Snub 24-cell honeycomb

Snub 24-cell honeycomb
(No image)
TypeUniform 4-honeycomb
Schläfli symbolss{3,4,3,3}
sr{3,3,4,3}
s2r{4,3,3,4}
s2r{4,3,31,1}
s{31,1,1,1}
Coxeter diagrams





=

4-face typesnub 24-cell
16-cell
5-cell
Cell type{3,3}
{3,5}
Face typetriangle {3}
Vertex figure
Irregular decachoron
Symmetry groups[3+,4,3,3], ½{\tilde{F}}_4
[3,4,(3,3)+], ½{\tilde{F}}_4
[4,(3,3)+,4], ½{\tilde{C}}_4
[4,(3,31,1)+], ½{\tilde{B}}_4
[31,1,1,1]+, ½{\tilde{D}}_4
PropertiesVertex transitive, nonWythoffian

In four-dimensional Euclidean geometry, the snub 24-cell honeycomb, or snub icositetrachoric honeycomb is a uniform space-filling tessellation (or honeycomb) by snub 24-cells, 16-cells, and 5-cells. It was discovered by Thorold Gosset with his 1900 paper of semiregular polytopes. It is not semiregular by Gosset's definition of regular facets, but all of its cells (ridges) are regular, either tetrahedra or icosahedra.

It can be seen as an alternation of a truncated 24-cell honeycomb, and can be represented by Schläfli symbol s{3,4,3,3}, s{31,1,1,1}, and 3 other snub constructions.

It is defined by an irregular decachoron vertex figure (10-celled 4-polytope), faceted by four snub 24-cells, one 16-cell, and five 5-cells. The vertex figure can be seen topologically as a modified tetrahedral prism, where one of the tetrahedra is subdivided at mid-edges into a central octahedron and four corner tetrahedra. Then the four side-facets of the prism, the triangular prisms become tridiminished icosahedra.

Symmetry constructions

There are five different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored snub 24-cell, 16-cell, and 5-cell facets. In all cases, four snub 24-cells, five 5-cells, and one 16-cell meet at each vertex, but the vertex figures have different symmetry generators.

Coxeter group Extended symmetry
group
Coxeter notation		 Coxeter diagram
Schläfli symbol		 Facets (on vertex figure)
Snub 24-cell
(4)		 16-cell
(1)		 5-cell
(5)
{\tilde{F}}_4 = [3,4,3,3] [3+,4,3,3]
s{3,4,3,3}		 4:
{\tilde{F}}_4 = [3,4,3,3] [3,4,(3,3)+]
sr{3,3,4,3}		 3:
1:
{\tilde{C}}_4 = [4,3,3,4] [[4,(3,3)+,4]]
s2r{4,3,3,4}		 2,2:
{\tilde{B}}_4 = [31,1,3,4] <[(31,1,3)+,4]> = [4,(3,3)+,4]
s2r{4,3,31,1}		 1,1:
2:
{\tilde{D}}_4 = [31,1,1,1] [(3,3)[31,1,1,1]+] = [3,3,4,3+]
[3[3,31,1,1]+] = [3,4,(3,3)+]
s{31,1,1,1}		 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