Bitruncated 16-cell honeycomb

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Bitruncated 16-cell honeycomb
(No image)
TypeUniform honeycomb
Schläfli symbolt1,2{3,3,4,3}
h2,3{4,3,3,4}
2t{3,31,1,1}
Coxeter-Dynkin diagram
=
4-face typeTruncated 24-cell
Bitruncated tesseract
Cell typeCube
Truncated octahedron
Truncated tetrahedron
Face type{3}, {4}, {6}
Vertex figuretriangular duopyramid
Coxeter group{{\tilde {F}}}_{4} = [3,3,4,3]
{{\tilde {B}}}_{4} = [4,3,31,1]
{{\tilde {D}}}_{4} = [31,1,1,1]
Dual?
Propertiesvertex-transitive

In four-dimensional Euclidean geometry, the bitruncated 16-cell honeycomb (or runcicantic tesseractic honeycomb) is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space.

Symmetry constructions

There are 3 different symmetry constructions, all with 3-3 duopyramid vertex figures. The {{\tilde {B}}}_{4} symmetry doubles on {{\tilde {D}}}_{4} in three possible ways, while {{\tilde {F}}}_{4} contains the highest symmetry.

Affine Coxeter group {{\tilde {F}}}_{4}
[3,3,4,3]		 {{\tilde {B}}}_{4}
[4,3,31,1]		 {{\tilde {D}}}_{4}
[31,1,1,1]
Coxeter diagram
4-faces



Related honeycombs

The [3,4,3,3], , Coxeter group generates 31 permutations of uniform tessellations, 28 are unique in this family and ten are shared in the [4,3,3,4] and [4,3,31,1] families. The alternation (13) is also repeated in other families.

Extended
symmetry		 Extended
diagram		 Order Honeycombs
[3,3,4,3]×1

1, 3, 5, 6, 8,
9, 10, 11, 12

[3,4,3,3]×1

2, 4, 7, 13,
14, 15, 16, 17,
18, 19, 20, 21,
22 23, 24, 25,
26, 27, 28, 29

[(3,3)[3,3,4,3*]]
=[(3,3)[31,1,1,1]]
=[3,4,3,3]
=
= 		×4

(2), (4), (7), (13)

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.

Extended
symmetry		 Extended
diagram		 Order Honeycombs
[4,3,31,1]: ×1

5, 6, 7, 8

<[4,3,31,1]>:
=[4,3,3,4]
= 		×2

9, 10, 11, 12, 13, 14,

(10), 15, 16, (13), 17, 18, 19

[3[1+,4,3,31,1]]
= [3[3,31,1,1]]
= [3,3,4,3]
=
= 		×3

1, 2, 3, 4

[(3,3)[1+,4,3,31,1]]
= [(3,3)[31,1,1,1]]
= [3,4,3,3]
=
= 		×12

20, 21, 22, 23

This honeycomb is one of ten uniform honeycombs constructed by the {{\tilde {D}}}_{4} 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:

Extended
symmetry		 Extended
diagram		 Order Honeycombs
[31,1,1,1] ×1 (none)
<[31,1,1,1]>
= [31,1,3,4]
= 		×2 (none)
<<[1,131,1]>>
= [4,3,3,4]
= 		×4 1, 2
[3[3,31,1,1]]
= [3,4,3,3]
= 		×6 3, 4, 5, 6
[<<[1,131,1]>>]
= [[4,3,3,4]]
= 		×8 7, 8, 9, 10
[(3,3)[31,1,1,1]]
= [3,3,4,3]
= 		×24

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]
    • George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
    • Richard Klitzing, 4D, Euclidean tesselations x3x3x *b3x *b3o, x3x3o *b3x4o, o3x3x4o3o - bithit - O107
    Fundamental convex regular and uniform honeycombs in dimensions 2–11
    Family {{\tilde {A}}}_{{n-1}} {{\tilde {C}}}_{{n-1}} {{\tilde {B}}}_{{n-1}} {{\tilde {D}}}_{{n-1}} {{\tilde {G}}}_{2} / {{\tilde {F}}}_{4} / {{\tilde {E}}}_{{n-1}}
    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 n-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
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