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{4,3,3^1,1}
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] ${\displaystyle {\tilde {B))_{4))$ , [4,3,3^1,1] ${\displaystyle {\tilde {D))_{4))$ , [3^1,1,1,1]
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	Facets	Vertex figure symmetry (order)
${\displaystyle {\tilde {F))_{4))$ = [3,4,3,3]	4: 1:	, [3,3,2] (48)
${\displaystyle {\tilde {F))_{4))$ = [3,4,3,3]	3: 1: 1:	, [3,2] (12)
${\displaystyle {\tilde {C))_{4))$ = [4,3,3,4]	2,2: 1:	, [2,2] (8)
${\displaystyle {\tilde {B))_{4))$ = [3^1,1,3,4]	1,1: 2: 1:	, [2] (4)
${\displaystyle {\tilde {D))_{4))$ = [3^1,1,1,1]	1,1,1,1: 1:	, [] (2)

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 [1]
- (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	{3^[3]}	δ₃	hδ₃	qδ₃	Hexagonal
E³	Uniform convex honeycomb	{3^[4]}	δ₄	hδ₄	qδ₄
E⁴	Uniform 4-honeycomb	{3^[5]}	δ₅	hδ₅	qδ₅	24-cell honeycomb
E⁵	Uniform 5-honeycomb	{3^[6]}	δ₆	hδ₆	qδ₆
E⁶	Uniform 6-honeycomb	{3^[7]}	δ₇	hδ₇	qδ₇	2₂₂
E⁷	Uniform 7-honeycomb	{3^[8]}	δ₈	hδ₈	qδ₈	1₃₃ • 3₃₁
E⁸	Uniform 8-honeycomb	{3^[9]}	δ₉	hδ₉	qδ₉	1₅₂ • 2₅₁ • 5₂₁
E⁹	Uniform 9-honeycomb	{3^[10]}	δ₁₀	hδ₁₀	qδ₁₀
E¹⁰	Uniform 10-honeycomb	{3^[11]}	δ₁₁	hδ₁₁	qδ₁₁
E^n-1	Uniform (n-1)-honeycomb	{3^[n]}	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

Alternate names

Symmetry constructions

See also

References