Cubitruncated cuboctahedron

Short description: Polyhedron with 20 faces

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
Dual polyhedron	Tetradyakis hexahedron
Vertex figure	6.8.8/3
Bowers acronym	Cotco

File:Cubitruncated cuboctahedron.stl 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, and has a shäfli symbol of tr{4,³/₂}

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

File:Tetradyakis hexahedron.stl 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 $\arccos (\frac{3}{4})\approx 41.40962210927^{\circ }$, one of $\arccos (\frac{1}{6}+\frac{7}{12}\sqrt{2})\approx 7.42069464742^{\circ }$ and one of $\arccos (\frac{1}{6}-\frac{7}{12}\sqrt{2})\approx 131.16968324331^{\circ }$. The dihedral angle equals $\arccos (-\frac{5}{7})\approx 135.58469140281^{\circ }$. 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

References

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

External links

• Weisstein, Eric W.. "Cubitruncated cuboctahedron". http://mathworld.wolfram.com/CubitruncatedCuboctahedron.html. 
• Weisstein, Eric W.. "Tetradyakis hexahedron". http://mathworld.wolfram.com/TetradyakisHexahedron.html. 
• http://gratrix.net Uniform polyhedra and duals