Truncated 24-cell honeycomb

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.

Truncated 24-cell honeycomb
(No image)
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	${\displaystyle {\tilde {F}}_{4}}$, [3,4,3,3] ${\displaystyle {\tilde {B}}_{4}}$, [4,3,3^1,1] ${\displaystyle {\tilde {C}}_{4}}$, [4,3,3,4] ${\displaystyle {\tilde {D}}_{4}}$, [3^1,1,1,1]
Properties	Vertex transitive

It has a uniform alternation, called the snub 24-cell honeycomb. It is a snub from the ${\displaystyle {\tilde {D}}_{4}}$ 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	Coxeter diagram	Facets	Vertex figure	Vertex figure symmetry (order)
${\displaystyle {\tilde {F}}_{4}}$ = [3,4,3,3]		4: 1:		, [3,3] (24)
${\displaystyle {\tilde {F}}_{4}}$ = [3,3,4,3]		3: 1: 1:		, [3] (6)
${\displaystyle {\tilde {C}}_{4}}$ = [4,3,3,4]		2,2: 1:		, [2] (4)
${\displaystyle {\tilde {B}}_{4}}$ = [3^1,1,3,4]		1,1: 2: 1:		, [ ] (2)
${\displaystyle {\tilde {D}}_{4}}$ = [3^1,1,1,1]		1,1,1,1: 1:		[ ]^+ (1)

See also

Regular and uniform honeycombs in 4-space:

• Tesseractic honeycomb
• 16-cell honeycomb
• 24-cell honeycomb
• Rectified 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 99
• Klitzing, Richard. "4D Euclidean tesselations". o4x3x3x4o, x3x3x *b3x4o, x3x3x *b3x *b3x, o3o3o4x3x, x3x3x4o3o - ticot - O99

Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space	Family	${\displaystyle {\tilde {A}}_{n-1}}$	${\displaystyle {\tilde {C}}_{n-1}}$	${\displaystyle {\tilde {B}}_{n-1}}$	${\displaystyle {\tilde {D}}_{n-1}}$	${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
E^2	Uniform tiling	{3^[3]}	δ3	hδ3	qδ3	Hexagonal
E^3	Uniform convex honeycomb	{3^[4]}	δ4	hδ4	qδ4
E^4	Uniform 4-honeycomb	{3^[5]}	δ5	hδ5	qδ5	24-cell honeycomb
E^5	Uniform 5-honeycomb	{3^[6]}	δ6	hδ6	qδ6
E^6	Uniform 6-honeycomb	{3^[7]}	δ7	hδ7	qδ7	222
E^7	Uniform 7-honeycomb	{3^[8]}	δ8	hδ8	qδ8	133 • 331
E^8	Uniform 8-honeycomb	{3^[9]}	δ9	hδ9	qδ9	152 • 251 • 521
E^9	Uniform 9-honeycomb	{3^[10]}	δ10	hδ10	qδ10
E^n-1	Uniform (n-1)-honeycomb	{3^[n]}	δn	hδn	qδn	1k2 • 2k1 • k21