3D Printer Templates for Aperiodic Wang Tiles

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This Demonstration shows a set of 11 Wang tiles that tile the plane, but only aperiodically. Prisms with a polygonal base (triangle, square, pentagon, hexagon) are used instead of colors (white, red, green, blue).

Contributed by: Izidor Hafner (March 2020)
Open content licensed under CC BY-NC-SA



Wang tiles (or Wang dominoes), were first proposed by mathematician, logician and philosopher Hao Wang in 1961; they are a class of formal systems. They are modeled visually by square tiles with a color on each edge. A set of such tiles is selected, and copies of the tiles are arranged side by side with colors matched, without changing their orientation by rotation or reflection.

The basic question about a set of Wang tiles is whether it can tile the plane, that is, whether the entire infinite plane can be filled in this way. The next question is whether this can be done in a periodic pattern [1].


[1] Wikipedia. "Wang Tile." (Mar 10, 2020) en.wikipedia.org/wiki/Wang_tile.

[2] R. M. Robinson, "Undecidability and Nonperiodicity for Tilings of the Plane," Inventiones Mathematicae, 12(3), 1971 pp. 177–209. doi:10.1007/BF01418780.

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