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By Einstein A.

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Record the number of times each pixel is hit, and color the pixel according to the value of that number. This process is no more than coloring by number. The actual colors are chosen according to which colors bring out best the underlying structure. 13, we show the result of coloring a figure with fivefold symmetry after 667,000,000 iterations on a 3,000 × 3,000 pixel grid. Since there are 9,000,000 pixels, it follows by the pigeonhole principle that some pixels must have been hit more than once.

We might call the number zero a repelling fixed point. As we apply the rule, numbers move away from zero. An attractor captures the opposite behavior. As we apply a rule, we move closer, or are attracted, to the attractor. For example, we may consider the halving rule, which takes a number and halves it. Since half of zero is still zero, we see that zero is a fixed point for the halving rule. However, unlike the doubling rule, all numbers move closer to zero under repeated applications of the halving rule.

There are so many different kinds of symmetry that it is virtually impossible to give one simple all-inclusive definition of this term. For example, one can discuss the symmetries of an equation or a physical law, not just the symmetries of a picture. Although the ideas of symmetry in its different manifestations are related, care must be taken when making the notion precise. In our discussion of symmetry, however, our goals are really quite modest. We want to know what are the possible symmetry groups for pictures in the plane.

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Contributions to Annalen der Physik 1901-1922 by Einstein A.

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