Bitruncated tesseractic honeycomb

Bitruncated tesseractic honeycomb
(No image)
Type	Uniform 4-honeycomb
Schläfli symbol	t12{4,3,3,4} t12{4,31,1} t23{4,31,1}
Coxeter-Dynkin diagram
4-face type	bitruncated tesseract truncated 16-cell
Cell type	octahedron truncated tetrahedron truncated octahedron
Face type	{3}, {4}, {6}
Vertex figure	square duopyramid
Coxeter group	${\tilde {C}}_{4}$ = [4,3,3,4] ${\tilde {B}}_{4}$ = [4,31,1] ${\tilde {D}}_{4}$ = [31,1,1,1]
Dual
Properties	vertex-transitive

In four-dimensional Euclidean geometry, the bitruncated tesseractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space. It is constructed by a bitruncation of a tesseractic honeycomb.

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]		${\tilde {D}}_{4}$	(none)
<[31,1,1,1]> ↔ [31,1,3,4]	↔	${\tilde {D}}_{4}$×2 = ${\tilde {B}}_{4}$	(none)
<2[1,131,1]> ↔ [4,3,3,4]	↔	${\tilde {D}}_{4}$×4 = ${\tilde {C}}_{4}$	1, 2
[3[3,31,1,1]] ↔ [3,3,4,3]	↔	${\tilde {D}}_{4}$×6 = ${\tilde {F}}_{4}$	3, 4, 5, 6
[4[1,131,1]] ↔ [[4,3,3,4]]	↔	${\tilde {D}}_{4}$×8 = ${\tilde {C}}_{4}$×2	7, 8, 9
[(3,3)[31,1,1,1]] ↔ [3,4,3,3]	↔	${\tilde {D}}_{4}$×24 = ${\tilde {F}}_{4}$
[(3,3)[31,1,1,1]]+ ↔ [3+,4,3,3]	↔	½${\tilde {D}}_{4}$×24 = ½${\tilde {F}}_{4}$	10

See also

Regular and uniform honeycombs in 4-space:

• Tesseractic honeycomb
• Demitesseractic honeycomb
• 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] See p318 [2]
• 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#4D". x3x3x *b3o *b3o , x3x3x *b3o4o , o3x3o *b3x4o , o4x3x3o4o - batitit - O92
• Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9.

Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
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