Cubitruncated cuboctahedron

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] and has a shäfli symbol of tr{4,³/₂}

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}
Coxeter diagram
Wythoff symbol	3 4 4/3 |
Symmetry group	Oh, [4,3], *432
Index references	U16, C52, W79
Dual polyhedron	Tetradyakis hexahedron
Vertex figure	6.8.8/3
Bowers acronym	Cotco

3D model of a cubitruncated cuboctahedron

Convex hull

Its convex hull is a nonuniform truncated cuboctahedron.

Convex hull

Cubitruncated cuboctahedron

Orthogonal projection

Cartesian coordinates

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

(±(√2−1), ±1, ±(√2+1))

Related polyhedra

Tetradyakis hexahedron

Tetradyakis hexahedron
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.

See also

• List of uniform polyhedra