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Regular Tessellation

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RegularTessellations

Consider a two-dimensional tessellation with q regular p-gons at each polygon vertex. In the plane,

(1-2/p)pi=(2pi)/q
(1)
1/p+1/q=1/2,
(2)

so

(p-2)(q-2)=4
(3)

(Ball and Coxeter 1987), and the only factorizations are

4=4·1=(6-2)(3-2)=>{6,3}
(4)
=2·2=(4-2)(4-2)=>{4,4}
(5)
=1·4=(3-2)(6-2)=>{3,6}.
(6)

Therefore, there are only three regular tessellations (composed of the hexagon, square, and triangle), illustrated above (Ghyka 1977, p. 76; Williams 1979, p. 36; Wells 1991, p. 213).

There do not exist any regular star polygon tessellations in the plane. Regular tessellations of the sphere by spherical triangles are called triangular symmetry groups.

See also

Demiregular Tessellation, Semiregular Tessellation, Tessellation

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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.Ghyka, M. The Geometry of Art and Life. New York: Dover, 1977.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.

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Regular Tessellation

Cite this as:

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

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