Bitruncated 16-cell honeycomb explained

bgcolor=#e7dcc3 colspan=2	Bitruncated 16-cell honeycomb
bgcolor=#ffffff align=center colspan=2	(No image)
Type	Uniform honeycomb
Schläfli symbol	t1,2 h2,3 2t
Coxeter-Dynkin diagram	= =
4-face type
Cell type
Face type	,,
Vertex figure
Coxeter group	{\tilde{F}} _{4} = [3,3,4,3] {\tilde{B}} _{4} = [4,3,3<sup>1,1</sup>] {\tilde{D}} _{4} = [3<sup>1,1,1,1</sup>]
Dual	?
Properties	vertex-transitive

In four-dimensional Euclidean geometry, the bitruncated 16-cell honeycomb (or runcicantic tesseractic honeycomb) is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space.

Symmetry constructions

There are 3 different symmetry constructions, all with 3-3 duopyramid vertex figures. The

{\tilde{B}} _{4} symmetry doubles on {\tilde{D}} _{4} in three possible ways, while {\tilde{F}} _{4} contains the highest symmetry.

Affine Coxeter group	{\tilde{F}} _{4} [3,3,4,3]	{\tilde{B}} _{4} [4,3,3<sup>1,1</sup>]	{\tilde{D}} _{4} [3<sup>1,1,1,1</sup>]
Coxeter diagram				- align=center	Vertex figure
Vertex figure symmetry	[3,2,3] (order 36)	[3,2] (order 12)	[3] (order 6)-->
4-faces

See also

Regular and uniform honeycombs in 4-space:

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

References

• 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)
• x3x3x *b3x *b3o, x3x3o *b3x4o, o3x3x4o3o - bithit - O107