The image below is a representation of audioactive decay. The iteration started with "1" and was iterated 22 times.
The colour code is as follows: 1 = white, 2 = grey, 3 = black. The iterations are laid around the center ("1") in such a way that the description of an inner substring covers the same angle as that substring (note the thick grey radial spokes of "22"). Because of the black borders, strings of "33" are not clearly distinguished from a single "3"; but it can be inferred from the inner ring if it is a double "3" or not.
The horizontal border that extends to the right from the center corresponds to an angle of 0 degrees - there counting starts in counterclockwise direction. Note the repetition of start and end patterns of the iterates (period 3 of the beginnings and period 2 of the ends). Note also the colour field borders that go unbroken like spokes outward. These are separating lines for Conway's elements.
Because of the exponential growth of the string length, the colored fields soon become indistinguishable. It is interesting that certain regions exhibit faster growth than others which is seen in more densely spaced colour fields. This effect is a consequence of the fact that some elements multiply faster than others.
The first part grows faster than the last part which leads to a conterclockwise spiralling design. The self-similarity of the Gleichniszahlenreihe adds to this pleasing diagram which reminds me of certain Indian baskets.
I discovered the possibility of this representation in December 1996. After some pencil sketches I finally decided to write a perl program that produced the appropriate postscript code. For aesthetic reasons the diameter of the center was made smaller than twice the radius increase for the outer rings.
book chapter: A Fractal Voyage With Conway's Audioactive Decay
Essays on mathematical themes.
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© Copyright 1993, Mario Hilgemeier, email: contact