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Concept
Convex uniform honeycomb
Résumé
In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.
Twenty-eight such honeycombs are known:
the familiar cubic honeycomb and 7 truncations thereof;
the alternated cubic honeycomb and 4 truncations thereof;
10 prismatic forms based on the uniform plane tilings (11 if including the cubic honeycomb);
5 modifications of some of the above by elongation and/or gyration.
They can be considered the three-dimensional analogue to the uniform tilings of the plane.
The Voronoi diagram of any lattice forms a convex uniform honeycomb in which the cells are zonohedra.
1900: Thorold Gosset enumerated the list of semiregular convex polytopes with regular cells (Platonic solids) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions, including one regular cubic honeycomb, and two semiregular forms with tetrahedra and octahedra.
1905: Alfredo Andreini enumerated 25 of these tessellations.
1991: Norman Johnson's manuscript Uniform Polytopes identified the list of 28.
1994: Branko Grünbaum, in his paper Uniform tilings of 3-space, also independently enumerated all 28, after discovering errors in Andreini's publication. He found the 1905 paper, which listed 25, had 1 wrong, and 4 being missing. Grünbaum states in this paper that Norman Johnson deserves priority for achieving the same enumeration in 1991. He also mentions that I. Alexeyev of Russia had contacted him regarding a putative enumeration of these forms, but that Grünbaum was unable to verify this at the time.
2006: George Olshevsky, in his manuscript Uniform Panoploid Tetracombs, along with repeating the derived list of 11 convex uniform tilings, and 28 convex uniform honeycombs, expands a further derived list of 143 convex uniform tetracombs (Honeycombs of uniform 4-polytopes in 4-space).
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