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Toothpick sequence

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The first three steps of the toothpick sequence and its emulation by a cellular automaton with the Margolus neighborhood
The 89th stage of the sequence, one of the stages at which T(n)/n2 is a local minimum

In geometry, the toothpick sequence is a sequence of 2-dimensional patterns which can be formed with a recursive procedure.

The first design is a single "toothpick", or line segment. Every design after the first can be formed by taking the previous design and, for every exposed toothpick end, placing another toothpick centered on that end at a right angle to the earlier toothpick.[1]

This process results in a pattern of growth in which the number of segments at stage n oscillates with a fractal pattern between 0.45n2 and 0.67n2. If T(n) denotes the number of segments at stage n, then the local maxima of T(n)/n2 occur when n is near a power of two, while the local minima occur near numbers that are approximately 1.43 times a power of two.[2] The structure of stages in the toothpick sequence often resemble the T-square fractal, or the arrangement of cells in the Ulam–Warburton cellular automaton.[1]

References

  1. ^ a b Applegate, David; Pol, Omar E.; Sloane, N. J. A. (2010). "The Toothpick Sequence and Other Sequences from Cellular Automata". Retrieved 18 September 2012. {{cite journal}}: Cite journal requires |journal= (help)
  2. ^ Cipra, Barry (2010). "What Comes Next?" (PDF). Science. 327. AAAS: 943. Retrieved 18 September 2012.
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