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.
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.
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.
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.)
