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