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Honeycomb

HoneycombTessellation

The regular tessellation {6,3} consisting of regular hexagons (i.e., a hexagonal grid).

In general, the term honeycomb is used to refer to a tessellation in n dimensions for n>=3. The only regular honeycomb in three dimensions is {4,3,4}, which consists of eight cubes meeting at each polyhedron vertex. The only quasiregular honeycomb (with regular cells and semiregular vertex figures) has each polyhedron vertex surrounded by eight tetrahedra and six octahedra and is denoted { 3; 3,4}.

Ball and Coxeter (1987) use the term "sponge" for a solid that can be parameterized by integers p, q, and n that satisfy the equation

2sin(pi/p)sin(pi/q)=cos(pi/n).

The possible sponges are {p,q|n}={6,6|3}, {6,4|4}, {4,6|4}, {3,6|6}, and {4,4|infty}.

There are many semiregular honeycombs, such as {3,3 ; 4 }, in which each polyhedron vertex consists of two octahedra {3,4} and four cuboctahedra {3; 4}.

See also

Hexagon, Hexagonal Grid, Honeycomb Conjecture, Menger Sponge, Regular Tessellation, Tessellation, Tetrix, Tiling

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References

Ball, W. W. R. and Coxeter, H. S. M. "Regular Sponges." In Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 152-153, 1987.Bulatov, V. "Infinite Regular Polyhedra." http://bulatov.org/polyhedra/infinite/.Coxeter, H. S. M. "Regular Honeycombs in Hyperbolic Space." Proc. International Congress of Math., Vol. 3. Amsterdam, Netherlands: pp. 155-169, 1954.Coxeter, H. S. M. "Space Filled with Cubes," "Other Honeycombs," and "Polytopes and Honeycombs." §4.6, 4.7, and 7.4 in Regular Polytopes, 3rd ed. New York: Dover, pp. 68-72 and 126-128, 1973.Cromwell, P. R. Polyhedra. New York: Cambridge University Press, p. 79, 1997.Gott, J. R. III "Pseudopolyhedrons." Amer. Math. Monthly 73, 497-504, 1967.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 104-106, 1991.Williams, R. The Geometrical Foundation of Natural Structure: A Source Book of Design. New York: Dover, 1979.

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Honeycomb

Cite this as:

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

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