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]

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

It is the dual of the uniform cubitruncated cuboctahedron.

References

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

Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208 p. 92

External links

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[1] Maeder, Roman. "16: cubitruncated cuboctahedron". MathConsult.