An Electronic Journal of
Geography and Mathematics.
(Major articles are refereed; full electronic archives available).
Persistent URL: http://deepblue.lib.umich.edu/handle/2027.42/58219
|Fractals Take A Non-Euclidean
Sandra L. Arlinghaus, Ph.D., The University of Michigan
Central place geometry looks at groupings of hexagons as they tile the Euclidean plane. Pattern within these groupings may be interpreted as representing various elements of how cities share space. The geometry plays out over the unbounded surface of the Euclidean plane; the interpretation surface, however, involves the bounded surface of the Earth (roughly a sphere). One might ask if tilings of regular polygons could suggest other surfaces as a basis for similar interpretation.
Euclid's Fifth ("Parallel") Postulate states that given a line, and a point not on that line, there is exactly one line through that point that does not intersect the given line. That is, the newly drawn line is "parallel" to the given line.
In "Non-Euclidean" geometries the Parallel postulate does not hold (Coxeter, 1961, 1965). One of these non-Euclidean geometries, that admits multiple rather than unique parallels, is called "hyperbolic" geometry. For inspiration for that name, consider the hyperbola: multiple branches through a single point not on a given line might all be asymptotic to the given line--they do not intersect it. Of course, as the process becomes infinite, one might imagine the asymptote getting to the line--of the "parallels" meeting at a point at infinity.
Most of us live in an Euclidean universe. We may function in other ways, but when we attempt to capture the geometry of our lives we do so, for the most part, in the Euclidean plane. To do otherwise, is jarring to most.
Place Geometry: Fractal Characterization
The patterns that emerge from grouping sets of hexagons according to K=3, K=4, and K=7 principles are familiar to even beginning geography students. What is perhaps not as familiar is the use of fractal geometry to produce these patterns from number theoretic properties (Arlinghaus, 1985; Arlinghaus and Arlinghaus, 1989). See the linked materials for a full explanation; Figures 1.a, 1.b, and 1.c show (respectively) how to create a generator to form appropriate pattern for the K=7, K=4, and K=3 hierarchies. Iteration of the generator at different geographic scales produces the entire landscape as suggested in Figure 1.d which shows the next level of generator scaling and iteration (Arlinghaus and Arlinghaus, 2005).
An advantage to the fractal
approach is that it permits
complete systematic extension of process to higher K-values and, at the
same time, allows complete solution to existing open questions.
One broad concern, however, that is not addressed in this approach or
in any other within the Euclidean plane is the issue of bounding the
process (rather than allowing it to continue to indefinite
extent). In the world of fine art, Maurits Escher addressed this
issue elegantly with his Circle Limit Series in which pattern that
replicates indefinitely is confined within the boundary of a "limiting"
Hyperbolic Plane: Poincaré Disk Model
The difference between the polygon on the sphere and a polygon in the hyperbolic plane is that the sphere has constant positive curvature whereas the hyperbolic plane has constant negative curvature, so one must imagine sides that are convex (bulge out) on the sphere would be sides that are concave (sink inward) in the hyperbolic plane (and vice-versa). With the curvature issue, and the shrinking of polygons with distance from the center, in mind, a natural question is to ask about the nature of tilings of this new space.
Data compression and the Fibonacci sequence
The Fibonacci sequence is a number pattern that has seen, and continues to see, application in a number of different arenas. The sequence itself is generated as follows:
Start with 1; add the preceding number to it (there is none) generating the next term which is also 1. Then add this new 1 to the previous term generating 2; then add 2 and 1, and so forth, to create the following number pattern:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,...
Some characterizations of this sequence simply have it begin as 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,... Whichever is used, its applications are deep and well known. Indeed, an entire journal, The Fibonacci Quarterly, is devoted to its theory and applications. Using it to represent phyllotaxis and the arrangement of spiral leaf patterns on branches or the spiral arrangement of cells on the skin of a pineapple have been with us for a long time. So too has an interpretation of the sequence in terms of the way that rabbits reproduce.
More recent applications employ the sequence in creating a universal computer code that will stem error propogation in computing systems. This property, and others, classify Fibonacci coding as a "Lossless" Entropy Encoding Data Compression Method.
Pattern compression in the hyperbolic plane and the Fibonacci sequence
Series and Wright note that tilings, as well, can create interesting methods of data compression through pattern compression. They note the following example involving similar tilings of both the Euclidean plane, Figure 3.a, and the Hyperbolic plane (realized in the Poincaré Disk Model), Figure 3.b.
Simple enumeration of bands of polygons surrounding a chosen central polygon reveals a numerical compression of pattern. Of particular interest, note that once again the Fibonacci sequence gives a systematic basis for the greater numerical compression available in the hyperbolic plane (Series and Wright). Figure 4.a shows the simple linear number pattern that emerges from the hexagonal tiling of the Euclidean plane while Figure 4.b captures systematically the pattern in the hyperbolic plane.
Pattern compression in the hyperbolic plane is more than 4500 fold greater than in the Euclidean plane at the level of n=12 and this compression rate appears to increase as n increases (Figure 5).
It remains to offer formal proof of this observation; however, it seems clear that the observation is true since the Fibonacci sequence itself, let alone the 2nth Fibonacci sequence, becomes large faster than does the sequence of natural numbers.
|Central Place Hierarchies in the
While the observation by Series and Wright concerning the Fibonacci sequence in association with an heptagonal tiling of the hyperbolic plane appears both interesting and important, one might ask about other tilings of the plane and such association with the Fibonacci sequence, or other ideas, as a compression tool. An article in Wikipedia is useful in classifying possible tilings of the hyperbolic plane as realized in the Poincaré Disk Model. The reader is encourage to take a look at the link above and study the beautiful and highly symmetric patterns that emerge.
Central Place Theory
One of the "other ideas" that seems natural to consider as a compression tool is that of the fractal. Thus, a logical next step might be to try to merge the fractal compression from central place theory (and hexagonal tilings of the Euclidean plane) with a corresponding construction in the hyperbolic plane (Poincaré Disk Model) in order to assign fractal dimension to these patterns as ways to analyze the extent to which space is "filled" by them in the limiting process.
Tiling by heptagons
In the spirit of the example from Series and Wright, begin by considering the heptagonal tiling of the hyperbolic plane. Figure 6.a shows selection of a generator, similar to the generator for K=7 in the Euclidean case above (Figure 1.a) that will generate, when scaled appropriately and applied throughout the Disk, the entire network mesh. Here, the generator has four sides; in the corresponding Euclidean case the generator had 3 sides.
To continue the central place theory extension, one might ask about the cases for K=4 and K=3. A natural appoach, for K=4 for example, is to consider using as a generator one that surrounds half the area of a heptagon in the hyperbolic plane. Then, apply this generator in an "in and out" fashion to eventually generate a set of polygons that will cover 7*0.5 + 1 amount of heptagonal cells. The problem with this approach is that the "in and out" approach requires an even number of sides in the basic cell; otherwise, two "outs" are adjacent and the inner central cell becomes the wrong shape. Thus, the hierarchy cannot be produced. Figure 7 shows the beginning of this application of a natural generator, with reference to the underlying base so that the difficulty with iteration is clearer.
Prime numbers and polygon shape
Clearly one cannot examine each individual case, as with Figures 6 and 7, to determine when a particular tiling might or might not be generated through fractal generation. One easy approach to this problem is to look at the number-theoretic properties of the base cell of the tiling. In the case of hexagons, the unique factorization of 6 is 3*2. Thus, one can look to generating 1/3 (K=3) and 1/2 (K=4) polygonal pieces in association with fractal iteration. In the case of a prime number, such as 7, one can find only the natural surrounding pattern of full tiles--no fractional tiles.
With this idea in mind, move now to considering other tilings of the hyperbolic plane as candidates for calculation of fractal dimension. A natural place to begin is with tiling by hexagons.
Tiling by hexagons
One of the characteristics that a central place style of tiling has is that cells on each side of a central cell touch other cells with a side in common with that central cell. Thus, the tiling by hexagons shown in Figure 8 is not this sort of tiling: tiles 1 and 2 are in correct position in relation to the central cell and to each other. So also are the polygon pairs 3 and 4, as well as 5 and 6. However 2 and 3 are separated by an x cell, as are 4 and 5, as well as 6 and 1. The required adjacency pattern is only partially present and so this case is discarded (for now) as not a true related central place case. Thus, one sees why Series and Wright chose the heptagonal tiling, rather than an hexagonal tiling, of the hyperbolic plane as the one naturally associated with the hexagonal tiling of the Euclidean plane. One could, of course, choose other tilings (Woldenberg, 1968). The point here is to indicate what tiling of the hyperbolic plane is, or is not, similar to a tiling of the Euclidean plane by hexagons.
Tiling by pentagons
As in the case above, there are gaps separating tiles. This time, not in pairs but between every tile surrounding the central tile. Figure 9 illustrates this difficulty (which is more extreme than in Figure 8) using notation parallel to that employed in Figure 8. Again, as with the hexagonal tiling, this case which might have appeared attractive at the outset, is discarded as well: its level of violation of a basic premise is even more extreme than is the case with the hexagonal tiling.
Tiling by octagons
The tiling of the hyperbolic plane shown in Figure 10 is not even as close to a central place net as those in either Figure 8 or Figure 9. Here, the central octagonal cell is surrounding by 8 cells with correct adjacency but they are hexagonal rather than octagonal cells. The next layer out has octagonal cells with no adjacency--each pair of octagonal cells is separated by an hexagonal cell.
It remains an open question to determine if there are any other true central place net styles of tiling of the hyperbolic plane (there were with the Euclidean plane however there only the case of square tiles (Arlinghaus, 1992) is of interest since a hexagon may be decomposed into triangles). Examination of cases did suffice in the Euclidean case and it may well be sufficient in the hyperbolic case, as well--a quick examination of the visual pattern displayed in the Wikipedia article suggests no others; however, a more systematic search, based on careful notational analysis rather than solely on visual pattern, is necessary to prove there are no others. Indeed, that is one direction for the future--to prove uniqueness of a single central place net for hyperbolic space.
|For the Future...
General links (last accessed June 15, 2010) and references:
An Electronic Journal of Geography and Mathematics,
Congratulations to all Solstice contributors.
|Remembering those who
are gone now but who contributed in various ways to Solstice or to IMaGe
projects, directly or indirectly, during the first 25 years of IMaGe:
Allen K. Philbrick | Donald F. Lach | Frank Harary | H. S. M. Coxeter | Saunders Mac Lane | Chauncy D. Harris | Norton S. Ginsburg | Sylvia L. Thrupp | Arthur L. Loeb | George Kish |