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ment (Fig. 5.7). If one keeps repeating the estimate, moving the divider's points closer together each time, the estimated length will converge onto a particular value, which is the curve's true length. Indeed, the exact length of the curve, and not merely a good approximation, can be had essentially by setting the points of the divider infinitesimally close. Those who remember their calculus will recognize this procedure as taking the limit of the estimate: in the jargon of the calculus, the curve is differentiable.

In traditional geometry, every differentiable object has a dimension, which we designate with a number D. A line or a curve, for example, has only a dimension of length, so D = 1. Planes or surfaces, on the other hand, because they have an area, have dimension of length squared (/2), which we can express as a dimension of D = 2. Solid objects, because they have dimensions of length cubed (l3), have a dimension of D = 3. Obviously, D is the power of the length dimension needed to describe the object.

The dimensions of simple curves, planes, and shapes are easy to grasp. Fractal objects require us to stretch the concept of a dimension a bit. You will have noticed that the dimensions of lines, surfaces, or volumes are, respectively, 1, 2, and 3—that is, they are whole numbers, or integers. Get ready to stretch, now: fractal objects, in contrast, are objects whose dimensions are not integers but fractions. This is a tricky concept to grasp, but again it is easily illustrated by the problem of how one measures the length of something.

This time, consider a curve found in nature—the usual example in most introductions to fractal geometry is a coastline, say of Papua New Guinea (Fig. 5.8). You can measure the coastline's length as we measured the differentiable curve's, by dividing it up into a series of linear segments and multiplying the segments' lengths (corrected, of course, for the scale of the map) by the number of lengths needed to span the coastline. You can take these measurements with increasing degrees of precision, either by altering the distance between the divider's points or by employing several maps drawn at smaller and smaller scales, get-

Figure 5.8 Estimating the length of a fractal curve. a: A map of a coastline reveals ever more finely detailed curves as its magnification is increased. b: Unlike a smooth curve, a fractal curve's estimated length increases continuously and without bound as segment length decreases.

log (segment length)

Figure 5.8 Estimating the length of a fractal curve. a: A map of a coastline reveals ever more finely detailed curves as its magnification is increased. b: Unlike a smooth curve, a fractal curve's estimated length increases continuously and without bound as segment length decreases.

ting, as we did before, increasingly fine estimates of the length. Once you run out of maps of sufficiently fine scale, you can go down to the shore with a surveyor's chain and keep repeating the measurements on still finer scales, all the way down to the level of individual grains of sand if we so chose. Plotting the estimate of coastline length against step length, you will see a remarkable result—no matter how fine the scale used for our measurement, no matter how infinitesimally small the lengths, your estimate will never approach a "true" value, as our previous example did for the length of the simple curve. Rather, the estimated length just keeps increasing ad infinitum, never converging on a "true" value (Fig. 5.8).

This inability to measure the true length of a natural shape like a coastline (or a leaf margin, or the lining of the lung, or ... I could go on forever) leads us to the central idea in fractal geometry. The fact that you cannot arrive at the coastline's "true" length means that it does not have a simple or integral dimension. Structures that have well-behaved dimensions, like 1, 2, or 3, are, by definition, differentiable, and so it is possible to apply the methods of the calculus to estimate their magnitude. The inability to define a length's "true" value means that the dimension of the structure or object is not an integral number, that is, it is neither a curve (D = 1) nor a surface (D = 2) but something in between. It has, in fact, a fractional dimension (or fractal), where 1 < D <2. In fact, most coastlines have fractional dimensions D that fall somewhere between 1.2 and 1.3.

The connection between measuring coastlines and the growth of sponges and corals is that modular growth often results in a fractal object. The similarity is easily illustrated by one of the simplest fractal objects, the Koch curve (Fig. 5.9). The Koch curve starts as a simple triangle, obviously with three corners, or apices. The curve is generated by repeated addition of a triangle in the middle of each of the original triangle's legs. With each step, the curve grows into an elaborately branched structure. The analogy with modular organisms is obvious. Anything that grows by repeated addition of some sort of modular unit will end up generating a fractal object of some sort. Indeed, the

Figure 5.9 The Koch curve.

growth of sponges and corals has been simulated by computer models that use fractal geometry: the results are startlingly lifelike.

### Accretion and Modular Growth

Modular growth of sponges and corals is a process akin to accretion, which is simply growth by addition of new material to an existing surface. The most familiar example of accretive growth is crystallization. A crystal of table salt (NaCl), for example, grows by the addition of sodium and chloride ions—the "modules" in this case—to an existing surface of a sodium chloride crystal. This surface serves as a faithful template for future growth: the arrangement of atoms in the crystal provides "niches" for the addition of new ions. Through the repetition of this process—niches open up and then are filled—the crystal grows. Because the arrangement of the niches arises from the arrangement of sodium and chloride atoms in the existing crystal, the shape of the crystal is maintained as it grows: NaCl crystals, no matter what their size, are always cubic.

Sponges and corals must have a different type of accretive growth, though: as we have seen, their shapes are variable and are strongly influenced by the conditions in which growth occurs. Why should the process of accretive growth in one case, crystallization, preserve shape, while in another, the growth of a sponge or coral, it does not?

Accretive growth is a two-step process. First, material must be delivered to the surface from the solution in which it is immersed. In the case of a sodium chloride crystal, sodium and chloride ions must travel from the solution to the surface of the growing crystal. Second, once the material is at the surface, it must find its appropriate "niche" there and settle in. Each step affects the shape of a growing surface differently;de-pending upon which step is more easily accomplished, accreting surfaces will either maintain their shapes as they grow or not. When the settling-in time of new material is slower than the rate at which new material can be delivered, accretion growth preserves shape. When delivery rate is slower than the settling-in time, however, the shape of an accreting surface will be influenced by conditions that affect the delivery rate. Areas that experience more rapid delivery will grow faster than areas that do not, resulting in a change of shape of the growing surface. The variable morphology of sponges and corals suggests that their growth is dominated by limitations on delivery of new material to the growing modules. To understand their growth, we must then understand what these limitations are.

The rate of delivery of material to a growing surface is ultimately determined by the process of diffusion, already encountered in an informal way in Chapter 4. Let us examine the concept formally now, for understanding diffusion is the key to understanding the growth forms of accretive growers like sponges and corals.

Diffusion is the spontaneous movement of matter down a concentration gradient. In textbooks, diffusion usually is illustrated by a box separated into two compartments by a permeable barrier (Fig. 5.10): let us designate them with the Roman numerals I and II. Each compartment contains some substance that is capable of moving by diffusion, either molecules in solution or components in a mixture of gases. Each compartment will therefore contain some concentration of this substance, C. If the concentration in compartment I differs from that in compartment II, there will be a concentration difference, C: — Cjj.

The concentration difference is essentially a potential energy difference, and as such it is capable of doing work. Work is done when the substance is moved across the barrier separating the compartments. For example, if compartment I is richer in oxygen than compartment II, oxygen molecules will move spontaneously from I to II. Because work is proportional to the potential energy driving it, we can express a proportionality: