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Cantitruncated tesseractic honeycomb

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Uniform space-filling tessellation in Euclidean 4-space
Cantitruncated tesseractic honeycomb
(No image)
Type Uniform 4-honeycomb
Schläfli symbol tr{4,3,3,4}
tr{4,3,3}
Coxeter-Dynkin diagram
4-face type t0,1,2{4,3,3}
t0,1{3,3,4}
{3,4}×{}
Cell type Truncated cuboctahedron
Octahedron
Truncated tetrahedron
Triangular prism
Face type {3}, {4}, {6}
Vertex figure Square double pyramid
Coxeter group C ~ 4 {\displaystyle {\tilde {C}}_{4}} =
B ~ 4 {\displaystyle {\tilde {B}}_{4}} =
Dual
Properties vertex-transitive

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

Related honeycombs

The , , 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 family. Two alternations (13) and (17), and the quarter tesseractic (2) are repeated in other families.

C4 honeycombs
Extended
symmetry 		Extended
diagram 		Order Honeycombs
: ×1

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

] ×2 (1), (2), (13), 18
(6), 19, 20
]
↔ ]
↔
↔
↔ 		×6

14, 15, 16, 17

The , , 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
Extended
symmetry 		Extended
diagram 		Order Honeycombs
: ×1

5, 6, 7, 8

<>:
↔
↔ 		×2

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

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

]
↔ ]
↔
↔
↔ 		×3

1, 2, 3, 4

]
↔ ]
↔
↔
↔ 		×12

20, 21, 22, 23

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, 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". o3x3o *b3x4x, x4x3x3o4o - grittit - O94
  • 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 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}}
E Uniform tiling 0 δ3 hδ3 qδ3 Hexagonal
E Uniform convex honeycomb 0 δ4 hδ4 qδ4
E Uniform 4-honeycomb 0 δ5 hδ5 qδ5 24-cell honeycomb
E Uniform 5-honeycomb 0 δ6 hδ6 qδ6
E Uniform 6-honeycomb 0 δ7 hδ7 qδ7 222
E Uniform 7-honeycomb 0 δ8 hδ8 qδ8 133 • 331
E Uniform 8-honeycomb 0 δ9 hδ9 qδ9 152 • 251 • 521
E Uniform 9-honeycomb 0 δ10 hδ10 qδ10
E Uniform 10-honeycomb 0 δ11 hδ11 qδ11
E Uniform (n-1)-honeycomb 0 δn hδn qδn 1k2 • 2k1 • k21
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