Chaos - James Gleick [46]
And on toward microscopic perspectives: twine turns to three-dimensional columns, the columns resolve themselves into one-dimensional fibers, the solid material dissolves into zero-dimensional points. Mandelbrot appealed, unmathematically, to relativity: “The notion that a numerical result should depend on the relation of object to observer is in the spirit of physics in this century and is even an exemplary illustration of it.”
But philosophy aside, the effective dimension of an object does turn out to be different from its mundane three dimensions. A weakness in Mandelbrot’s verbal argument seemed to be its reliance on vague notions, “from far away” and “a little closer.” What about in between? Surely there was no clear boundary at which a ball of twine changes from a three-dimensional to a one-dimensional object. Yet, far from being a weakness, the ill-defined nature of these transitions led to a new idea about the problem of dimensions.
Mandelbrot moved beyond dimensions 0,1,2,3…to a seeming impossibility: fractional dimensions. The notion is a conceptual high-wire act. For nonmathematicians it requires a willing suspension of disbelief. Yet it proves extraordinarily powerful.
Fractional dimension becomes a way of measuring qualities that otherwise have no clear definition: the degree of roughness or brokenness or irregularity in an object. A twisting coastline, for example, despite its immeasurability in terms of length, nevertheless has a certain characteristic degree of roughness. Mandelbrot specified ways of calculating the fractional dimension of real objects, given some technique of constructing a shape or given some data, and he allowed his geometry to make a claim about the irregular patterns he had studied in nature. The claim was that the degree of irregularity remains constant over different scales. Surprisingly often, the claim turns out to be true. Over and over again, the world displays a regular irregularity.
One wintry afternoon in 1975, aware of the parallel currents emerging in physics, preparing his first major work for publication in book form, Mandelbrot decided he needed a name for his shapes, his dimensions, and his geometry. His son was home from school, and Mandelbrot found himself thumbing through the boy’s Latin dictionary. He came across the adjective fractus, from the verb frangere, to break. The resonance of the main English cognates—fracture and fraction—seemed appropriate. Mandelbrot created the word (noun and adjective, English and French) fractal.
IN THE MIND’S EYE, a fractal is a way of seeing infinity.
Imagine a triangle, each of its sides one foot long. Now imagine a certain transformation—a particular, well-defined, easily repeated set of rules. Take the middle one-third of each side and attach a new triangle, identical in shape but one-third the size.
The result is a star of David. Instead of three one-foot segments, the outline of this shape is now twelve four-inch segments. Instead of three points, there are six.
THE KOCH SNOWFLAKE. “A rough but vigorous model of a coastline,” in Mandelbrot’s words. To construct a Koch curve, begin with a triangle with sides of length 1. At the middle of each side, add a new triangle one-third the size; and so on. The length of the boundary is 3 × 4/3 × 4/3 × 4/3…—infinity. Yet the area remains less than the area of a circle drawn around the original triangle. Thus an infinitely long line surrounds a finite area.
Now take each of the twelve sides and repeat the transformation, attaching a smaller triangle onto the middle third. Now again, and so on to infinity. The outline becomes more and more detailed, just as a Cantor set becomes more and more sparse. It resembles a sort of ideal snowflake. It is known as a Koch curve—a curve being any connected line, whether straight or round—after Helge von Koch, the Swedish mathematician who first described it in 1904.
On reflection,