# Cubitruncated cuboctahedron

Cubitruncated cuboctahedron
Type Uniform star polyhedron
Elements F = 20, E = 72
V = 48 (χ = −4)
Faces by sides 8{6}+6{8}+6{8/3}
Wythoff symbol 3 4 4/3 |
Symmetry group Oh, [4,3], *432
Index references U16, C52, W79
Vertex figure
6.8.8/3
Bowers acronym Cotco

In geometry, the cubitruncated cuboctahedron or cuboctatruncated cuboctahedron is a nonconvex uniform polyhedron, indexed as U16. It has 20 faces (8 hexagons, 6 octagons, and 6 octagrams), 72 edges, and 48 vertices.[1]

3D model of a cubitruncated cuboctahedron

## Convex hull

Its convex hull is a nonuniform truncated cuboctahedron.

 Convex hull Cubitruncated cuboctahedron

## Cartesian coordinates

Cartesian coordinates for the vertices of a cubitruncated cuboctahedron are all the permutations of

(±(2−1), ±1, ±(2+1))

## Related polyhedra

Type Star polyhedron
Face
Elements F = 48, E = 72
V = 20 (χ = −4)
Symmetry group Oh, [4,3], *432
Index references DU16
dual polyhedron Cubitruncated cuboctahedron

3D model of a tetradyakis hexahedron

The tetradyakis hexahedron (or great disdyakis dodecahedron) is a nonconvex isohedral polyhedron. It has 48 intersecting scalene triangle faces, 72 edges, and 20 vertices.

#### Proportions

The triangles have one angle of ${\displaystyle \arccos({\frac {3}{4}})\approx 41.409\,622\,109\,27^{\circ }}$ , one of ${\displaystyle \arccos({\frac {1}{6}}+{\frac {7}{12}}{\sqrt {2}})\approx 7.420\,694\,647\,42^{\circ }}$  and one of ${\displaystyle \arccos({\frac {1}{6}}-{\frac {7}{12}}{\sqrt {2}})\approx 131.169\,683\,243\,31^{\circ }}$ . The dihedral angle equals ${\displaystyle \arccos(-{\frac {5}{7}})\approx 135.584\,691\,402\,81^{\circ }}$ . Part of each triangle lies within the solid, hence is invisible in solid models.

It is the dual of the uniform cubitruncated cuboctahedron.

## References

1. ^ Maeder, Roman. "16: cubitruncated cuboctahedron". MathConsult.