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)
TypeUniform 4-honeycomb
Schläfli symbolr{3,4,3,3}
rr{3,3,4,3}
r2r{4,3,3,4}
r2r{4,3,31,1}
Coxeter-Dynkin diagrams

=
=
=

4-face typeTesseract
Rectified 24-cell
Cell typeCube
Cuboctahedron
Face typeSquare
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]
PropertiesVertex 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)

Regular and uniform honeycombs in 4-space:

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
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 133331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152251521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k22k1k21