*[From: Ulam, Stansilaw. "Patterns of Growth of Figures: Mathematical Aspects." In Module, Proportion, Symmetry, Rhythm. Vision and Value series. Gyorgy Kepes, ed. New York: George Braziller, 1966.*

Figures: Mathematical Aspects

The simplest patterns observed, for example, in crystals, are periodic and the properties of such have been very extensively studied mathematically. The rules which we shall employ will lead to much more complicated and in general nonperiodic structures, whose properties are more difficult to establish, despite the relative simplicity of our recursion relations. The objects defined in that way seem to be, so to say, intermediate in complexity between inorganic patterns like those of crystals and the more varied intricacies of organic molecules and structures. In fact, one of the aims of the present note is to show, by admittedly somewhat artificial examples, an enormous variety of objects which may be obtained by means of rather simple inductive definitions and to throw a sidelight on the question of how much "information" is necessary to describe the seemingly enormously elaborate structures of living objects . . . . We have used electronic computing machines at the Los Alamos Scientific Laboratory to produce a great number of such patterns and to survey certain properties of their morphology, both in time and space. Most of the results are empirical in nature, and so far there are very few general properties which can be obtained theoretically.

l. In the simplest case we have the subdivision of the infinite plane into squares . . . . [p. 64]

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