The Heesch Tile: An Interesting Non-Tiler

Steven Dutch, Professor Emeritus, Natural and Applied Sciences, University of Wisconsin - Green Bay

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The Heesch Tile consists of a square joined to an equilateral triangle and a 30-60-90 half triangle as shown by the yellow tile at lower right. It can be completely surrounded by copies of itself, yet it does not tile the plane.

The Heesch Tile can even cover fairly complex patches. Here a ring of tiles in red and purple is completely enclosed by other Heesch Tiles.

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References

Grunbaum, B and Shephard, G. C., Tilings and Patterns, Freeman, 1987. Just about everything there was to know on the subject at the time.

It turns out there are a lot (actually an infinity) of Heesch tiles. In fact the Heesch number of a tile is the number of times a single tile can be surrounded before the tiling breaks down. Two good sites are:

Peter RaeDschelders at http://home.planetinternet.be/~praeDsch/heersch.htm

Erich Friedman at http://www.stetson.edu/~efriedma/papers/heesch/heesch.html

Mark Thompson at http://www.flash.net/~markthom/html/self-surrounding_tiles.html

A. Fontaine, "An infinite number of plane figures with Heesch number two". Journal of Combinatorial Theory A 57 (1991) 151-156.

P. RaeDschelders, "Heesch Tiles Based on Regular polygons". Geombinatorics, 7 (1998), 101-106.

M. Senechal, Quasicrystals and Geometry, Cambridge Univ. Press, 1995.

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Created 14 October 1999, Last Update 20 January 2000