Extension To Multiple Colors
Greg Turk and Jim Propp considered a simple extension to Langton's ant where instead of just two colors, more colors are used. The colors are modified in a cyclic fashion. A simple naming scheme is used: for each of the successive colors, a letter 'L' or 'R' is used to indicate whether a left or right turn should be taken. Langton's ant has the name 'RL' in this naming scheme.
Some of these extended Langton's ants produce patterns that become symmetric over and over again. One of the simplest examples is the ant 'RLLR'. One sufficient condition for this to happen is that the ant's name, seen as a cyclic list, consists of consecutive pairs of identical letters 'LL' or 'RR' (the term "cyclic list" indicates that the last letter may pair with the first one.) The proof involves Truchet tiles.
- Some example patterns in the multiple-color extension of Langton's Ants:
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RLR: grows chaotically. It is not known if this ant ever produces a highway.
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LLRR: grows symmetrically.
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LRRRRRLLR: fills space in a square around itself.
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LLRRRLRLRLLR: creates a convoluted highway.
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RRLLLRLLLRRR: creates a filled triangle shape that grows and moves.
Read more about this topic: Langton's Ant
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