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 .
, Hexagonal Grid
, Honeycomb Conjecture
, Regular Tessellation
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ReferencesBall, 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|AlphaHoneycomb
Cite this as:
Weisstein, Eric W. "Honeycomb." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Honeycomb.html