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Pentagon Tiling


PentagonTilings

There are at least 15 classes of convex pentagonal tilings, as illustrated above. The first five were discovered during investigations of German mathematician Karl Reinhardt in 1918. After a gap of 50 years, R. B. Kershner found three more in 1968. Richard James subsequently discovered a ninth type of pentagonal tiling in 1975 and over the next few years, Marjorie Rice discovered another four types. Rolf Stein found a 14th tiling in 1985. The most recently discovered 15th tiling was found by Casey Mann, Jennifer McLoud and David Von Derau of the University of Washington Bothell in 2015 using a computer to exhaustively search through a large but finite set of possibilities (Bellos 2015).

It has not been proven whether these 15 cases exhaust all possible tilings, but no others are known.

Note that the tile in the 14th tiling is essentially different from the others because it is unique (up to similarity), while all the others form families with at least one parameter. For this special tile, all angles are determined, and all sides are in fixed ratios. Its angles A and E are 90 degrees and 110.67667... degrees, respectively. The tiling by this tile is also unique (B. Grünbaum, pers. comm.).


See also

Pentagon, Tiling

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References

Bellos, A. "Attack on the Pentagon Results in Discovery of New Mathematical Tile." The Guardian, Aug. 11, 2015. http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/aug/10/attack-on-the-pentagon-results-in-discovery-of-new-mathematical-tile.Bowers, P. L. and Stephenson, K. "A 'Regular' Pentagonal Tiling of the Plane." Conformal Geom. Dyn. 1, 58-68, 1997.Pegg, E. Jr. "The 14 Different Types of Pentagons that Tile the Plane." http://www.mathpuzzle.com/tilepent.html.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 75, 1999.ten Have, M. "Pentagons and the Golden Section." http://www.mtenhave.net/5/pentagon.htm.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 177-179, 208, and 211, 1991.

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Pentagon Tiling

Cite this as:

Weisstein, Eric W. "Pentagon Tiling." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PentagonTiling.html

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