Birectified 16-cell honeycomb

Birectified 16-cell honeycomb
(No image)
Type	Uniform honeycomb
Schläfli symbol	t2{3,3,4,3}
Coxeter-Dynkin diagram	Template:CDD Template:CDD = Template:CDD Template:CDD
4-face type	Rectified tesseract Rectified 24-cell
Cell type	Cube Cuboctahedron Tetrahedron
Face type	{3}, {4}
Vertex figure	{3}×{3} duoprism
Coxeter group	${\tilde{F}}_4$ = [3,3,4,3] ${\tilde{B}}_4$ = [4,3,31,1] ${\tilde{D}}_4$ = [31,1,1,1]
Dual	?
Properties	vertex-transitive

In four-dimensional Euclidean geometry, the birectified 16-cell honeycomb (or runcic 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 duoprism 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]	${\tilde{B}}_4$ [4,3,31,1]	${\tilde{D}}_4$ [31,1,1,1]
Coxeter diagram	Template:CDD	Template:CDD	Template:CDD
Vertex figure
Vertex figure symmetry	[3,2,3] (order 36)	[3,2] (order 12)	[3] (order 6)
4-faces	Template:CDD Template:CDD	Template:CDD Template:CDD Template:CDD	Template:CDD Template:CDD
Cells	Template:CDD Template:CDD Template:CDD	Template:CDD Template:CDD Template:CDD Template:CDD	Template:CDD Template:CDD Template:CDD

Related honeycombs

There are ten uniform honeycombs constructed by the ${\tilde{D}}_4$ 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:

D4 honeycombs
Extended symmetry	Extended diagram	Extended group	Honeycombs
[31,1,1,1]	Template:CDD	${\tilde{D}}_4$	(none)
<[31,1,1,1]> ↔ [31,1,3,4]	Template:CDD ↔ Template:CDD	${\tilde{D}}_4$×2 = ${\tilde{B}}_4$	(none)
<2[1,131,1]> ↔ [4,3,3,4]	Template:CDD ↔ Template:CDD	${\tilde{D}}_4$×4 = ${\tilde{C}}_4$	Template:CDD 1, Template:CDD 2
[3[3,31,1,1]] ↔ [3,3,4,3]	Template:CDD ↔ Template:CDD	${\tilde{D}}_4$×6 = ${\tilde{F}}_4$	Template:CDD3, Template:CDD 4, Template:CDD 5, Template:CDD 6
[4[1,131,1]] ↔ [[4,3,3,4]]	Template:CDD ↔ Template:CDD	${\tilde{D}}_4$×8 = ${\tilde{C}}_4$×2	Template:CDD 7, Template:CDD 8, Template:CDD 9
[(3,3)[31,1,1,1]] ↔ [3,4,3,3]	Template:CDD ↔ Template:CDD	${\tilde{D}}_4$×24 = ${\tilde{F}}_4$
[(3,3)[31,1,1,1]]+ ↔ [3+,4,3,3]	Template:CDD ↔ Template:CDD	½${\tilde{D}}_4$×24 = ½${\tilde{F}}_4$	Template:CDD 10

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, Template:ISBN [1] Template:Webarchive
  • (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)
• Template:KlitzingPolytopes x3o3x *b3x *b3o, x3o3o *b3x4o, o3o3x4o3o - bricot - O106