Rectified 24-cell honeycomb

Rectified 24-cell honeycomb
(No image)
Type	Uniform 4-honeycomb
Schläfli symbol	r{3,4,3,3} rr{3,3,4,3} r2r{4,3,3,4} r2r{4,3,31,1}
Coxeter-Dynkin diagrams	Template:CDD Template:CDD Template:CDD Template:CDD = Template:CDD Template:CDD = Template:CDD Template:CDD = Template:CDD
4-face type	Tesseract Rectified 24-cell
Cell type	Cube Cuboctahedron
Face type	Square Triangle
Vertex figure	Tetrahedral prism
Coxeter groups	${\tilde{F}}_4$, [3,4,3,3] ${\tilde{C}}_4$, [4,3,3,4] ${\tilde{B}}_4$, [4,3,31,1] ${\tilde{D}}_4$, [31,1,1,1]
Properties	Vertex transitive

In four-dimensional Euclidean geometry, the rectified 24-cell honeycomb is a uniform space-filling honeycomb. It is constructed by a rectification of the regular 24-cell honeycomb, containing tesseract and rectified 24-cell cells.

Alternate names

• Rectified icositetrachoric tetracomb
• Rectified icositetrachoric honeycomb
• Cantellated 16-cell honeycomb
• Bicantellated tesseractic honeycomb

Symmetry constructions

There are five different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored rectified 24-cell and tesseract facets. The tetrahedral prism vertex figure contains 4 rectified 24-cells capped by two opposite tesseracts.

Coxeter group	Coxeter diagram	Facets	Vertex figure	Vertex figure symmetry (order)
${\tilde{F}}_4$ = [3,4,3,3]	Template:CDD	4: Template:CDD 1: Template:CDD		Template:CDD, [3,3,2] (48)
	Template:CDD	3: Template:CDD 1: Template:CDD 1: Template:CDD		Template:CDD, [3,2] (12)
${\tilde{C}}_4$ = [4,3,3,4]	Template:CDD	2,2: Template:CDD 1: Template:CDD		Template:CDD, [2,2] (8)
${\tilde{B}}_4$ = [31,1,3,4]	Template:CDD	1,1: Template:CDD 2: Template:CDD 1: Template:CDD		Template:CDD, [2] (4)
${\tilde{D}}_4$ = [31,1,1,1]	Template:CDD	1,1,1,1: Template:CDD 1: Template:CDD		Template:CDD, [] (2)

See also

Regular and uniform honeycombs in 4-space:

• Tesseractic honeycomb
• 16-cell honeycomb
• 24-cell honeycomb
• Truncated 24-cell honeycomb
• Snub 24-cell honeycomb
• 5-cell honeycomb
• Truncated 5-cell honeycomb
• Omnitruncated 5-cell honeycomb

References

• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, Template:Isbn p. 296, Table II: Regular honeycombs
• 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) Model 93
• Template:KlitzingPolytopes, o3o3o4x3o, o4x3o3x4o - ricot - O93