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The Geometry Junkyard: Penrose Tiling
Penrose TilesPenrose was not the first to discover aperiodictilings, 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 upagainst each other only in certain patterns. It can also be formedby tiles in the shape of "kites" and "darts" or even by deformedchickens (see the "perplexing poultry" entry below).Part of the interest in this tiling stems from the factthat it has a five-fold symmetry impossible inperiodic crystals, and has been used to explain the structure of certain"quasicrystal" substances.AncientIslamic Penrose Tiles. Peter Lu uncovers evidence that thearchitects of a 500-year-old Iranian shrine used Penrose tiling to layout the decorative patterns on its archways. From Ivars Peterson'sMathTrek.Aperiodictiling and Penrose tiles, Steve Edwards.TheArt and Science of Tiling.Penrose tiles at Carleton College.Cellularautomaton run on Penrose tiles, D. Griffeath.See also EricWeeks' page on cellular automata over quasicrystals.Clusters anddecagons, new rules for using overlapping shapes to constructPenrose 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 generatingPenrose tilings, making periodic drawings a la Escher in the Euclideanand hyperbolic planes, playing pinball in negatively curved spaces,viewing 3d objects, exploring the space of angle geometries, andvisualizing Riemann surfaces.GoldeneSchnittmuster. Article in German on Penrose tiling and related topics.Irrationaltiling 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.Mathematical imagery by Jos Leys.Knots, Escher tilings, spirals, fractals, circle inversions, hyperbolictilings, Penrose tilings, and more.Making your own set of Penrose rhombs, N. Casey.Math, Penrose,Plato, and Pentagrams. Philip Marsh claims that Penrose's pentagonaltilings connect him to the numerology of the Pythagoreans.Dave Molnar's researchon non-Euclidean symmetry and long-range order, Penrose and substitutiontilings, L-systems, and cellular automata.Nonperiodic tiling of the plane.Including Penrose tiles, Pinhweel tiling, and more. Paul Bourke.Ozbird Escher-like tessellationsby John Osborn, including several based on Penrose tilings.Patterns withinrhombic Penrose tilings. Stephen Collins' program "Bob" generatesthese tilings and explores the patterns formed by geodesic walks in them.Peneschertiling, Marc Pickett.Penrosemandala and five-way Borromean rings.Das Penrose Parkett. (In German.)PenrosePavers in Penngrove. Pat Walp shows off a path of concrete Penroserhombs he made for his garden.Penrose quilt on asnow bank, M.&S. Newbold. See alsoLisbethClemens' Penrose quilt.The penrosetile and the golden mean: towards hyperdimensional intergeometry.Penrose-tiledlace doily.Penrose-tiled swallowPenrose tilesand how their visualization leads to strange looks from priests andsmall children. Drew Olbrich.Penrosetilings and quasitiler.PenroseTilings 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.PenroseTiles entry from E. Weisstein's treasure trove.Penrose tiles at Storey Hall, RMIT, Melbourne, Australia.See alsoAperiodicTiling, Penrose Tiling and the Generation of Architectural Forms,M. J. Ostwald, Nexus '98.Penrose tilings and the golden mean, K. Wiedman.See also this closely related Wheatoncollege course project.Penrose-Wangtilings. Tony Smithdescribes 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?"Penrose'sinfluence on Escher.Perplexing poultryPenrose pieces from pentaplex. Also comes with alien space dogs.Pentagonal coffee table with rhombic bronze casting related to the Penrose tiling, by Greg Frederickson.PrettyPenrose picture, J. Beale, Stanford.Quasicrystalsand color symmetry. Ron Lifshitz provides a light introduction tothe symmetry of periodic and aperiodic crystals, and the complicationsintroduced by including permutations of colors in a coloring as part ofa symmetry operation. Hispublicationlist includes more technical material on the same subject.QuasiGfreeware Windows Penrose tiling program.Quasitiler image, E. Durand.SaintsAmong Us.Anna Chupamakes kaleidoscopic photomontages based on the geometry of the Penrose tiling.Santa Fe Ribbon,painting by Connie Simon featuring a rhombic Penrose tiling.Shawn'smathematical gallery. Penrose tilings,Newton-iteration convergence-domain fractals,Schlegel diagrams of four-dimensional polyhedra,and more.Some planar tilingsgenerated by the lattice projection method(of which the Penrose tiling is a special case)by Andrew Lewis, Queens U.Tessellations,a company which makes Puzzellations puzzles, posters, prints, andkaleidoscopes inspired in part by Escher, Penrose, and Mendelbrot.Three-color the Penrose tiling?Mark Bickford asks if this tiling is always three-colorable.IvarsPeterson reports on a new proof by Tom Sibley and Stan Wagonthat the rhomb version of the tiling is 3-colorable;A proof of 3-colorability for kites and dartswas recently published by Robert Babilon [Discrete Mathematics 235(1-3):137-143, May 2001].This is closely related to my page on linearrangement coloring, since every Penrose tiling is dual toa "multigrid", which is just an arrangement of lines in parallel families.But my page only deals with finite arrangements, while Penrose tilings areinfinite.Tilings.Lecture notes from the Clay Math Institute, by Richard Stanley andFederico Ardila, discussing polyomino tilings, coloring arguments forproving the nonexistence of tilings, counting how many tilings a regionhas, 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 annotatedbibliography. A nice but necessarily concise introduction to the subject.(Via Andrei Lopatenko.)Toiletpaper plagiarism. A big tissue company tries to rip off Sir Roger P.Thetrouble with five. Craig Kaplan explains why five-fold symmetrydoesn't work in regular plane tilings, but does work for the Penrose tiling.Voronoidiagram of a Penrose tiling (rhomb version), Cliff Reiter.Z2section of a Penrose tiling.Robbie Robinsonexplains his work on the dynamical theory of tilings.From the Geometry Junkyard,computationaland recreational geometry pointers.Send email if youknow of an appropriate page not listed here.David Eppstein,Theory Group,ICS,UC Irvine.Semi-automaticallyfilteredfrom a common source file. |
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