Toothpick sequence
In geometry, the toothpick sequence is a sequence of 2-dimensional patterns which can be formed by repeatedly adding line segments ("toothpicks") to the previous pattern in the sequence.
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 values of n for which T(n)/n2 is near its maximum occur when n is near a power of two, while the values for which it is near its minimum 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
- ^ 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) - ^ Cipra, Barry (2010). "What Comes Next?" (PDF). Science. 327. AAAS: 943. Retrieved 18 September 2012.
External links
- A list of integer sequences related to the Toothpick Sequence from the On-line Encyclopedia of Integer Sequences. (note: the IDs like A139250 are IDs within the OEIS, and descriptions of the sequences can be located by entering these IDs in the OEIS search page.)
- A java applet demonstrating the sequence