Chapter 5: Two Dimensions and Beyond

Section 7: Systems Based on Constraints


An example of a tiling problem that is in some respects particularly close to the grid-based constraint systems discussed in the main text concerns covering the plane with polyominoes that are formed by gluing collections of squares together. Tiling by polyominoes has been investigated since at least the late 1950s, particularly by Solomon Golomb, but it is only very recently that sets of polyominoes which force non-periodic patterns have been found. The set (a) below was announced by Roger Penrose in 1994; the slightly smaller set (b) was found by Matthew Cook as part of the development of this book.

Both of these sets yield nested patterns. Steps in the construction of the pattern for set (b) are shown below. At stage n the number of polyominoes of each type is Fibonacci[2n - {2, 0, 1}]/{1, 2, 1}. Set (a) works in a roughly similar way, but with a considerably more complicated recursion.

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From Stephen Wolfram: A New Kind of Science [citation]