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Truncated 16-cell honeycomb
Truncated 16-cell honeycomb
(No image)
TypeUniform honeycomb
Schläfli symbolst{3,3,4,3}h2{4,3,3,4}t{3,31,1,1}
Coxeter diagrams =
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 F ~ 4 {\displaystyle {\tilde {F}}_{4}} = [3,3,4,3] B ~ 4 {\displaystyle {\tilde {B}}_{4}} = [4,3,31,1] D ~ 4 {\displaystyle {\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.

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Alternate names

  • Truncated hexadecachoric tetracomb / Truncated hexadecachoric honeycomb

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.

F4 honeycombs
ExtendedsymmetryExtendeddiagramOrderHoneycombs
[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.

C4 honeycombs
ExtendedsymmetryExtendeddiagramOrderHoneycombs
[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.

B4 honeycombs
ExtendedsymmetryExtendeddiagramOrderHoneycombs
[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

There are ten uniform honeycombs constructed by the D ~ 4 {\displaystyle {\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:

D4 honeycombs
ExtendedsymmetryExtendeddiagramExtendedgroupHoneycombs
[31,1,1,1] D ~ 4 {\displaystyle {\tilde {D}}_{4}} (none)
<[31,1,1,1]>↔ [31,1,3,4]↔ D ~ 4 {\displaystyle {\tilde {D}}_{4}} ×2 = B ~ 4 {\displaystyle {\tilde {B}}_{4}} (none)
<2[1,131,1]>↔ [4,3,3,4]↔ D ~ 4 {\displaystyle {\tilde {D}}_{4}} ×4 = C ~ 4 {\displaystyle {\tilde {C}}_{4}} 1, 2
[3[3,31,1,1]]↔ [3,3,4,3]↔ D ~ 4 {\displaystyle {\tilde {D}}_{4}} ×6 = F ~ 4 {\displaystyle {\tilde {F}}_{4}} 3, 4, 5, 6
[4[1,131,1]]↔ [[4,3,3,4]]↔ D ~ 4 {\displaystyle {\tilde {D}}_{4}} ×8 = C ~ 4 {\displaystyle {\tilde {C}}_{4}} ×2 7, 8, 9
[(3,3)[31,1,1,1]]↔ [3,4,3,3]↔ D ~ 4 {\displaystyle {\tilde {D}}_{4}} ×24 = F ~ 4 {\displaystyle {\tilde {F}}_{4}}
[(3,3)[31,1,1,1]]+↔ [3+,4,3,3]↔ ½ D ~ 4 {\displaystyle {\tilde {D}}_{4}} ×24 = ½ F ~ 4 {\displaystyle {\tilde {F}}_{4}} 10

See also

Regular and uniform honeycombs in 4-space:

Notes

  • 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 [1]
    • (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)
  • Klitzing, Richard. "4D Euclidean tesselations". (x3x3o *b3o4o), (x3x3o *b3o *b3o), x3x3o4o3o - thext - O105
  • v
  • t
  • e
Fundamental convex regular and uniform honeycombs in dimensions 2–9
SpaceFamily A ~ n − 1 {\displaystyle {\tilde {A}}_{n-1}} C ~ n − 1 {\displaystyle {\tilde {C}}_{n-1}} B ~ n − 1 {\displaystyle {\tilde {B}}_{n-1}} D ~ n − 1 {\displaystyle {\tilde {D}}_{n-1}} G ~ 2 {\displaystyle {\tilde {G}}_{2}} / F ~ 4 {\displaystyle {\tilde {F}}_{4}} / E ~ n − 1 {\displaystyle {\tilde {E}}_{n-1}}
E2Uniform tiling0[3]δ3hδ3qδ3Hexagonal
E3Uniform convex honeycomb0[4]δ4hδ4qδ4
E4Uniform 4-honeycomb0[5]δ5hδ5qδ524-cell honeycomb
E5Uniform 5-honeycomb0[6]δ6hδ6qδ6
E6Uniform 6-honeycomb0[7]δ7hδ7qδ7222
E7Uniform 7-honeycomb0[8]δ8hδ8qδ8133 • 331
E8Uniform 8-honeycomb0[9]δ9hδ9qδ9152 • 251 • 521
E9Uniform 9-honeycomb0[10]δ10hδ10qδ10
E10Uniform 10-honeycomb0[11]δ11hδ11qδ11
En−1Uniform (n−1)-honeycomb0[n]δnhδnqδn1k2 • 2k1 • k21