Truncated 24-cell honeycomb explained

bgcolor=#e7dcc3 colspan=2	Truncated 24-cell honeycomb
bgcolor=#ffffff align=center colspan=2	(No image)
Type	Uniform 4-honeycomb
Schläfli symbol	t tr t2r t2r t
Coxeter-Dynkin diagrams
4-face type
Cell type
Face type	Square Triangle
Vertex figure	Tetrahedral pyramid
Coxeter groups	{\tilde{F}} _{4} , [3,4,3,3] {\tilde{B}} _{4} , [4,3,3<sup>1,1</sup>] {\tilde{C}} _{4} , [4,3,3,4] {\tilde{D}} _{4} , [3<sup>1,1,1,1</sup>]
Properties	Vertex 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 snub from the

{\tilde{D}} _{4} construction. This truncated 24-cell has Schläfli symbol t, and its snub is represented as s.

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} = [3<sup>1,1</sup>,3,4]		1,1: 2: 1:		, [] (2)
{\tilde{D}} _{4} = [3<sup>1,1,1,1</sup>]		1,1,1,1: 1:		[]^+ (1)

See also

Regular and uniform honeycombs in 4-space:

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

References

• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, 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, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
  • (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
• o4x3x3x4o, x3x3x *b3x4o, x3x3x *b3x *b3x, o3o3o4x3x, x3x3x4o3o - ticot - O99