The Geometry Junkyard: Penrose Tiling

Penrose Tiling

Penrose Tiles, adresses http :

Penrose was not the first to discover aperiodic tilings, but his is probably the most well-known. In its simplest form, it consists of 54- 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.

The Art and Science of Tiling. Penrose tiles at Carleton College.
Duane Bailey's color postscript Penrose tiler
Cellular automaton run on Penrose tiles, D. Griffeath.
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. dans Le Site :
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.
Heureka, the Finnish science center uses Penrose tiles to pave the area in front of its main entrance. (Unfortunately, the picture included here is not very good -- see the Mathematical Intelligencer 18(4), Fall 1996, p. 65 for a better photo.)
Irrational tiling by logical quantifiers. LICS proceedings cover art by Alvy Ray Smith, based on the Penrose tiling.
Java Penrose Tiler, Geert-Jan van Opdorp. Shuxiang Zeng has written another Java applet to play with Penrose tiles.
Kadon Enterprises, makers of games and puzzles including polyominoes and Penrose tiles.
Making your own set of Penrose rhombs, N. Casey.
Math, Penrose, Plato, and Pentagrams. Philip Marsh claims that Penrose's pentagonal tilings connect him to the numerology of the Pythagoreans.
Dave Molnar's research on non-Euclidean symmetry and long-range order, Penrose and substitution tilings, L-systems, and cellular automata.
Penrose mandala and five-way Borromean rings.
Das Penrose Parkett. (In German.)
Penrose Pavers in Penngrove. Pat Walp shows off a path of concrete Penrose rhombs he made for his garden.
Penrose quilt on a snow bank, M.&S. Newbold. See also Lisbeth Clemens' Penrose quilt.
Penrose-tiled swallow
Penrose Tilings at Miami Univ.
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 at Storey Hall, RMIT, Melbourne, Australia.
Penrose tilings and the golden mean, K. Wiedman.
Penrose-Wang tilings. Tony Smith describes some of the mathematics behind these aperiodic tilings, somehow leading to the concluding question "Can musical sequences also simulate the operation of any Turing machine?"
Perplexing poultry Penrose pieces from pentaplex. Also comes with alien space dogs.
PentaBlocks, 1 cm thick regular pentagons, 36-degree isosceles triangles and rhombi, and five-point stars.
Pentagonal coffee table with rhombic bronze casting related to the Penrose tiling, by Greg Frederickson. dans Le Site :
Pretty Penrose picture, J. Beale, Stanford.
Quasicrystals and color symmetry. Ron Lifshitz provides a light introduction to the symmetry of periodic and aperiodic crystals, and the complications introduced by including permutations of colors in a coloring as part of a symmetry operation. His publication list includes more technical material on the same subject.
Quasitiler image, E. Durand.
Saints Among Us. Anna Chupa makes kaleidoscopic photomontages based on the geometry of the Penrose tiling.
Shawn's mathematical gallery. Penrose tilings, Newton-iteration convergence-domain fractals, Schlegel diagrams of four-dimensional polyhedra, and more.
Tesselations, 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.
Toilet paper plagiarism. A big tissue company tries to rip off Sir Roger P.
Z² section of a Penrose tiling. Robbie Robinson explains his work on the dynamical theory of tilings.

From the Geometry Junkyard, computational and recreational geometry pointers.
Send email if you know of an appropriate page not listed here.
David Eppstein, Theory Group, ICS, UC Irvine.
Semi-automatically filtered from a common source file. Last update: 26 May 1999, 16:16:11 PDT.