Snub 24-cell honeycomb

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.

Snub 24-cell honeycomb
(No image)
Type	Uniform 4-honeycomb
Schläfli symbols	s{3,4,3,3} sr{3,3,4,3} 2sr{4,3,3,4} 2sr{4,3,31,1} s{31,1,1,1}
Coxeter diagrams	=
4-face type	snub 24-cell 16-cell 5-cell
Cell type	{3,3} {3,5}
Face type	triangle {3}
Vertex figure	Irregular decachoron
Symmetries	[3+,4,3,3] [3,4,(3,3)+] [4,(3,3)+,4] [4,(3,31,1)+] [31,1,1,1]+
Properties	Vertex transitive, nonWythoffian

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{3^1,1,1,1}, 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.

Symmetry	Coxeter Schläfli	Facets (on vertex figure)
		Snub 24-cell (4)	16-cell (1)	5-cell (5)
[3+,4,3,3]	s{3,4,3,3}	4:
[3,4,(3,3)+]	sr{3,3,4,3}	3: 1:
[[4,(3,3)+,4]]	2sr{4,3,3,4}	2,2:
[(31,1,3)+,4]	2sr{4,3,31,1}	1,1: 2:
[31,1,1,1]+	s{31,1,1,1}	1,1,1,1:

See also

Regular and uniform honeycombs in 4-space:

• Tesseractic honeycomb
• 16-cell honeycomb
• 24-cell honeycomb
• Truncated 24-cell honeycomb
• 5-cell honeycomb
• Truncated 5-cell honeycomb
• Omnitruncated 5-cell honeycomb

References

• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
• 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 133
• Klitzing, Richard. "4D Euclidean tesselations"., o4s3s3s4o, s3s3s *b3s4o, s3s3s *b3s *b3s, o3o3o4s3s, s3s3s4o3o - sadit - O133

v t e Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	${\displaystyle {\tilde {A}}_{n-1}}$	${\displaystyle {\tilde {C}}_{n-1}}$	${\displaystyle {\tilde {B}}_{n-1}}$	${\displaystyle {\tilde {D}}_{n-1}}$	${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
E2	Uniform tiling	0[3]	δ3	hδ3	qδ3	Hexagonal
E3	Uniform convex honeycomb	0[4]	δ4	hδ4	qδ4
E4	Uniform 4-honeycomb	0[5]	δ5	hδ5	qδ5	24-cell honeycomb
E5	Uniform 5-honeycomb	0[6]	δ6	hδ6	qδ6
E6	Uniform 6-honeycomb	0[7]	δ7	hδ7	qδ7	222
E7	Uniform 7-honeycomb	0[8]	δ8	hδ8	qδ8	133 • 331
E8	Uniform 8-honeycomb	0[9]	δ9	hδ9	qδ9	152 • 251 • 521
E9	Uniform 9-honeycomb	0[10]	δ10	hδ10	qδ10
E10	Uniform 10-honeycomb	0[11]	δ11	hδ11	qδ11
En−1	Uniform (n−1)-honeycomb	0[n]	δn	hδn	qδn	1k2 • 2k1 • k21