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Penrose Tiles


PenroseTiles

The Penrose tiles are a pair of shapes that tile the plane only aperiodically (when the markings are constrained to match at borders). These two tiles, illustrated above, are called the "kite" and "dart," respectively. In strict Penrose tiling, the tiles must be placed in such a way that the colored markings agree; in particular, the two tiles may not be combined into a rhombus (Hurd).

Two additional types of Penrose tiles known as the rhombs (of which there are two varieties: fat and skinny) and the pentacles (or which there are six type) are sometimes also defined that have slightly more complicated matching conditions (McClure 2002).

In 1997, Penrose sued the Kimberly Clark Corporation over their quilted toilet paper, which allegedly resembles a Penrose aperiodic tiling (Mirsky 1997). The suit was apparently settled out of court.

PenroseTilesAcuteObtuse

To see how the plane may be tiled aperiodically using the kite and dart, divide the kite into acute and obtuse tiles, shown above (Hurd).

PenroseTilesInflationDeflation

Now define "deflation" and "inflation" operations. The deflation operator takes an acute triangle to the union of two acute triangles and one obtuse, and the obtuse triangle goes to an acute and an obtuse triangle. These operations are illustrated above. Note that the operators do not respect tile boundaries, but do respect half-tiles.

PenroseTilesStarSun

When applied to a collection of tiles, the deflation operator leads to a more refined collection. The operators do not respect tile boundaries, but do respect the half tiles defined above. There are two ways to obtain aperiodic tilings with 5-fold symmetry about a single point. They are known as the "star" and "sun" configurations, and are shown above (Hurd).

PenroseTilesStarSun3

Higher order versions can then be obtained by deflation. For example, the illustrations above depict the third-order deflations (Hurd).

John Conway has asked if Penrose tilings are three colorable in such a way that adjacent tiles receive different colors. Sibley and Wagon (2000) proved that tilings by rhombs are three-colorable, and Babilon (2001) proved that tilings by kites and darts are three-colorable. McClure then found an algorithm that appears to three-color tilings by kites and darts, rhombs, and pentacles.


See also

Kepler's Monsters, Tiling

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References

Babilon, R. "3-Colourability of Penrose Kite-and-Dart Tilings." Disc. Math. 235, 137-143, 2001.Gardner, M. "Extraordinary Nonperiodic Tiling that Enriches the Theory of Tiles." Sci. Amer. 237, 110-119, Dec. 1977.Gardner, M. "Penrose Tiling" and "Penrose Tiling II." Chs. 1-2 in Penrose Tiles and Trapdoor Ciphers... and the Return of Dr. Matrix, reissue ed. New York: W. H. Freeman, pp. 1-29, 1989.Gardner, M. The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems. New York: W. W. Norton, pp. 216 and 218, 2001.Grünbaum, B. and Shephard, G. C. Tilings and Patterns. New York: W. H. Freeman, 1986. Hurd, L. P. "Penrose Tiles." http://library.wolfram.com/infocenter/MathSource/595/.McClure, M. "Three-Coloring Penrose Tiles." http://www.unca.edu/~mcmcclur/mathematicaGraphics/PenroseColoring/.McClure, M. "A Stochastic Cellular Automaton for Three-Coloring Penrose Tiles." Computers & Graphics 26, 519-524, 2002. http://www.unca.edu/~mcmcclur/professional/Penrose3Color.pdf.Mirsky, S. "The Emperor's New Toilet Paper." Sci. Amer. 277, 24, July 1997.Pegg, E. Jr. "Math Games: Melbourne, City of Math." Sep. 5, 2006. http://www.maa.org/editorial/mathgames/mathgames_09_05_06.html.Peterson, I. The Mathematical Tourist: Snapshots of Modern Mathematics. New York: W. H. Freeman, pp. 86-95, 1988.Radin, C. Miles of Tiles. Providence, RI: Amer. Math. Soc., pp. 2 and 34-36, 1999.Sibley, T. and Wagon, S. "Rhombic Penrose Tilings Can Be 3-Colored." Amer. Math. Monthly 107, 251-253, 2000.Smith, T. "Penrose Tilings and Wang Tilings." http://www.innerx.net/personal/tsmith/pwtile.html.Vichera, M. "Penrose Tiling." http://www.vicher.cz/puzzle/penrose/penr.htm.Wagon, S. "Penrose Tiles." §4.3 in Mathematica in Action. New York: W. H. Freeman, pp. 108-117, 1991.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 175-177, 1991.

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Penrose Tiles

Cite this as:

Weisstein, Eric W. "Penrose Tiles." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PenroseTiles.html

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