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In geometry, a convex uniform honeycomb is a uniform tessellation which fills threedimensional Euclidean space with nonoverlapping convex uniform polyhedral cells.
Twentyeight such honeycombs exist:
They can be considered the threedimensional analogue to the uniform tilings of the plane.
Contents
History
 1900: Thorold Gosset enumerated the list of semiregular convex polytopes with regular cells (Platonic solids) in his publication On the Regular and SemiRegular 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 complete list of 28.
 1994: Branko Grünbaum, in his paper Uniform tilings of 3space, 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 polychorons in 4space).
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