Truncated 16-cell honeycomb

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Truncated 16-cell honeycomb
(No image)
TypeUniform honeycomb
Schläfli symbolt{3,3,4,3}
h2{4,3,3,4}
t{3,31,1,1}
Coxeter-Dynkin diagram
=
4-face type{3,4,3}
t{3,3,4}
Cell type{3,3}
t{3,3}
Face type{3}
{6}
Vertex figurecubic pyramid
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 truncated 16-cell honeycomb (or cantic tesseractic honeycomb) is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space. It is constructed by 24-cell and truncated 16-cell facets.

Alternate names

  • Truncated hexadecachoric tetracomb / Truncated hexadecachoric honeycomb

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 [4,3,3,4], , 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.

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

1, 2, 3, 4,
5, 6, 7, 8,
9, 10, 11, 12,
13

[[4,3,3,4]] ×2 (1), (2), (13), 18
(6), 19, 20
[(3,3)[1+,4,3,3,4,1+]]
= [(3,3)[31,1,1,1]]
= [3,4,3,3]
=
= 		×6

14, 15, 16, 17

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 (x3x3o *b3o4o), (x3x3o *b3o *b3o), x3x3o4o3o - thext - O105
    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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