Tiling
A plane-filling arrangement of plane figures or its generalization to higher dimensions. Formally, a tiling is a collection of disjoint open sets, the closures of which cover
the plane. Given a single tile, the so-called first corona
is the set of all tiles that have a common boundary point with the tile (including
the original tile itself).
Wang's conjecture (1961) stated that if a set of tiles tiled the plane, then they could always be arranged to do so periodically.
A periodic tiling of the plane by polygons
or space by polyhedra is
called a tessellation. The conjecture was refuted
in 1966 when R. Berger showed that an aperiodic set of 
tiles exists.
By 1971, R. Robinson had reduced the number to six and, in 1974, R. Penrose
discovered an aperiodic set (when color-matching rules are included) of two tiles:
the so-called Penrose tiles. It is not known if
there is a single aperiodic tile.
A spiral tiling using a single piece is illustrated on the cover of Grünbaum and Shephard (1986).
The number of tilings possible for convex irregular polygons
are given in the following table.
There are no tilings for identical convex 
-gons for 
, although
non-identical convex heptagons can tile the plane (Steinhaus 1999, p. 77; Gardner
1984, pp. 248-249).
SEE ALSO: Anisohedral Tiling,
Aperiodic Tiling,
Corona,
Domino Tiling,
Gosper
Island,
Harborth's Tiling,
Heesch
Number,
Heesch's Problem,
Honeycomb
Conjecture,
Isohedral Tiling,
Koch
Snowflake,
Monohedral Tiling,
Penrose
Tiles,
Polygon Tiling,
Polyomino
Tiling,
Space-Filling Polyhedron,
Square Tiling,
Tessellation,
Tiling Theorem,
Triangle
Tiling,
Wallpaper Groups
REFERENCES:
Eppstein, D. "Tiling." http://www.ics.uci.edu/~eppstein/junkyard/tiling.html.
Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University
of Chicago Press, pp. 248-249, 1984.
Gardner, M. "Tilings with Convex Polygons." Ch. 13 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman,
pp. 162-176, 1988.
Gardner, M. "Penrose Tiling" and "Penrose Tiling II." Chs. 1-2 in Penrose
Tiles and Trapdoor Ciphers... and the Return of Dr. Matrix, reissue ed.
New York: W. H. Freeman, pp. 1-29, 1989.
Grünbaum, B. and Shepard, G. C. "Some Problems on Plane Tilings." In The
Mathematical Gardner (Ed. D. Klarner). Boston, MA: Prindle, Weber, and
Schmidt, pp. 167-196, 1981.
Grünbaum, B. and Shephard, G. C. Tilings
and Patterns. New York: W. H. Freeman, 1986.
Pappas, T. "Mathematics & Moslem Art." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 178,
1989.
Peterson, I. The Mathematical Tourist: Snapshots of Modern Mathematics. New York: W. H.
Freeman, pp. 82-85, 1988.
Rawles, B. Sacred Geometry Design Sourcebook: Universal Dimensional Patterns. Nevada City,
CA: Elysian Pub., 1997.
Schattschneider, D. "In Praise of Amateurs." In The Mathematical Gardner (Ed. D. Klarner). Boston, MA: Prindle, Weber, and
Schmidt, pp. 140-166, 1981.
Seyd, J. A. and Salman, A. S. Symmetries
of Islamic Geometrical Patterns. River Edge, NJ: World Scientific, 1995.
Stein, S. and Szabó, S. Algebra and Tiling: Homomorphisms in the Service of Geometry. Washington, DC: Math.
Assoc. Amer., 1994.
Steinhaus, H. Mathematical
Snapshots, 3rd ed. New York: Dover, 1999.
Stevens, P. S. Handbook of Regular Patterns: An Introduction to Symmetry in Two Dimensions. Cambridge,
MA: MIT Press, 1992.
Weisstein, E. W. "Books about Tilings." http://www.ericweisstein.com/encyclopedias/books/Tilings.html.
Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin,
pp. 177-179, 208, and 211, 1991.
Referenced on Wolfram|Alpha:
Tiling
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