Truncated 24-cell honeycomb

Truncated 24-cell honeycomb
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Type	Uniform 4-honeycomb
Schläfli symbol	t{3,4,3,3} tr{3,3,4,3} t2r{4,3,3,4} t2r{4,3,3^1,1} t{3^1,1,1,1}
Coxeter-Dynkin diagrams
4-face type	Tesseract Truncated 24-cell
Cell type	Cube Truncated octahedron
Face type	Square Triangle
Vertex figure	Tetrahedral pyramid
Coxeter groups	${\tilde {F}}_{4}$ , [3,4,3,3] ${\tilde {B}}_{4}$ , [4,3,3^1,1] ${{\tilde {C}}}_{4}$ , [4,3,3,4] ${\tilde {D}}_{4}$ , [3^1,1,1,1]
Properties	Vertex transitive

In four-dimensional Euclidean geometry, the truncated 24-cell honeycomb is a uniform space-filling honeycomb. It can be seen as a truncation of the regular 24-cell honeycomb, containing tesseract and truncated 24-cell cells.

It has a uniform alternation, called the snub 24-cell honeycomb. It is a snub from the ${\tilde {D}}_{4}$ construction. This truncated 24-cell has Schläfli symbol t{3^1,1,1,1}, and its snub is represented as s{3^1,1,1,1}.

Alternate names

Truncated icositetrachoric tetracomb
Truncated icositetrachoric honeycomb
Cantitruncated 16-cell honeycomb
Bicantitruncated tesseractic honeycomb

Symmetry constructions

There are five different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored truncated 24-cell facets. In all cases, four truncated 24-cells, and one tesseract meet at each vertex, but the vertex figures have different symmetry generators.

Coxeter group	Facets	Vertex figure symmetry (order)
${\tilde {F}}_{4}$ = [3,4,3,3]	4: 1:	, [3,3] (24)
${\tilde {F}}_{4}$ = [3,3,4,3]	3: 1: 1:	, [3] (6)
${{\tilde {C}}}_{4}$ = [4,3,3,4]	2,2: 1:	, [2] (4)
${\tilde {B}}_{4}$ = [3^1,1,3,4]	1,1: 2: 1:	, [ ] (2)
${\tilde {D}}_{4}$ = [3^1,1,1,1]	1,1,1,1: 1:	[ ]⁺ (1)

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 99
Klitzing, Richard. "4D Euclidean tesselations". o4x3x3x4o, x3x3x *b3x4o, x3x3x *b3x *b3x, o3o3o4x3x, x3x3x4o3o - ticot - O99

Fundamental convex regular and uniform honeycombs in dimensions 3–10 (or 2-9)
Family	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
Uniform tiling	{3^[3]}	δ₃	hδ₃	qδ₃	Hexagonal
Uniform convex honeycomb	{3^[4]}	δ₄	hδ₄	qδ₄
Uniform 5-honeycomb	{3^[5]}	δ₅	hδ₅	qδ₅	24-cell honeycomb
Uniform 6-honeycomb	{3^[6]}	δ₆	hδ₆	qδ₆
Uniform 7-honeycomb	{3^[7]}	δ₇	hδ₇	qδ₇	2₂₂
Uniform 8-honeycomb	{3^[8]}	δ₈	hδ₈	qδ₈	1₃₃ • 3₃₁
Uniform 9-honeycomb	{3^[9]}	δ₉	hδ₉	qδ₉	1₅₂ • 2₅₁ • 5₂₁
Uniform 10-honeycomb	{3^[10]}	δ₁₀	hδ₁₀	qδ₁₀
Uniform n-honeycomb	{3^[n]}	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

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Truncated 24-cell honeycomb

Alternate names

Symmetry constructions

See also

References