Uniform Honeycombs in 3-Space

  • Here is a collection of uniform regular-polyhedral honeycombs. George Olshevsky defines a uniform honeycomb in three-space as "one all of whose cells are uniform polyhedra and whose vertices are transitive on its symmetry group, and which covers three-space exactly once". Banko Grünbaum lists all 28 of these in his 1994 paper: "Uniform tilings of 3-space," Geombinatorics 4, 49-56. I have not read seen the paper yet, so any errors with the figures are my own. Gurnbaum's paper does not represent the original discovery of these figures, as they appear to have been known for some time (dating back at least to the 1950's). But in any case, here is a simple presentation of these figures for those who are interested.

    Note: Each figure shows all of the polyhedra surrounding a single vertex, with the exceptions of figures 24, 25, 27 and 28. See the explanation below the table for those figures.
    Model.StelLong Name and Alternate NamesComponents per vertex
    01-Chon.StelCubic honeycomb8 Cubes
    02-Rich.StelRectified cubic honeycomb4 cuboctahedra, 2 octahedra
    03-Tich.StelTruncated cubic honeycomb4 truncated cubes, 1 octahedron
    04-Srich.StelCantellated cubic honeycomb (small rhombated)2 rhombicuboctahedra, 1 cuboctahedron, 2 cubes
    05-Batch.StelBitruncated cubic honeycomb; truncated octahedral honeycomb4 truncated octahedra
    06-Grich.StelCantitruncated cubic honeycomb (great rhombated)2 truncated cuboctahedra, 1 truncated octahedron, 1 cube
    07-Prich.StelRuncitruncated cubic honeycomb (prismatorhombated)1 truncated cube, 1 rhombicuboctahedron, 2 octagonal prisms, 1 cube
    08-Otch.StelOmnitruncated cubic honeycomb2 truncated cuboctahedra, 2 octahedral prisms
    09-Tassiph.StelTruncated square prismatic honeycomb4 octagonal prisms, 2 cubes
    10-Sassiph.StelSnub square prismatic honeycomb6 triangular prisms, 4 cubes
    11-Octet.StelOctahedral-Tetrahedral honeycomb (oct-tet struss)8 tetrahedra, 6 octahedra (see 26-Gytoh)
    12-Ratoh.StelRuncinated alternated cubic honeycomb; runcinated tetrahedral-octahedral honeycomb3 rhombicuboctahedra, 1 tetrahedron, 1 cube
    13-Tatoh.StelTruncated alternated cubic honeycomb; truncated tetrahedral-octahedral honeycomb2 truncated octahedra, 2 truncated tetrahedra, 1 cuboctahedron
    14-Gratoh.StelCantitruncated alternated cubic honeycomb; cantitruncated tetrahedral-octahedral honeycomb2 truncated cuboctahedra, 1 truncated cube, 1 truncated tetrahedron
    15-Batatoh.StelBitruncated alternated cubic honeycomb; bitruncated tetrahedral-octahedral honeycomb6 truncated tetrahedra, 2 tetrahedra
    16-Hiph.StelHexagonal prismatic honeycomb6 hexagonal prisms
    17-Thiph.StelTriangular-hexagonal prismatic honeycomb4 hexagonal prisms, 4 triangular prisms
    18-Tiph.StelTriangular prismatic honeycomb12 triangular prisms
    19-Thaph.StelTruncated hexagonal prismatic honeycomb4 dodecagonal prisms, 2 triangular prisms
    20-Rothaph.StelRhombitriangular-hexagonal prismatic honeycomb2 hexagonal prisms, 2 triangular prisms, 4 cubes
    21-Otathaph.StelOmnitruncated triangular-hexagonal prismatic honeycomb2 dodecagonal prisms, 2 hexagonal prisms, 2 cubes
    22-Snathaph.StelSnub triangular-hexagonal prismatic honeycomb2 hexagonal prisms, 8 triangular prisms
    23-Gytoph.StelGyrated triangular prismatic honeycomb12 triangular prisms
    24-Etoph=x.StelElongated triangular prismatic honeycomb6 triangular prisms, 4 cubes (see note below)
    25-Gyetaph=x.StelGyroelongated triangular prismatic honeycomb6 triangular prisms, 4 cubes (see note below)
    26-Gytoh.StelGyrated Tetrahedral-Octahedral honeycomb8 tetrahedra, 6 octahedra (see 11-Octet)
    27-Etoh=x.StelElongated cubic honeycomb; elongated tetrahedral-octahedral honeycomb3 octahedra, 4 tetrahedra, 6 triangular prisms (see note below)
    28-Gyetoh=x.StelGyroelongated cubic honeycomb; gyroelongated tetrahedral-octahedral honeycomb3 octahedra, 4 tetrahedra, 6 triangular prisms (see note below)

    Note: Figures labeled with "=x", (numbers 24, 25, 27 and 28), include the polyhedra around 2 vertices instead of 1, so as to differentiate between related pairs.

    Thanks to George Olshevsky and Jonathan Bowers for introducing me to this collection, and the names associated with them.

    Thanks also to Robert Webb for his very cool "Stella" program. I have used it extensively to generate VRML files for my site. It is a great tool for rapid exploration of augmentations and excavations of polyhedra (among other things).


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