Penrose Tiles
Penrose was not the first to discover aperiodic tilings, but his is probably the most well-known. In its simplest form, it consists of 36- and 72-degree rhombi, with "matching rules" forcing the rhombi to line up against each other only in certain patterns. It can also be formed by tiles in the shape of "kites" and "darts" or even by deformed chickens (see the "perplexing poultry" entry below). Part of the interest in this tiling stems from the fact that it has a five-fold symmetry impossible in periodic crystals, and has been used to explain the structure of certain "quasicrystal" substances.
Ancient Islamic Penrose Tiles. Peter Lu uncovers evidence that the architects of a 500-year-old Iranian shrine used Penrose tiling to lay out the decorative patterns on its archways. From Ivars Peterson's MathTrek.
Aperiodic tiling and Penrose tiles, Steve Edwards.
The Art and Science of Tiling. Penrose tiles at Carleton College.
Cellular automaton run on Penrose tiles, D. Griffeath. See also Eric Weeks' page on cellular automata over quasicrystals.
Clusters and decagons, new rules for using overlapping shapes to construct Penrose tilings. Ivars Peterson, Science News, Oct. 1996.
Five-fold symmetry in crystalline quasicrystal lattices, Donald L. D. Caspar and Eric Fontano.
Gallery of interactive on-line geometry. The Geometry Center's collection includes programs for generating Penrose tilings, making periodic drawings a la Escher in the Euclidean and hyperbolic planes, playing pinball in negatively curved spaces, viewing 3d objects, exploring the space of angle geometries, and visualizing Riemann surfaces.
Goldene Schnittmuster. Article in German on Penrose tiling and related topics.
Irrational tiling by logical quantifiers. LICS proceedings cover art by Alvy Ray Smith, based on the Penrose tiling.
Kadon Enterprises, makers of games and puzzles including polyominoes and Penrose tiles.
Mathematical imagery by Jos Leys. Knots, Escher tilings, spirals, fractals, circle inversions, hyperbolic tilings, Penrose tilings, and more.
Non periodic tiling of the plane. Including Penrose tiles, Pinhweel tiling, and more. Paul Bourke.
Ozbird Escher-like tessellations by John Osborn, including several based on Penrose tilings.
Patterns within rhombic Penrose tilings. Stephen Collins' program "Bob" generates these tilings and explores the patterns formed by geodesic walks in them.
Penrose mandala and five-way Borromean rings.
Penrose quilt on a snow bank, M.&S. Newbold. See also Lisbeth Clemens' Penrose quilt.
The penrose tile and the golden mean: towards hyperdimensional intergeometry.
Penrose tiles and how their visualization leads to strange looks from priests and small children. Drew Olbrich.
Penrose tiles and worse. This article from Dave Rusin's known math pages discusses the difficulty of correctly placing tiles in a Penrose tiling, as well as describing other tilings such as the pinwheel.
Penrose Tiles entry from E. Weisstein's treasure trove.
Pentagonal coffee table with rhombic bronze casting related to the Penrose tiling, by Greg Frederickson.
Quasitiler image, E. Durand.
Santa Fe Ribbon, painting by Connie Simon featuring a rhombic Penrose tiling.
Tessellations, a company which makes Puzzellations puzzles, posters, prints, and kaleidoscopes inspired in part by Escher, Penrose, and Mendelbrot.
Three-color the Penrose tiling? Mark Bickford asks if this tiling is always three-colorable. Ivars Peterson reports on a new proof by Tom Sibley and Stan Wagon that the rhomb version of the tiling is 3-colorable; A proof of 3-colorability for kites and darts was recently published by Robert Babilon [Discrete Mathematics 235(1-3):137-143, May 2001]. This is closely related to my page on line arrangement coloring, since every Penrose tiling is dual to a "multigrid", which is just an arrangement of lines in parallel families. But my page only deals with finite arrangements, while Penrose tilings are infinite.
Tilings. Lecture notes from the Clay Math Institute, by Richard Stanley and Federico Ardila, discussing polyomino tilings, coloring arguments for proving the nonexistence of tilings, counting how many tilings a region has, the arctic circle theorem for domino tilings of diamonds, tiling the unit square with unit-fraction rectangles, symmetry groups, penrose tilings, and more. In only 21 pages, including the annotated bibliography. A nice but necessarily concise introduction to the subject. (Via Andrei Lopatenko.)
Toilet paper plagiarism. A big tissue company tries to rip off Sir Roger P.
The trouble with five. Craig Kaplan explains why five-fold symmetry doesn't work in regular plane tilings, but does work for the Penrose tiling.
Voronoi diagram of a Penrose tiling (rhomb version), Cliff Reiter.
