) and then moves to more sion. W. H. Freeman, San Francisco. 365 p. nets associated with K-values - 1983: The Fractal Geometry of Nature. W. H. Freeman, San complicated higher Francisco. 468 The of 2 p. (Figure 2). general strategy Figure Moellering, H., 1978: Professor of Geography, The Ohio State extends to any point of a triangular lattice with Univ., Columbus, OH. Personal Communication. integer coordinates; the procedure for generating Nystuen, J. D., 1963: Identification of some fundamental spatial K-value central nets from fractal ite- concepts. Papers of the Michigan Academy of Science, Arts, higher place 373-384. ration rests in a fractal Letters, 48, pp. sequences choosing genera- - 1967: Boundary shapes and boundary problems. Papers, tor that forces K to emerge from k2. The separa- Peace Research Society International, 7, 107-128. tion of such generators into three mutually exclu- - 1978, 1984. Personal communication. sive, but classes shows that the set of Polya, G., 1954: Patterns of Plausible Inference, Vol. II of exhaustive, Mathematics and Plausible Princeton Univ. central nets is a subset of the set of fractal Reasoning. place Press, Princeton. 190p. iteration sequences. It is a proper subset as it relies Raisz, E., 1948: General Cartography. Mc Graw-Hill, New only on one basic initiator (the hexagon) and on York. 354p. L. A. three basic T4, and Steen, and Seebach, J. A. Jr., 1970: Counterexamples in generator types (T3, T7). (The Holt, Rinehart and New York. and economic of nets Topology. Winston, Inc., geographic implications 210p. formed from other initiators and/or generators is Sullivan D., 1982: Geometry, iteration and group theory: collo- an open issue). quium lectures, 88th annual American Mathematical So- as the calculations and above ciety Meetings, Jan. 13-16. Personal communication via at- Moreover, figures tendance at lectures. the of central nets exhibit, precise alignment place Tobler, W., 1984: Professor of Geography, Univ. of California, with overlays of fractal teragons along the seam K Santa Barbara. Personal communication.

88 GEOGRAFISKA ANNALER .67 B (1985) - 2