Art in Los Angeles: Seventeen Artists in the Sixties by Maurice Tuchman

By Maurice Tuchman

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Fn (x) = An x + bn on R2 . The attractor S of the system is the set of all points of the form bk1 + Ak1 bk2 + Ak1 Ak2 bk3 +   , where each matrix Akj , and each vector bkj , is one of those listed above. That is, S = fbk1 + Ak1 bk2 + Ak1 Ak2 bk3 +    j 1  kj  n for all j g: Now let's show that the attractor S really does \attract" points. Suppose we have an iterated function system like that given in Item I, and suppose we have a starting point x0 2 R2 . We begin creating a sequence of points x0 ; x1 ; x2 ; .

Vine and Tablecloth The graphic entitled Vine and Tablecloth in Figure 1 is a visual metaphor for the contrast that many people feel exists between mathematics and the natural world. The pattern of the tablecloth reminds us of the coordinate grid of the xy-plane, with its neat rows and columns of little boxes. The vine has a completely di erent character. It exhibits a wild, scraggly, complex form, beautiful in a way that is both rugged and delicate. In no way does it t into the rigid order of the coordinate grid, and it seems to defy us to describe it with mathematics.

To draw the canvas in perspective, we make the sides vertical for convenience, and make the lines determined by the top and bottom edges converge to v. Since the sides of the canvas were located rather arbitrarily, how can we be sure that the outline of the canvas corresponds to a rectangle seen in perspective which is twice as wide as it is high? The answer is given by Theorem 3, which says that the \shape ratio" of the canvas is proportional to the viewing distance. We have chosen the shape ratio rst, so the correct viewing distance is therefore completely determined, and from this distance the outline will appear just as it should.

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