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Tessellation


A tiling of regular polygons (in two dimensions), polyhedra (three dimensions), or polytopes (n 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.

RegularTessellations

There are exactly three regular tessellations composed of regular polygons symmetrically tiling the plane.

SemiregularTessellations

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).

DemiregularTessellations

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 n-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

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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.

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Tessellation

Cite this as:

Weisstein, Eric W. "Tessellation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Tessellation.html

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