Snub 24cell honeycomb
In fourdimensional Euclidean geometry, the snub 24cell honeycomb, or snub icositetrachoric honeycomb is a uniform spacefilling tessellation (or honeycomb) by snub 24cells, 16cells, and 5cells. 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 24cell honeycomb  

(No image)  
Type  Uniform 4honeycomb 
Schläfli symbols  s{3,4,3,3} sr{3,3,4,3} 2sr{4,3,3,4} 2sr{4,3,3^{1,1}} s{3^{1,1,1,1}} 
Coxeter diagrams 

4face type  snub 24cell 16cell 5cell 
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,(3,3^{1,1})^{+}] [3^{1,1,1,1}]^{+} 
Properties  Vertex transitive, nonWythoffian 
It can be seen as an alternation of a truncated 24cell 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 (10celled 4polytope), faceted by four snub 24cells, one 16cell, and five 5cells. The vertex figure can be seen topologically as a modified tetrahedral prism, where one of the tetrahedra is subdivided at midedges into a central octahedron and four corner tetrahedra. Then the four sidefacets 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 24cell, 16cell, and 5cell facets. In all cases, four snub 24cells, five 5cells, and one 16cell meet at each vertex, but the vertex figures have different symmetry generators.
Symmetry  Coxeter Schläfli 
Facets (on vertex figure)  

Snub 24cell (4) 
16cell (1) 
5cell (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: 

[(3^{1,1},3)^{+},4]  2sr{4,3,3^{1,1}} 
1,1: 2: 

[3^{1,1,1,1}]^{+}  s{3^{1,1,1,1}} 
1,1,1,1: 
See also
Regular and uniform honeycombs in 4space:
References
 T. Gosset: On the Regular and SemiRegular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
 Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0486614808 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, WileyInterscience Publication, 1995, ISBN 9780471010036
 (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345]
 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
Fundamental convex regular and uniform honeycombs in dimensions 29  

Space  Family  / /  
E^{2}  Uniform tiling  {3^{[3]}}  δ_{3}  hδ_{3}  qδ_{3}  Hexagonal 
E^{3}  Uniform convex honeycomb  {3^{[4]}}  δ_{4}  hδ_{4}  qδ_{4}  
E^{4}  Uniform 4honeycomb  {3^{[5]}}  δ_{5}  hδ_{5}  qδ_{5}  24cell honeycomb 
E^{5}  Uniform 5honeycomb  {3^{[6]}}  δ_{6}  hδ_{6}  qδ_{6}  
E^{6}  Uniform 6honeycomb  {3^{[7]}}  δ_{7}  hδ_{7}  qδ_{7}  2_{22} 
E^{7}  Uniform 7honeycomb  {3^{[8]}}  δ_{8}  hδ_{8}  qδ_{8}  1_{33} • 3_{31} 
E^{8}  Uniform 8honeycomb  {3^{[9]}}  δ_{9}  hδ_{9}  qδ_{9}  1_{52} • 2_{51} • 5_{21} 
E^{9}  Uniform 9honeycomb  {3^{[10]}}  δ_{10}  hδ_{10}  qδ_{10}  
E^{n1}  Uniform (n1)honeycomb  {3^{[n]}}  δ_{n}  hδ_{n}  qδ_{n}  1_{k2} • 2_{k1} • k_{21} 