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	triangular duopyramid
Coxeter group	= [3,3,4,3] = [4,3,31,1] = [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 symmetry doubles on in three possible ways, while contains the highest symmetry.

Affine Coxeter group	[3,3,4,3]	[4,3,31,1]	[31,1,1,1]
Coxeter diagram
4-faces

Related honeycombs

The [3,4,3,3], , Coxeter group generates 31 permutations of uniform tessellations, 28 are unique in this family and ten are shared in the [4,3,3,4] and [4,3,3^1,1] families. The alternation (13) is also repeated in other families.

Extended symmetry	Extended diagram	Order	Honeycombs
[3,3,4,3]		×1	1, 3, 5, 6, 8, 9, 10, 11, 12
[3,4,3,3]		×1	2, 4, 7, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 23, 24, 25, 26, 27, 28, 29
[(3,3)[3,3,4,3*]] =[(3,3)[31,1,1,1]] =[3,4,3,3]	= =	×4	(2), (4), (7), (13)

The [4,3,3^1,1], , Coxeter group generates 31 permutations of uniform tessellations, 23 with distinct symmetry and 4 with distinct geometry. There are two alternated forms: the alternations (19) and (24) have the same geometry as the 16-cell honeycomb and snub 24-cell honeycomb respectively.

Extended symmetry	Extended diagram	Order	Honeycombs
[4,3,31,1]:		×1	5, 6, 7, 8
<[4,3,31,1]>: =[4,3,3,4]	=	×2	9, 10, 11, 12, 13, 14, (10), 15, 16, (13), 17, 18, 19
[3[1+,4,3,31,1]] = [3[3,31,1,1]] = [3,3,4,3]	= =	×3	1, 2, 3, 4
[(3,3)[1+,4,3,31,1]] = [(3,3)[31,1,1,1]] = [3,4,3,3]	= =	×12	20, 21, 22, 23

This honeycomb is one of ten uniform honeycombs constructed by the Coxeter group, all repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 10th is constructed as an alternation. As subgroups in Coxeter notation: [3,4,(3,3)^*] (index 24), [3,3,4,3^*] (index 6), [1⁺,4,3,3,4,1⁺] (index 4), [3^1,1,3,4,1⁺] (index 2) are all isomorphic to [3^1,1,1,1]. The ten permutations are listed with its highest extended symmetry relation:

Extended symmetry	Extended diagram	Order	Honeycombs
[31,1,1,1]		×1	(none)
<[31,1,1,1]> = [31,1,3,4]	=	×2	(none)
<<[1,131,1]>> = [4,3,3,4]	=	×4	1, 2
[3[3,31,1,1]] = [3,4,3,3]	=	×6	3, 4, 5, 6
[<<[1,131,1]>>] = [[4,3,3,4]]	=	×8	7, 8, 9, 10
[(3,3)[31,1,1,1]] = [3,3,4,3]	=	×24

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

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

Fundamental convex regular and uniform honeycombs in dimensions 2–10
Family					/ /
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 n-honeycomb	{3[n]}	δn	hδn	qδn	1k2 • 2k1 • k21