Truncated icositetrachoron

Truncated icositetrachoron
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Tico
Coxeter diagram	x3x4o3o ()
Elements
Cells	24 cubes, 24 truncated octahedra
Faces	144 squares, 96 hexagons
Edges	96+288
Vertices	192
Vertex figure	Triangular pyramid, edge lengths √2 (base) and √3 (sides)
Measures (edge length 1)
Circumradius	${\sqrt {7}}\approx 2.64575$
Hypervolume	159
Dichoral angles	Toe–4–cube: 135°
	Toe–6–toe: 120°
Central density	1
Number of external pieces	48
Level of complexity	4
Related polytopes
Army	Tico
Regiment	Tico
Dual	Octakis icositetrachoron
Conjugate	None
Abstract & topological properties
Flag count	4608
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	F4, order 1152
Convex	Yes
Nature	Tame

The truncated icositetrachoron, or tico, also commonly called the truncated 24-cell, is a convex uniform polychoron that consists of 24 cubes and 24 truncated octahedra. 1 cube and 3 truncated octahedra join at each vertex. As the name suggests, it can be obtained as the truncation of an icositetrachoron.

It is also the cantitruncated hexadecachoron, and the omnitruncate of the D₄ group. It can be alternated into a snub disicositetrachoron.

Cross-sections

Vertex coordinates

The vertices of a truncated icositetrachoron of edge length 1 are given by all permutations of:

• $\left(\pm {\frac {3{\sqrt {2}}}{2}},\,\pm {\sqrt {2}},\,\pm {\frac {\sqrt {2}}{2}},\,0\right)$.

The vertices of the truncation of the dual icositetrachoron can be given by all permutations of:

• $\left(\pm {\frac {5}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right)$,
• $\left(\pm {\frac {3}{2}},\,\pm {\frac {3}{2}},\,\pm {\frac {3}{2}},\,\pm {\frac {1}{2}}\right)$,
• $\left(\pm 2,\,\pm 1,\,\pm 1,\,\pm 1\right)$.

Representations

A truncated icositetrachoron has the following Coxeter diagrams:

• x3x4o3o () (full symmetry)
• o4x3x3x () (B₄ symmetry, great rhombated hexadecachoron)
• x3x3x *b3x () (D₄ symmetry, omnitruncated demitesseract)
• s4x3x3x () (as snub)
• oooqooo3xxuxuxx3xuxxxux&#xt (B₃ axial, truncated octahedron-first)
• xux3qoo3ooo3oqQ&#zx (D₄ symmetry, truncated dual ico)
• xuxu(xd)(xd)uxux4ooqo(oo)(oo)oqoo3oooq(Qo)(Qo)qooo&#xt (B₃ axial, cube-first)

Semi-uniform variant

The truncated icositetrachoron has a semi-uniform variant of the form x3y4o3o that maintains its full symmetry. This variant uses 24 cubes of size y and 24 semi-uniform truncated octahedra of form x3y4o as cells, with 2 edge lengths.

With edges of length a (surrounded by truncated octahedra only) and b (of cubes), its circumradius is given by ${\sqrt {a^{2}+3b^{2}+3ab}}$ and its hypervolume is given by $2a^{4}+16a^{3}b+48a^{2}b^{2}+64ab^{3}+29b^{4}$.

Variations

The truncated icositetrachoron has two semi-uniform subsymmetrical variations:

• Great rhombated hexadecachoron - 8 truncated octahedra, 16 great rhombitetratetrahedra, and 24 square prisms with a sphenoid verf
• Great prismated demitesseract - 3 sets of 8 great rhombitetratetrahedra and 24 cuboids, with an irregular tetrahedron verf

Related polychora

Uniform polychoron compounds composed of truncated icositetrachora include:

• Truncated stellated tetracontoctachoron (2)
• Truncated dodecahedronary hexacosichoron (25)

External links

• Bowers, Jonathan. "Category 2: Truncates" (#21).

• Bowers, Jonathan. "Tessic Isogonals".

• Klitzing, Richard. "tico".
• Quickfur. "The Truncated 24-cell".

• Wikipedia contributors. "The Truncated 24-cell".