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The figure above illustrates the "toothpick sequence" described in 2010 by David Applegate, Omar E. Pol and Neil J. A. Sloane. We start with a blank slate, then place a single toothpick at the origin. At every subsequent step we find all exposed ends of toothpicks and place new toothpicks there. The new toothpick is centered on the endpoint of the old toothpick and oriented perpendicular to it.
To learn more, see sequence A139250 in Sloane's Online Encyclopedia of Integer Sequences or the paper by Applgate et al. The toothpick process is also discussed in the bit-player article Joshua Trees and Toothpicks.
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