The Geometry Junkyard: Tilings
TilingOne way to define a tiling is a partition of an infinite space (usuallyEuclidean) into pieces having a finite number of distinct shapes.Tilings can be divided into two types,periodic and aperiodic, depending on whether they have anytranslational symmetries.If these symmetries exist, they form a lattice.However there hasbeen much recent research and excitement on aperiodic tilings(which lack such symmetries) and their possible realizationin certain crystal structures. Tilings also have connectionsto much of pure mathematics including operator K-theory, dynamicalsystems, and non-commutative geometry.Aperiodiccolored tilings, F. Gähler.Alsoavailable in postscript.An aperiodic set of Wang cubes, J. UCS 1:10 (1995).Culik and Kari describe how to increase the dimension of sets ofaperiodic tilings, turning a 13-square set of tiles into a 21-cube set.Aperiodic space-filling tiles:John Conway describes a way of glueing two prisms together to form a shape that tiles space onlyaperiodically.Ludwig Danzer speaks at NYU onvarious aperiodic 3d tilings including Conway'sbiprism.Chaotic tilingof two kinds of equilateral pentagon, with30degree symmetry, Ed Pegg Jr.Cognitive EngineeringLab, Java applets for exploring tilings, symmetry, polyhedra, andfour-dimensional polytopes.Complexregular tesselations on the Euclid plane, Hironori Sakamoto.Acomputational approach to tilings. Daniel Huson investigates thecombinatorics of periodic tilings in two and three dimensions, includinga classification of the tilings by shapes topologically equivalent tothe five Platonic solids.Cool math: tessellationsAndrew Crompton.Grotesque geometry, Tessellations, Lifelike Tilings, Escher style drawings,Dissection Puzzles, Geometrical Graphics, Mathematical Art. Anamorphic Mirrors, Aperiodic tilings, Optical Machines.Delta Blocks.Hop David discusses ideas for manufacturing building blocks based onthe tetrahedron-octahedron space tiling depicted in Escher's "Flatworms".Dissection and dissection tiling.This page describes problems of partitioning polygonsinto pieces that can be rearranged to tile the plane.(With references to publications on dissection.) The downstairs half bath.Bob Jenkins decorated his bathroom with ceramic and painted pentagonal tiles.Equilateralpentagons that tile the plane, Livio Zucca.Theequivalence of two face-centered icosahedral tilings with respect tolocal derivability, J. Phys. A26 (1993) 1455. J. Roth dissects anaperiodic three-dimensional tiling involving zonohedra into anothertiling involving tetrahedra and vice versa.Escher-like tilings of interlocking animaland human figures, by various artists.Fisher Pavers.A convex heptagon and some squares produce an interesting four-waysymmetric tiling system.Fivespace-filling polyhedra. And not the ones you're likely thinking of,either.Guy Inchbald, reproduced from Math. Gazette 80, November 1996.The fractal art ofWolter Schraa. Includes some nice reptiles and sphere packings.Fractalreptiles and othertilings by IFSattractors, Stewart Hinsley.Fractal tilings.Fractiles,multicolored magnetic rhombs with angles based on multiples of pi/7.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.GeometricArts. Knots, fractals, tesselations, and op art.Formerly QuincyKim's World of Geometry.Ghostdiagrams, Paul Harrison's software for finding tilings withWang-tile-like hexagonal tiles, specified by matching rules on theiredges. These systems are Turing-complete, so capable of forming allsorts of complex patterns; the web site shows binary circuitry, fractals,1d cellular automaton simulation, Feynman diagrams, and more.Heesch's problem. How many times can a shapebe completely surrounded by copies of itself, without being able to tilethe entire plane? W. R. Marshall and C. Mann have recently madesignificant progress on this problem using shapes formed by indentingand outdenting the edges of polyhexes. Infect.Eric Weeks generates interesting colorings of aperiodic tilings.InvestigatingPatterns: Symmetry and Tessellations.Companion site to a middle school text by Jill Britton,with links to many other web sites involving symmetry or tiling.Irreptiles.Karl Scherer and Erich Friedman generalize the concept of a reptile(tiling of a shapeby smaller copies of itself) to allow the copies to have different scales.See alsoKarl Scherer's two-part irreptile puzzle. The isoperimetric problem for pinwheel tilings.In these aperiodic tilings (generated by a substitution system involvingsimilar triangles) vertices are connected by paths almost as goodas the Euclidean straight-line distance.Jovo Click 'n Construct.Plastic click-together triangular, square, and pentagonal tiles forbuilding models of polyhedra and polygonal tilings.Includes a mathematical modelgalleryshowing examples of shapes constructable from Jovo.Kaleidotilesoftware for visualizing tilings of the sphere, Euclidean plane, andhyperbolic plane.Keller's cube-tiling conjecture is false in high dimensions,J. Lagarias and P. Shor, Bull. AMS 27 (1992).Constructs a tiling of ten-dimensional space by unit hypercubesno two of which meet face-to-face, contradicting aconjectureof Kellerthat any tiling included two face-to-face cubes.RichardKenyon's Gallery of tilings by squares and equilateral triangles ofvarying sizes.Mike Kolountzakis' publications include several recent papers on lattice tiling.Labyrinth tiling.This aperiodic substitution tiling by equilateral and isosceles trianglesforms fractal space-filling labyrinths. Lenses,rational-angled equilateral hexagons can tile the plane in variousinteresting patterns. See also Jorge Mireles' nicelenspuzzle applet: rotate decagons and stars to get the pieces into theright places.Log-spiral tiling,and other radialand spiral tilings, S. Dutch.Mathematicalorigami, Helena Verrill. Includes constructions of a shape withgreater perimeter than the original square, tessellations, hyperbolicparaboloids, and more.Mitre Tiling.Ed Pegg describes the discovery of the versatile tiling system(with Adrian Fisher and Miroslav Vicher), also discussing manyother interesting tilings including a tile that can fill the plane witheither five-fold or six-fold symmetry.Modularity in art.Slavik Jablan explores connections between art, tiling, knotwork, andother mathematical topics.Dave Molnar's researchon non-Euclidean symmetry and long-range order, Penrose and substitutiontilings, L-systems, and cellular automata.Newdirections in aperiodic tilings, L. Danzer, Aperiodic '94.Nonperiodic tiling of the plane.Including Penrose tiles, Pinhweel tiling, and more. Paul Bourke.Nontrivialconvexity. Ed Pegg asks about partitions of convex regions intoequal tiles, other than the "trivial" ones in which some rotational ortranslational symmetry group relates all the tile positions to each other.See also MiroslavVicher's page on nontrivial convexityOrigamitessellations andpaper mosaics, Alex Bateman.Parquetdeformations.Craig Kaplan involves continuous spatial transformations of one tiling to another.Penrose tilings.This five-fold-symmetric tiling by rhombs or kites and darts is probably the most well known aperiodic tiling.Perplexingpentagons, Doris Schattschneider, from the Discovering GeometryNewsletter.A brief introduction to the problem of tiling the plane by pentagons.PentagonalTessellations. John Savard experiments with substitution systems toproduce tilings resembling Kepler's.Pentagons that tile the plane, Bob Jenkins.See alsoEd Pegg's page onpentagon tiles.PerronNumber Tiling Systems.Mathematica software for computing fractals that tile the plane fromPerron numbers.Platonictesselations of Riemann surfaces, Gerard Westendorp.Polygonswith angles of different k-gons.Leroy Quet asks whether polygons formed by combining the angles ofdifferent regular polygons can tile the plane.The answer turns out to be related toEgyptian fractiondecompositions of 1 and 1/2.PolyMultiForms.L. Zucca uses pinwheel tilers to dissect an illustration of the Pythagoreantheorem into few congruent triangles.Polyomino tiling.Joseph Myers classifies the n-ominoes up to n=15 according to howsymmetrically they can tile the plane.Polyominoes, figures formed from subsetsof the square lattice tiling of the plane. Interesting problemsassociated with these shapes include finding all of them, determiningwhich ones tile the plane, and dissecting rectangles or other shapesinto sets of them. Also includes relatedmaterial on polyiamonds, polyhexes, and animals.ProtoZoneinteractive shockwave museum exhibits for exploring geometric conceptssuch as symmetry, tiling, and wallpaper groups.Publications on quasicrystals and aperiodic tilings, F. Gähler.A Puzzling Journey To The Reptiles And Related Animals, andNew Mosaics.Books on tiling by Karl Scherer.QuaquaversalTilings and Rotations. John Conway and Charles Radin describe athree-dimensional generalization of the pinwheel tiling, the mathematicsof which is messier due to the noncommutativity of three-dimensionalrotations.Quasicrystalsand aperiodic tilings, A. Zerhusen, U. Kentucky.Includes a nice description of how to make 3d aperiodic tilesfrom zometool pieces.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.Reptileproject-of-the-month from the Geometry Forum.Form tilings by dividing polygons into copies of themselves.Rhombicspirallohedra, concave rhombus-faced polyhedra that tile space,R. Towle.Rhombictilings. Abstract of Serge Elnitsky's thesis, "Rhombic tilings ofpolygons and classes of reduced words in Coxeter groups". He also supplied thepicture below of a rhombically tiled 48-gon, available with better colorresolution from his website. Self-affine tiles, J. Lagarias and Y. Wang, DIMACS.Mathematics of a class of generalized reptiles.Semi-regulartilings of the plane, K. Mitchell, Hobart and William Smith Colleges.Some generalizations of the pinwheel tiling, L. Sadun, U. Texas.Some planar tilingsgenerated by the lattice projection method(of which the Penrose tiling is a special case)by Andrew Lewis, Queens U.SpaceBric building blocksand Windows software based on a tiling of 3d space by congruenttetrahedra.Spidron,a triangulated double spiral shape tiles the plane and various othersurfaces. With photos of related paperfolding experiments.Spiral tilings.These similarity tilings are formed by applying the exponential functionto a lattice in the complex number plane. Symmetry,tilings, and polyhedra, S. Dutch.Symmetry and Tilings. Charles Radin, Not. AMS, Jan. 1995.See also hisSymmetryof Tilings of the Plane, Bull. AMS 29 (1993), which proves that thepinwheel tiling is ergodic and can be generated by matching rules.TapratsJava software for generating symmetric Islamic-style star patterns.Tesselatinglocking polyominos, Bob Newman.Tessellationlinks, S. Alejandre.Tessellationresources. Compiled for the Geometry Center by D. Schattschneider.3D-Geometrie.T. E. Dorozinski provides a gallery of images of 3d polyhedra,2d and 3d tilings, and subdivisions of curved surfaces.Tilableperspectives.Patrick Snels creates two-dimensional images which tile the plane toform 3d-looking views including some interesting Escher-like warpedperspectives.See also his even more Escherian tesselations page.Tiling plane& fancy, Steven Edwards, SPSU.Tiling the infinite grid with finite clusters.Mario Szegedy describes an algorithm for determining whether a (possiblydisconnected) polyomino will tile the plane by translation,in the case where the number of squares in the polyomino is a primeor four.Tiling the integers with one prototile.Talk abstract by Ethan Coven on a one-dimensional tiling problem on theboundary betweengeometry and number theory, with connections to factorization of finitecyclic groups.See also Coven's paper with Aaron Meyerowitz,Tiling the integerswith translates of one finite set.Tiling problems.Collected at a problem session at Smith College, 1993, byMarjorie Senechal.Tilingtransformer. Java applet for subdividing tilings (starting from asquare or hexagonal tiling) in various different ways. Tiling dynamical systems.Chris Hillman describes his researchon topological spaces in which each point represents a tiling.Ona tiling scheme by M. C. Escher, D. Davis, Elect. J. Combinatorics.Tilings of hyperbolic space.Tilings and visual symmetry, Xah Lee.Toroidal tile for tessellating three-space, C. Séquin, UC Berkeley.Totally Tessellated.Mosaics, tilings, Escher, and beyond.Triangle tiling. Geom. Ctr. exhibit at the Science Museum of Minnesota.Federation Square.This building in Melbourne uses the pinwheeltiling as a design motif. Thanks to Khalad Karim for identifying it.Photos by Dick Hess, scanned by Ed Pegg Jr.See this Flickrphotopool for many more photos.  true_tilemailing list for discussion of Euclidean and non-Euclidean tilings.Tysenloves hexagons. And supplies ascii, powerpoint, and png graphics forseveral styles of hexagonal grid graph paper.Unbalancedanisohedral tiling.Joseph Myers andJohn Berglund find a polyhex that must be placed two different ways ina tiling of a plane, such that one placement occurs twice as often asthe other.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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