Rectified tesseractic honeycomb

quarter cubic 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}, File:Schlegel wireframe 16-cell.png h3{4,32}, Error creating thumbnail:
Cell type	{3,3}, File:Tetrahedron.png t1{4,3}, Error creating thumbnail:
Face type	{3} {4}
Edge figure	File:Square pyramid.png Square pyramid
Vertex figure	Error creating thumbnail: 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]
  • (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

Template:Navbar-collapsible
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