# Truncated icositetrachoron

Truncated icositetrachoron
Rank4
TypeUniform
Notation
Bowers style acronymTico
Coxeter diagramx3x4o3o ()
Elements
Cells24 cubes, 24 truncated octahedra
Faces144 squares, 96 hexagons
Edges96+288
Vertices192
Vertex figureTriangular pyramid, edge lengths 2 (base) and 3 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {7}}\approx 2.64575}$
Hypervolume159
Dichoral anglesToe–4–cube: 135°
Toe–6–toe: 120°
Central density1
Number of external pieces48
Level of complexity4
Related polytopes
ArmyTico
RegimentTico
DualOctakis icositetrachoron
ConjugateNone
Abstract & topological properties
Flag count4608
Euler characteristic0
OrientableYes
Properties
SymmetryF4, order 1152
Flag orbits4
ConvexYes
NatureTame

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.

## Vertex coordinates

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

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

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

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

## 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 ${\displaystyle {\sqrt {a^{2}+3b^{2}+3ab}}}$ and its hypervolume is given by ${\displaystyle 2a^{4}+16a^{3}b+48a^{2}b^{2}+64ab^{3}+29b^{4}}$.

## Variations

The truncated icositetrachoron has two semi-uniform subsymmetrical variations:

## Related polychora

Uniform polychoron compounds composed of truncated icositetrachora include: