Rectified tesseractic honeycomb

quarter tesseractic honeycomb
(No image)
Type	Uniform 4-honeycomb
Family	Quarter hypercubic honeycomb
Schläfli symbol	r{4,3,3,4} r{4,31,1} r{4,31,1} q{4,3,3,4}
Coxeter-Dynkin diagram	Template:CDD Template:CDD = Template:CDD Template:CDD = Template:CDD Template:CDD = Template:CDD Template:CDD = Template:CDD
4-face type	h{4,32}, h3{4,32},
Cell type	{3,3}, t1{4,3},
Face type	{3} {4}
Edge figure	Square pyramid
Vertex figure	Elongated {3,4}×{}
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 rectified tesseractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space. It is constructed by a rectification of a tesseractic honeycomb which creates new vertices on the middle of all the original edges, rectifying the cells into rectified tesseracts, and adding new 16-cell facets at the original vertices. Its vertex figure is an octahedral prism, {3,4}×{}.

It is also called a quarter tesseractic honeycomb since it has half the vertices of the 4-demicubic honeycomb, and a quarter of the vertices 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]	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
• 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, Template:ISBN [1] Template:Webarchive
  • (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)
• Template:KlitzingPolytopes o4x3o3o4o, o3o3o *b3x4o, x3o3x *b3o4o, x3o3x *b3o *b3o - rittit - O87
• Template:Cite book