Rectified 24-cell honeycomb

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.

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	= = =
4-face type	Tesseract Rectified 24-cell
Cell type	Cube Cuboctahedron
Face type	Square Triangle
Vertex figure	Tetrahedral prism
Coxeter groups	${\displaystyle {\tilde {F}}_{4}}$, [3,4,3,3] ${\displaystyle {\tilde {C}}_{4}}$, [4,3,3,4] ${\displaystyle {\tilde {B}}_{4}}$, [4,3,31,1] ${\displaystyle {\tilde {D}}_{4}}$, [31,1,1,1]
Properties	Vertex transitive

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)
${\displaystyle {\tilde {F}}_{4}}$ = [3,4,3,3]		4: 1:		, [3,3,2] (48)
		3: 1: 1:		, [3,2] (12)
${\displaystyle {\tilde {C}}_{4}}$ = [4,3,3,4]		2,2: 1:		, [2,2] (8)
${\displaystyle {\tilde {B}}_{4}}$ = [31,1,3,4]		1,1: 2: 1:		, [2] (4)
${\displaystyle {\tilde {D}}_{4}}$ = [31,1,1,1]		1,1,1,1: 1:		, [] (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, ISBN 0-486-61480-8 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, ISBN 978-0-471-01003-6
  • (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
• Klitzing, Richard. "4D Euclidean tesselations"., o3o3o4x3o, o4x3o3x4o - ricot - O93

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