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 20px Truncated tetrahedron
Face type	{3}, {4}, {6}
Vertex figure
Coxeter group	[math]\displaystyle{ {\tilde{F}}_4 }[/math] = [3,3,4,3] [math]\displaystyle{ {\tilde{B}}_4 }[/math] = [4,3,31,1] [math]\displaystyle{ {\tilde{D}}_4 }[/math] = [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.

Symmetry constructions

There are 3 different symmetry constructions, all with 3-3 duopyramid vertex figures. The [math]\displaystyle{ {\tilde{B}}_4 }[/math] symmetry doubles on [math]\displaystyle{ {\tilde{D}}_4 }[/math] in three possible ways, while [math]\displaystyle{ {\tilde{F}}_4 }[/math] contains the highest symmetry.

Affine Coxeter group	[math]\displaystyle{ {\tilde{F}}_4 }[/math] [3,3,4,3]	[math]\displaystyle{ {\tilde{B}}_4 }[/math] [4,3,31,1]	[math]\displaystyle{ {\tilde{D}}_4 }[/math] [31,1,1,1]
Coxeter diagram
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, 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". https://bendwavy.org/klitzing/dimensions/flat.htm.  x3x3x *b3x *b3o, x3x3o *b3x4o, o3x3x4o3o - bithit - O107

Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space	Family	[math]\displaystyle{ {\tilde{A}}_{n-1} }[/math]	[math]\displaystyle{ {\tilde{C}}_{n-1} }[/math]	[math]\displaystyle{ {\tilde{B}}_{n-1} }[/math]	[math]\displaystyle{ {\tilde{D}}_{n-1} }[/math]	[math]\displaystyle{ {\tilde{G}}_2 }[/math] / [math]\displaystyle{ {\tilde{F}}_4 }[/math] / [math]\displaystyle{ {\tilde{E}}_{n-1} }[/math]
E2	Uniform tiling	{3[3]}	δ3	hδ3	qδ3	Hexagonal
E3	Uniform convex honeycomb	{3[4]}	δ4	hδ4	qδ4
E4	Uniform 4-honeycomb	{3[5]}	δ5	hδ5	qδ5	24-cell honeycomb
E5	Uniform 5-honeycomb	{3[6]}	δ6	hδ6	qδ6
E6	Uniform 6-honeycomb	{3[7]}	δ7	hδ7	qδ7	222
E7	Uniform 7-honeycomb	{3[8]}	δ8	hδ8	qδ8	133 • 331
E8	Uniform 8-honeycomb	{3[9]}	δ9	hδ9	qδ9	152 • 251 • 521
E9	Uniform 9-honeycomb	{3[10]}	δ10	hδ10	qδ10
En-1	Uniform (n-1)-honeycomb	{3[n]}	δn	hδn	qδn	1k2 • 2k1 • k21