Convex uniform honeycomb
Twenty-eight such honeycombs are known: 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.
Only 14 of the convex uniform polyhedra appear in these patterns: The icosahedron, snub cube, and square antiprism appear in some alternations, but those honeycombs cannot be realised with all edges unit length.
If cells are colored based on unique positions within each Wythoff construction, these different symmetries can be shown.
The total unique prismatic honeycombs above (excluding the cubic counted previously) are 10.
, [4,3] group offers 11 derived forms via truncation operations, four being unique uniform honeycombs.
There is one index 2 subgroup [3[4]]+ which generates the snub form, which is not uniform, but included for completeness.
Three more uniform honeycombs are generated by breaking one or another of the above honeycombs where its faces form a continuous plane, then rotating alternate layers by 60 or 90 degrees (gyration) and/or inserting a layer of prisms (elongation).
The elongated and gyroelongated alternated cubic tilings have the same vertex figure, but are not alike.
In the elongated form, each prism meets a tetrahedron at one triangular end and an octahedron at the other.
Green backgrounds are shown on repeated honeycombs, with the relations are expressed in the extended symmetry diagrams.
The space-filling truss of packed octahedra and tetrahedra was apparently first discovered by Alexander Graham Bell and independently re-discovered by Buckminster Fuller (who called it the octet truss and patented it in the 1940s).
Some scaliforms can be generated by an alternation process, leaving, for example, pyramid and cupola gaps.
From these 9 families, there are a total of 76 unique honeycombs generated: Several non-Wythoffian forms outside the list of 76 are known; it is not known how many there are.
These families can produce uniform honeycombs with unbounded facets or vertex figure, including ideal vertices at infinity:
