- Aperiodic
colored tilings, F. Gähler.
Also
available in postscript.
- An aperiodic set of Wang cubes, J. UCS 1:10 (1995).
Culik and Kari describe how to increase the dimension of sets of
aperiodic 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 only
aperiodically.
Ludwig Danzer speaks at NYU on
various aperiodic 3d tilings including Conway's
biprism.
- Chaotic tiling
of two kinds of equilateral pentagon, with
30degree symmetry, Ed Pegg Jr.
- Cognitive Engineering
Lab, Java applets for exploring tilings, symmetry, polyhedra, and
four-dimensional polytopes.
- Complex
regular tesselations on the Euclid plane, Hironori Sakamoto.
- a
computational approach to tilings. Daniel Huson investigates the
combinatorics of periodic tilings in two and three dimensions, including
a classification of the tilings by shapes topologically equivalent to
the five Platonic solids.
- Cool math: tessellations
- Andrew 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 on
the tetrahedron-octahedron space tiling depicted in Escher's "Flatworms".
- Dérivés
de l'hexagone. Art by Jerome Pierre based on modifications to the edges in a hexagonal tiling of the plane.
- Dissection and dissection tiling.
This page describes problems of partitioning polygons
into 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.
- Equilateral
pentagons that tile the plane, Livio Zucca.
- The
equivalence of two face-centered icosahedral tilings with respect to
local derivability, J. Phys. A26 (1993) 1455. J. Roth dissects an
aperiodic three-dimensional tiling involving zonohedra into another
tiling involving tetrahedra and vice versa.
- Escher-like tilings of interlocking animal
and human figures, by various artists.
- Fisher Pavers.
A convex heptagon and some squares produce an interesting four-way
symmetric tiling system.
- Five
space-filling polyhedra. And not the ones you're likely thinking of,
either.
Guy Inchbald, reproduced from Math. Gazette 80, November 1996.
- The fractal art of
Wolter Schraa. Includes some nice reptiles and sphere packings.
- Fractal
reptiles and other
tilings by IFS
attractors, 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 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.
- Geometric
Arts. Knots, fractals, tesselations, and op art.
Formerly Quincy
Kim's World of Geometry.
- Ghost
diagrams, Paul Harrison's software for finding tilings with
Wang-tile-like hexagonal tiles, specified by matching rules on their
edges. These systems are Turing-complete, so capable of forming all
sorts 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 shape
be completely surrounded by copies of itself, without being able to tile
the entire plane? W. R. Marshall and C. Mann have recently made
significant progress on this problem using shapes formed by indenting
and outdenting the edges of polyhexes.
- Infect.
Eric Weeks generates interesting colorings of aperiodic tilings.
- Investigating
Patterns: 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 shape
by smaller copies of itself) to allow the copies to have different scales.
See also
Karl Scherer's two-part irreptile puzzle.
- The isoperimetric problem for pinwheel tilings.
In these aperiodic tilings (generated by a substitution system involving
similar triangles) vertices are connected by paths almost as good
as the Euclidean straight-line distance.
- Jovo Click 'n Construct.
Plastic click-together triangular, square, and pentagonal tiles for
building models of polyhedra and polygonal tilings.
Includes a mathematical model
gallery
showing examples of shapes constructable from Jovo.
- Kaleidotile
software for visualizing tilings of the sphere, Euclidean plane, and
hyperbolic 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 hypercubes
no two of which meet face-to-face, contradicting a
conjecture
of Keller
that any tiling included two face-to-face cubes.
- Richard
Kenyon's Gallery of tilings by squares and equilateral triangles of
varying sizes.
- Mike Kolountzakis' publications include several recent papers on lattice tiling.
- Labyrinth tiling.
This aperiodic substitution tiling by equilateral and isosceles triangles
forms fractal space-filling labyrinths.
- Lenses,
rational-angled equilateral hexagons can tile the plane in various
interesting patterns. See also Jorge Mireles' nice
lens
puzzle applet: rotate decagons and stars to get the pieces into the
right places.
- Log-spiral tiling,
and other radial
and spiral tilings, S. Dutch.
- Mathematical
origami, Helena Verrill. Includes constructions of a shape with
greater perimeter than the original square, tessellations, hyperbolic
paraboloids, and more.
- Mitre Tiling.
Ed Pegg describes the discovery of the versatile tiling system
(with Adrian Fisher and Miroslav Vicher), also discussing many
other interesting tilings including a tile that can fill the plane with
either five-fold or six-fold symmetry.
- Modularity in art.
Slavik Jablan explores connections between art, tiling, knotwork, and
other mathematical topics.
- New
directions in aperiodic tilings, L. Danzer, Aperiodic '94.
- Non
periodic tiling of the plane.
Including Penrose tiles, Pinhweel tiling, and more. Paul Bourke.
- Nontrivial
convexity. Ed Pegg asks about partitions of convex regions into
equal tiles, other than the "trivial" ones in which some rotational or
translational symmetry group relates all the tile positions to each other.
See also Miroslav
Vicher's page on nontrivial convexity
- Origami
tessellations and
paper mosaics, Alex Bateman.
- Parquet
deformations.
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.
- Perplexing
pentagons, Doris Schattschneider, from the Discovering Geometry
Newsletter.
A brief introduction to the problem of tiling the plane by pentagons.
- Pentagonal
Tessellations. John Savard experiments with substitution systems to
produce tilings resembling Kepler's.
- Pentagons that tile the plane, Bob Jenkins.
See also
Ed Pegg's page on
pentagon tiles.
- Perron
Number Tiling Systems.
Mathematica software for computing fractals that tile the plane from
Perron numbers.
- Platonic
tesselations of Riemann surfaces, Gerard Westendorp.
- Polygons
with angles of different k-gons.
Leroy Quet asks whether polygons formed by combining the angles of
different regular polygons can tile the plane.
The answer turns out to be related to
Egyptian fraction
decompositions of 1 and 1/2.
- PolyMultiForms.
L. Zucca uses pinwheel tilers to dissect an illustration of the Pythagorean
theorem into few congruent triangles.
- Polyomino tiling.
Joseph Myers classifies the n-ominoes up to n=15 according to how
symmetrically they can tile the plane.
- Polyominoes, figures formed from subsets
of the square lattice tiling of the plane. Interesting problems
associated with these shapes include finding all of them, determining
which ones tile the plane, and dissecting rectangles or other shapes
into sets of them. Also includes related
material on polyiamonds, polyhexes, and animals.
- ProtoZone
interactive shockwave museum exhibits for exploring geometric concepts
such as symmetry, tiling, and wallpaper groups.
- Publications on quasicrystals and aperiodic tilings, F. Gähler.
- A Puzzling Journey To The Reptiles And Related Animals, and
New Mosaics.
Books on tiling by Karl Scherer.
- Quaquaversal
Tilings and Rotations. John Conway and Charles Radin describe a
three-dimensional generalization of the pinwheel tiling, the mathematics
of which is messier due to the noncommutativity of three-dimensional
rotations.
Quasicrystals and aperiodic tilings, A. Zerhusen, U. Kentucky. Includes a nice description of how to make 3d aperiodic tiles from zometool pieces.

- Reptile
project-of-the-month from the Geometry Forum.
Form tilings by dividing polygons into copies of themselves.
- Rhombic
spirallohedra, concave rhombus-faced polyhedra that tile space,
R. Towle.
- Rhombic
tilings. Abstract of Serge Elnitsky's thesis, "Rhombic tilings of
polygons and classes of reduced words in Coxeter groups". He also supplied the
picture below of a rhombically tiled 48-gon, available with better color
resolution from his website.
- Self-affine tiles, J. Lagarias and Y. Wang, DIMACS.
Mathematics of a class of generalized reptiles.
- Semi-regular
tilings of the plane, K. Mitchell, Hobart and William Smith Colleges.
- Some generalizations of the pinwheel tiling, L. Sadun, U. Texas.
- SpaceBric building blocks
and Windows software based on a tiling of 3d space by congruent
tetrahedra.
- Spidron,
a triangulated double spiral shape tiles the plane and various other
surfaces. With photos of related paperfolding experiments.
- Spiral tilings.
These similarity tilings are formed by applying the exponential function
to a lattice in the complex number plane.
- Symmetry,
tilings, and polyhedra, S. Dutch.
- Symmetry and Tilings. Charles Radin, Not. AMS, Jan. 1995.
See also his
Symmetry
of Tilings of the Plane, Bull. AMS 29 (1993), which proves that the
pinwheel tiling is ergodic and can be generated by matching rules.
- Taprats
Java software for generating symmetric Islamic-style star patterns.
- Tesselating
locking polyominos, Bob Newman.
- Tessellation
links, S. Alejandre.
- Tessellation
resources. 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.
- Tilable
perspectives.
Patrick Snels creates two-dimensional images which tile the plane to
form 3d-looking views including some interesting Escher-like warped
perspectives.
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 (possibly
disconnected) polyomino will tile the plane by translation,
in the case where the number of squares in the polyomino is a prime
or four.
- Tiling the integers with one prototile.
Talk abstract by Ethan Coven on a one-dimensional tiling problem on the
boundary between
geometry and number theory, with connections to factorization of finite
cyclic groups.
See also Coven's paper with Aaron Meyerowitz,
Tiling the integers
with translates of one finite set.
- Tiling problems.
Collected at a problem session at Smith College, 1993, by
Marjorie Senechal.
- Tiling
transformer. Java applet for subdividing tilings (starting from a
square or hexagonal tiling) in various different ways.
- Tiling dynamical systems.
Chris Hillman describes his research
on topological spaces in which each point represents a tiling.
- On
a 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 pinwheel
tiling as a design motif. Thanks to Khalad Karim for identifying it.
Photos by Dick Hess, scanned by Ed Pegg Jr.
See this Flickr
photopool for many more photos.
- true_tile
mailing list for discussion of Euclidean and non-Euclidean tilings.
- Tysen
loves hexagons. And supplies ascii, powerpoint, and png graphics for
several styles of hexagonal grid graph paper.
- Unbalanced
anisohedral tiling.
Joseph Myers and
John Berglund find a polyhex that must be placed two different ways in
a tiling of a plane, such that one placement occurs twice as often as
the other.

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.