| Cubitruncated cuboctahedron | |
|---|---|
| Type | Uniform star polyhedron |
| Elements | F = 20, E = 72V = 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 |
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 = 72V = 20 (χ = −4) |
| Symmetry group | Oh, [4,3], *432 |
| Index references | DU16 |
| dual polyhedron | Cubitruncated cuboctahedron |
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 ( 3 4 ) ≈ 41.409 622 109 27 ∘ {\displaystyle \arccos({\frac {3}{4}})\approx 41.409\,622\,109\,27^{\circ }} , one of arccos ( 1 6 + 7 12 2 ) ≈ 7.420 694 647 42 ∘ {\displaystyle \arccos({\frac {1}{6}}+{\frac {7}{12}}{\sqrt {2}})\approx 7.420\,694\,647\,42^{\circ }} and one of arccos ( 1 6 − 7 12 2 ) ≈ 131.169 683 243 31 ∘ {\displaystyle \arccos({\frac {1}{6}}-{\frac {7}{12}}{\sqrt {2}})\approx 131.169\,683\,243\,31^{\circ }} . The dihedral angle equals arccos ( − 5 7 ) ≈ 135.584 691 402 81 ∘ {\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
- Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208 p. 92
External links
- Weisstein, Eric W. "Cubitruncated cuboctahedron". MathWorld.
- Weisstein, Eric W. "Tetradyakis hexahedron". MathWorld.
- http://gratrix.net Uniform polyhedra and duals
References
Maeder, Roman. "16: cubitruncated cuboctahedron". MathConsult. Archived from the original on 2015-03-29. https://www.mathconsult.ch/static/unipoly/16.html ↩