Bitruncated tesseractic honeycomb

Bitruncated tesseractic honeycomb
(No image)
TypeUniform 4-honeycomb
Schläfli symbolt1,2{4,3,3,4} or 2t{4,3,3,4}
t1,2{4,31,1} or 2t{4,31,1}
t2,3{4,31,1}
q2{4,3,3,3,4}
Coxeter-Dynkin diagram




=

4-face typeBitruncated tesseract
Truncated 16-cell
Cell typeOctahedron
Truncated tetrahedron
Truncated octahedron
Face type{3}, {4}, {6}
Vertex figure
Square-pyramidal pyramid
Coxeter group = [4,3,3,4]
= [4,31,1]
= [31,1,1,1]
Dual
Propertiesvertex-transitive

In four-dimensional Euclidean geometry, the bitruncated tesseractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space. It is constructed by a bitruncation of a tesseractic honeycomb. It is also called a cantic quarter tesseractic honeycomb from its q2{4,3,3,4} construction.

Other names

Related honeycombs

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.

The [4,3,31,1], , Coxeter group generates 31 permutations of uniform tessellations, 23 with distinct symmetry and 4 with distinct geometry. There are two alternated forms: the alternations (19) and (24) have the same geometry as the 16-cell honeycomb and snub 24-cell honeycomb respectively.

There are ten uniform honeycombs constructed by the Coxeter group, all repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 10th is constructed as an alternation. As subgroups in Coxeter notation: [3,4,(3,3)*] (index 24), [3,3,4,3*] (index 6), [1+,4,3,3,4,1+] (index 4), [31,1,3,4,1+] (index 2) are all isomorphic to [31,1,1,1].

The ten permutations are listed with its highest extended symmetry relation:

See also

Regular and uniform honeycombs in 4-space:

Notes

    References

    Fundamental convex regular and uniform honeycombs in dimensions 3–10 (or 2-9)
    Family / /
    Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
    Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
    Uniform 5-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
    Uniform 6-honeycomb {3[6]} δ6 hδ6 qδ6
    Uniform 7-honeycomb {3[7]} δ7 hδ7 qδ7 222
    Uniform 8-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
    Uniform 9-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
    Uniform 10-honeycomb {3[10]} δ10 hδ10 qδ10
    Uniform n-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
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