Chaos - James Gleick [47]
Yet the curve itself is infinitely long, as long as a Euclidean straight line extending to the edges of an unbounded universe. Just as the first transformation replaces a one-foot segment with four four-inch segments, every transformation multiplies the total length by four-thirds. This paradoxical result, infinite length in a finite space, disturbed many of the turn-of–the-century mathematicians who thought about it. The Koch curve was monstrous, disrespectful to all reasonable intuition about shapes and—it almost went without saying—pathologically unlike anything to be found in nature.
Under the circumstances, their work made little impact at the time, but a few equally perverse mathematicians imagined other shapes with some of the bizarre qualities of the Koch curve. There were Peano curves. There were Sierpiński carpets and Sierpiński gaskets. To make a carpet, start with a square, divide it three-by–three into nine equal squares, and remove the central one. Then repeat the operation on the eight remaining squares, putting a square hole in the center of each. The gasket is the same but with equilateral triangles instead of squares; it has the hard-to–imagine property that any arbitrary point is a branching point, a fork in the structure. Hard to imagine, that is, until you have thought about the Eiffel Tower, a good three-dimensional approximation, its beams and trusses and girders branching into a lattice of ever-thinner members, a shimmering network of fine detail. Eiffel, of course, could not carry the scheme to infinity, but he appreciated the subtle engineering point that allowed him to remove weight without also removing structural strength.
The mind cannot visualize the whole infinite self-embedding of complexity. But to someone with a geometer’s way of thinking about form, this kind of repetition of structure on finer and finer scales can open a whole world. Exploring these shapes, pressing one’s mental fingers into the rubbery edges of their possibilities, was a kind of playing, and Mandelbrot took a childlike delight in seeing variations that no one had seen or understood before. When they had no names, he named them: ropes and sheets, sponges and foams, curds and gaskets.
Fractional dimension proved to be precisely the right yardstick. In a sense, the degree of irregularity corresponded to the efficiency of the object in taking up space. A simple, Euclidean, one-dimensional line fills no space at all. But the outline of the Koch curve, with infinite length crowding into finite area, does fill space. It is more than a line, yet less than a plane. It is greater than one-dimensional, yet less than a two-dimensional form. Using techniques originated by mathematicians early in the century and then all but forgotten, Mandelbrot could characterize the fractional dimension precisely. For the Koch curve, the infinitely extended multiplication by four-thirds gives a dimension of 1.2618.
CONSTRUCTING WITH HOLES. A few mathematicians in the early twentieth century conceived monstrous-seeming objects made by the technique of adding or removing infinitely many parts. One such shape is the Sierpinski carpet, constructed by cutting the center one-ninth of a square; then cutting out the centers of the eight smaller squares that remain; and so on. The three-dimensional analogue is the Menger sponge, a solid-looking lattice that has an infinite surface area, yet zero volume.
In pursuing this path, Mandelbrot had two great advantages over the few other mathematicians who had thought about such shapes. One was his access to the computing resources