IFS images also can be rendered by the Random IFS algorithm, an extension of the Chaos Game to general linear transformations. A natural question, as far as we know first posed by Ian Stewart, is what picture does the Random IFS Algorithm generate if the driving sequence is not random? Not surprisingly, the answer depends on how the sequence departs from randomness. Stewart asked the nonrandomness question for a Sierpinski gasket IFS. We prefer the IFS
T1(x, y) = (x/2, y/2)
T2(x, y) = (x/2, y/2) + (1/2, 0)
T3(x, y) = (x/2, y/2) + (0, 1/2)
T4(x, y) = (x/2, y/2) + (1/2, 1/2)
because, driven by a uniform random sequence, the Random IFS Algorithm fills in the unit square uniformly. Departures from uniform randomness are revealed by departures from uniform fill.
To motivate this excursion, here is an example first explored by H. Joel Jeffrey. The genetic code is written in an alphabet of four characters: C, A, T, and G. A sequence of several billion of these makes each of us. A sequence of 3957 symbols is needed to encode the production of the enzyme amylase. How can we convert a DNA sequence into an IFS picture? Read the sequence in order, and apply T1 whenever C is encountered, T2 for A, T3 for T, and T4 for G (for example). On the left is the picture that results. On the right is a picture that results when 3957 points are generated randomly, except that T4 never immediately follows T1.
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Not an exact match, but certainly suggestive. Here are a few more examples.
With this motivation, we investigate the effect of forbidden combinations on driving this IFS.
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