Tessellation
A tiling of regular polygons (in two dimensions), polyhedra (three dimensions),
or polytopes (
dimensions) is
called a tessellation. Tessellations can be specified using a Schläfli
symbol.
The breaking up of self-intersecting polygons into simple polygons is also called tessellation (Woo
et al. 1999), or more properly, polygon
tessellation.
There are exactly three regular tessellations
composed of regular polygons symmetrically tiling the plane.
Tessellations of the plane by two or more convex regular polygons such that the same polygons in the same order surround
each polygon vertex are called semiregular
tessellations, or sometimes Archimedean tessellations. In the plane, there are
eight such tessellations, illustrated above (Ghyka 1977, pp. 76-78; Williams
1979, pp. 37-41; Steinhaus 1999, pp. 78-82; Wells 1991, pp. 226-227).
There are 14 demiregular (or polymorph) tessellations which are orderly compositions of the three regular and eight semiregular
tessellations (Critchlow 1970, pp. 62-67; Ghyka 1977, pp. 78-80; Williams
1979, p. 43; Steinhaus 1999, pp. 79 and 81-82).
In three dimensions, a polyhedron which is capable of tessellating space is called a space-filling
polyhedron. Examples include the cube, rhombic
dodecahedron, and truncated octahedron.
There is also a 16-sided space-filler and a convex polyhedron
known as the Schmitt-Conway biprism which
fills space only aperiodically.
A tessellation of 
-dimensional polytopes is called a honeycomb.
SEE ALSO: Archimedean Solid,
Cairo Tessellation,
Cell,
Demiregular Tessellation,
Dual
Tessellation,
Hexagonal Grid,
Hinged
Tessellation,
Honeycomb,
Honeycomb
Conjecture,
Kepler's Monsters,
Regular
Tessellation,
Schläfli Symbol,
Semiregular
Polyhedron,
Semiregular Tessellation,
Space-Filling Polyhedron,
Spiral
Similarity,
Square Grid,
Symmetry,
Tiling,
Triangular Grid,
Triangular Symmetry Group,
Triangulation,
Wallpaper Groups
REFERENCES:
Ball, W. W. R. and Coxeter, H. S. M. Mathematical
Recreations and Essays, 13th ed. New York: Dover, pp. 105-107, 1987.
Bhushan, A.; Kay, K.; and Williams, E. "Totally Tessellated." http://library.thinkquest.org/16661/.
Britton, J. Symmetry and Tessellations: Investigating Patterns. Englewood Cliffs, NJ: Prentice-Hall,
1999.
Critchlow, K. Order
in Space: A Design Source Book. New York: Viking Press, 1970.
Cundy, H. and Rollett, A. Mathematical
Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 60-63, 1989.
Gardner, M. Martin Gardner's New Mathematical Diversions from Scientific American. New York:
Simon and Schuster, pp. 201-203, 1966.
Gardner, M. "Tilings with Convex Polygons." Ch. 13 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman,
pp. 162-176, 1988.
Ghyka, M. The
Geometry of Art and Life. New York: Dover, 1977.
Kraitchik, M. "Mosaics." §8.2 in Mathematical
Recreations. New York: W. W. Norton, pp. 199-207, 1942.
Lines, L. Solid Geometry, with Chapters on Space-Lattices, Sphere-Packs, and Crystals. New
York: Dover, pp. 199 and 204-207 1965.
Pappas, T. "Tessellations." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 120-122,
1989.
Peterson, I. The Mathematical Tourist: Snapshots of Modern Mathematics. New York: W. H.
Freeman, p. 75, 1988.
Radin, C. Miles
of Tiles. Providence, RI: Amer. Math. Soc., 1999.
Rawles, B. Sacred Geometry Design Sourcebook: Universal Dimensional Patterns. Nevada City,
CA: Elysian Pub., 1997.
Steinhaus, H. Mathematical
Snapshots, 3rd ed. New York: Dover, pp. 75-76 and 78-82, 1999.
Vichera, M. "Archimedean Polyhedra." http://www.vicher.cz/puzzle/telesa/telesa.htm.
Walsh, T. R. S. "Characterizing the Vertex Neighbourhoods of Semi-Regular
Polyhedra." Geometriae Dedicata 1, 117-123, 1972.
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. 121, 213, and 226-227, 1991.
Williams, R. The Geometrical Foundation of Natural Structure: A Source Book of Design. New
York: Dover, pp. 35-43, 1979.
Woo, M.; Neider, J.; Davis, T.; and Shreiner, D. Ch. 11 in OpenGL 1.2 Programming Guide, 3rd ed.: The Official Guide to Learning OpenGL, Version 1.2.
Reading, MA: Addison-Wesley, 1999.
Referenced on Wolfram|Alpha:
Tessellation
CITE THIS AS:
Weisstein, Eric W. "Tessellation." From
MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Tessellation.html
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