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Heesch Number

The Heesch number of a closed plane figure is the maximum number of times that figure can be completely surrounded by copies of itself. The determination of the maximum possible (finite) Heesch number is known as Heesch's problem. The Heesch number of a triangle, quadrilateral, regular hexagon, or any other shape that can tile or tessellate the plane, is infinity. Conversely, any shape with infinite Heesch number must tile the plane (Eppstein).

HeeschNumber5

A tile invented by R. Ammann has Heesch number three (Senechal 1995), and Mann has found an infinite family of tiles with Heesch number five (illustrated above), the largest (finite) number known.

A database of Heesch tilings is maintained by Mann (2008).

See also

Heesch's Problem, Tiling

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References

Eppstein, D. "Heesch's Problem." http://www.ics.uci.edu/~eppstein/junkyard/heesch/.Fontaine, A. "An Infinite Number of Plane Figures with Heesch Number Two." J. Comb. Th. A 57, 151-156, 1991.Friedman, E. "Heesch Tiles with Surround Numbers 3 and 4." http://www.stetson.edu/~efriedma/papers/heesch/heesch.html.Grünbaum, B. and Shephard, G. C. Tilings and Patterns. New York: W. H. Freeman, 1986.Mann, C. "Heesch's Problem." http://www.math.unl.edu/~cmann/math/heesch/heesch.htm.Mann, C. "The Edge-Marked Polyform Database." Aug. 12, 2008. http://www.math.uttyler.edu/polyformDB/.Raedschelders, P. "Heesch Tiles Based on Regular Polygons." Combinatorics 7, 101-106, 1998.Raedschelders, P. "Heesch-Tiles Based on n-gons." http://home.planetinternet.be/~praedsch/heersch.htm.Senechal, M. Quasicrystals and Geometry. New York: Cambridge University Press, 1995.Thompson, M. "Self-Surrounding Tiles." http://home.flash.net/~markthom/html/self-surrounding_tiles.html.

Referenced on Wolfram|Alpha

Heesch Number

Cite this as:

Weisstein, Eric W. "Heesch Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HeeschNumber.html

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