Bitruncated 16-cell honeycomb

Bitruncated 16-cell honeycomb
(No image)
Type	Uniform honeycomb
Schläfli symbol	t1,2{3,3,4,3} h2,3{4,3,3,4} 2t{3,31,1,1}
Coxeter-Dynkin diagram	= =
4-face type	Truncated 24-cell Bitruncated tesseract
Cell type	Cube Truncated octahedron Truncated tetrahedron
Face type	{3}, {4}, {6}
Vertex figure
Coxeter group	${\displaystyle {\tilde {F}}_{4}}$ = [3,3,4,3] ${\displaystyle {\tilde {B}}_{4}}$ = [4,3,31,1] ${\displaystyle {\tilde {D}}_{4}}$ = [31,1,1,1]
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.

There are 3 different symmetry constructions, all with 3-3 duopyramid vertex figures. The ${\displaystyle {\tilde {B}}_{4}}$ symmetry doubles on ${\displaystyle {\tilde {D}}_{4}}$ in three possible ways, while ${\displaystyle {\tilde {F}}_{4}}$ contains the highest symmetry.

Affine Coxeter group	${\displaystyle {\tilde {F}}_{4}}$ [3,3,4,3]	${\displaystyle {\tilde {B}}_{4}}$ [4,3,31,1]	${\displaystyle {\tilde {D}}_{4}}$ [31,1,1,1]
Coxeter diagram
4-faces

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

• 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 [1]
  • (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)
• Klitzing, Richard. "4D Euclidean tesselations". x3x3x *b3x *b3o, x3x3o *b3x4o, o3x3x4o3o - bithit - O107

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