Truncated 24-cell honeycomb

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Truncated 24-cell honeycomb
(No image)
TypeUniform 4-honeycomb
Schläfli symbolt{3,4,3,3}
tr{3,3,4,3}
t2r{4,3,3,4}
t2r{4,3,31,1}
t{31,1,1,1}
Coxeter-Dynkin diagrams





4-face typeTesseract
Truncated 24-cell
Cell typeCube
Truncated octahedron
Face typeSquare
Triangle
Vertex figure
Tetrahedral pyramid
Coxeter groups{{\tilde {F}}}_{4}, [3,4,3,3]
{{\tilde {B}}}_{4}, [4,3,31,1]
{{\tilde {C}}}_{4}, [4,3,3,4]
{{\tilde {D}}}_{4}, [31,1,1,1]
PropertiesVertex transitive

In four-dimensional Euclidean geometry, the truncated 24-cell honeycomb is a uniform space-filling honeycomb. It can be seen as a truncation of the regular 24-cell honeycomb, containing tesseract and truncated 24-cell cells.

It has a uniform alternation, called the snub 24-cell honeycomb. It is a true snub from the {{\tilde {D}}}_{4} construction. This truncated 24-cell has Schläfli symbol t{31,1,1,1}, and its snub is represented as s{31,1,1,1}.

Alternate names

  • Truncated icositetrachoric tetracomb
  • Truncated icositetrachoric honeycomb
  • Cantitruncated 16-cell honeycomb
  • Bicantitruncated tesseractic honeycomb

Symmetry constructions

There are five different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored truncated 24-cell facets. In all cases, four truncated 24-cells, and one tesseract meet at each vertex, but the vertex figures have different symmetry generators.

Coxeter group Coxeter
diagram		 Facets Vertex figure Vertex
figure
symmetry
(order)
{{\tilde {F}}}_{4}
= [3,4,3,3]		 4:
1: 		, [3,3]
(24)
{{\tilde {F}}}_{4}
= [3,3,4,3]		 3:
1:
1: 		, [3]
(6)
{{\tilde {C}}}_{4}
= [4,3,3,4]		 2,2:
1: 		, [2]
(4)
{{\tilde {B}}}_{4}
= [31,1,3,4]		 1,1:
2:
1: 		, [ ]
(2)
{{\tilde {D}}}_{4}
= [31,1,1,1]		 1,1,1,1:

1: 		[ ]+
(1)

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:

References

  • Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
  • 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) Model 99
  • Richard Klitzing, 4D, Euclidean tesselations o4x3x3x4o, x3x3x *b3x4o, x3x3x *b3x *b3x, o3o3o4x3x, x3x3x4o3o - ticot - O99
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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