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 |
| 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,[1] 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 [math]\displaystyle{ \arccos(\frac{3}{4})\approx 41.409\,622\,109\,27^{\circ} }[/math], one of [math]\displaystyle{ \arccos(\frac{1}{6}+\frac{7}{12}\sqrt{2})\approx 7.420\,694\,647\,42^{\circ} }[/math] and one of [math]\displaystyle{ \arccos(\frac{1}{6}-\frac{7}{12}\sqrt{2})\approx 131.169\,683\,243\,31^{\circ} }[/math]. The dihedral angle equals [math]\displaystyle{ \arccos(-\frac{5}{7})\approx 135.584\,691\,402\,81^{\circ} }[/math]. 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
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
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