Abstract
FRACTALS: A NEW LOOK AT BIOLOGICAL SHAPE AND PATTERNING ANASTASIOS A. TSONIS* and PANAGIOTIS A. TSONISf It is accepted beyond any doubt that biological shape is related to function. Numerous examples could be mentioned. Folding of a protein is important for its function [I]. The shape of a cell accounts for differential DNA synthesis [2]. Left-handed DNA relates to specific function of genes [3]. The patterning of biological form has intrigued developmental biologists over the last decades. However, there are no satisfactory qualitative and quantitative models from which to infer statistical properties and theories that directly relate biological shape to function. Part of the problem lies in the fact that biological structures often cannot be described within a straightforward Euclidean framework. Similar problems are encountered in many other fields of science. For example, clouds are not cubes or spheres, coastlines are not circles, lightning is not straight lines, and so forth. Euclidean geometry leaves these structures without a framework that can be used to quantitatively describe them. Nature is full of such structures. As a matter of fact, it is the Euclidean structures that are rarely found in nature. In the absence of a mathematical framework for the "amorphous" patterns, theories or models that are devised in order to explain and/or describe those patterns are inadequate. Lately, a new geometry has been developed in order to describe the irregular and fragmented non-Euclidean patterns of nature. It is called fractal geometry. In order to introduce the reader to the concept of fractals , some familiarity with the notion of dimension is needed. We therefore suggest at this point that the reader consult figure 1. The authors thank Daniel Fishman for excellent photography and Dr. P. Meakin for sending and permitting the publication of figure 3. *Department of Geological and Geophysical Sciences, University of Wisconsin— Milwaukee, Milwaukee, Wisconsin 53201. tCancer Research Center, La Jolla Cancer Research Foundation, 10901 N. Torrey Pines Rd., La Jolla, California 92037.© 1987 by The University of Chicago. All rights reserved. 003 1 -5982/87/3003-053 1 $0 1 .00 Perspectives in Biology and Medicine, 30, 3 ¦ Spring 1987 \ 355 (a) DT= 1, DE =1, D= 1 (b) ??/VV (c) DT =1, DE = 3, 1£D£3 S0!3 Fig. 1.—There are two basic definitions of dimension: the Euclidean (D6) and the topological (DT). They both can assume only the integer values 0, 1,2, 3, but, for a specific object, they may not be the same. In order to divide space, cuts that are called surfaces are necessary. Similarly, to divide surfaces, curves are necessary. A point cannot be divided since it is not a continuum. Topology tells us that, since curves can be divided by points which are not continua, they are continua of dimension one. Similarly, surfaces are continua of dimension two, and space is a continuum of dimension three. Apparently, the topological dimension of a point is zero. According to the Euclidean definition, a configuration is called E-dimensional if the least number of real parameters needed to describe its points is£. For example, to describe the points ofa straight line, one needs only the x, say, coordinates of those points. Therefore, D€ of a straight line is one. On the other hand, to describe the points of the curve (b) above, one needs the coordinates of the points in ? and y. Therefore, Dt of that curve is two. Similarly, in order to describe the points of the surface (d), one needs the coordinate of the points in x, y, and z. Therefore, D€ of that surface is three. The quantity D is the fractal dimension (see text for explanation). The concept of fractals can be introduced by a classic example given by the founder of fractals, Mandelbrot [4]. Let us assume that we measure the length of a given segment of a straight line by employing a measuring unit (yardstick) of length e. With this yardstick we walk along the line, each new step starting where the previous step leaves off. If the number of steps is ?(?), then e x 7V(e) is a measure of the length, L(e), of that segment. As we repeat the same procedure...