Bitruncated 16-cell honeycomb

Bitruncated 16-cell honeycomb
(No image)
Type	Uniform honeycomb
Schläfli symbol	t_1,2{3,3,4,3} h_2,3{4,3,3,4} 2t{3,3^1,1,1}
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	triangular duopyramid
Coxeter group	${\tilde {F}}_{4}$ = [3,3,4,3] ${\tilde {B}}_{4}$ = [4,3,3^1,1] ${\tilde {D}}_{4}$ = [3^1,1,1,1]
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.

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]	${\tilde {B}}_{4}$ [4,3,3^1,1]	${\tilde {D}}_{4}$ [3^1,1,1,1]
Coxeter diagram
4-faces

Notes

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

Fundamental convex regular and uniform honeycombs in dimensions 3–10 (or 2-9)
Family	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
Uniform tiling	{3^[3]}	δ₃	hδ₃	qδ₃	Hexagonal
Uniform convex honeycomb	{3^[4]}	δ₄	hδ₄	qδ₄
Uniform 5-honeycomb	{3^[5]}	δ₅	hδ₅	qδ₅	24-cell honeycomb
Uniform 6-honeycomb	{3^[6]}	δ₆	hδ₆	qδ₆
Uniform 7-honeycomb	{3^[7]}	δ₇	hδ₇	qδ₇	2₂₂
Uniform 8-honeycomb	{3^[8]}	δ₈	hδ₈	qδ₈	1₃₃ • 3₃₁
Uniform 9-honeycomb	{3^[9]}	δ₉	hδ₉	qδ₉	1₅₂ • 2₅₁ • 5₂₁
Uniform 10-honeycomb	{3^[10]}	δ₁₀	hδ₁₀	qδ₁₀
Uniform n-honeycomb	{3^[n]}	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

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Bitruncated 16-cell honeycomb

Symmetry constructions

See also

Notes

References