The regular tessellation consisting of regular hexagons
(i.e., a hexagonal grid ).

In general, the term honeycomb is used to refer to a tessellation in dimensions for . The only regular honeycomb in three dimensions is , 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 .

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

The possible sponges are ,
, , , and .

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

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. Referenced on Wolfram|Alpha Honeycomb
Cite this as:
Weisstein, Eric W. "Honeycomb." From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/Honeycomb.html

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