Truncated tesseractic honeycomb

In four-dimensional Euclidean geometry, the truncated tesseractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space. It is constructed by a truncation of a tesseractic honeycomb creating truncated tesseracts, and adding new 16-cell facets at the original vertices.

Truncated tesseractic honeycomb
(No image)
TypeUniform 4-honeycomb
Schläfli symbolt{4,3,3,4}
t{4,3,31,1}
Coxeter-Dynkin diagram
4-face typetruncated tesseract
16-cell
Cell typeTruncated cube
Tetrahedron
Face type{3}, {8}
Vertex figureoctahedral pyramid
Coxeter group = [4,3,3,4]
= [4,3,31,1]
Dual
Propertiesvertex-transitive

The [4,3,3,4], , Coxeter group generates 31 permutations of uniform tessellations, 21 with distinct symmetry and 20 with distinct geometry. The expanded tesseractic honeycomb (also known as the stericated tesseractic honeycomb) is geometrically identical to the tesseractic honeycomb. Three of the symmetric honeycombs are shared in the [3,4,3,3] family. Two alternations (13) and (17), and the quarter tesseractic (2) are repeated in other families.

See also

Regular and uniform honeycombs in 4-space:

Notes

    References
    • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
      • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] See p318
    • George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
    • Klitzing, Richard. "4D Euclidean tesselations#4D". o3o3o *b3x4x, x4x3o3o4o - tattit - O89
    • Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9.
    Fundamental convex regular and uniform honeycombs in dimensions 2-9
    Space Family / /
    E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
    E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
    E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
    E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
    E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
    E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
    E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
    E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
    En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
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