@MISC{Culik_anaperiodic, author = {Karel Culik and II}, title = {An aperiodic set of 13 Wang tiles}, year = {} }

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Abstract

A new aperiodic tile set containing only 13 tiles over 5 colors is presented. Its construction is based on a technique recently developed by J. Kari. The tilings simulate behavior of sequential machines that multiply real numbers in balanced representations by real constants. 1 Introduction Wang tiles are unit square tiles with colored edges. A tile set is a finite set of Wang tiles. We consider tilings of the infinite Euclidean plane using arbitrarily many copies of the tiles in the given tile set. The tiles are placed on the integer lattice points of the plane with their edges oriented horizontally and vertically. The tiles may not be rotated. A tiling is valid if everywhere the contiguous edges have the same color. Let T be a finite tile set, and f : ZZ 2 ! T a tiling. Tiling f is periodic with period (a; b) 2 ZZ 2 \Gamma f(0; 0)g iff f(x; y) = f(x + a; y + b) for every (x; y) 2 ZZ 2 . If there exists a periodic valid tiling with tiles of T , then there exists a doubly period...