Truncated cuboctahedron
From Wikipedia, the free encyclopedia
| Great rhombicuboctahedron or truncated cuboctahedron |
|
|---|---|
 (Click here for rotating model) |
|
| Type | Archimedean solid |
| Elements | F = 26, E = 72, V = 48 (χ = 2) |
| Faces by sides | 12{4}+8{6}+6{8} |
| Schläfli symbol |  |
| Wythoff symbol | 2 3 4 | |
| Coxeter-Dynkin |    |
| Symmetry | Oh |
| References | U11, C23, W15 |
| Properties | Semiregular convex zonohedron |
 Colored faces |
 4.6.8 (Vertex figure) |
 Disdyakis dodecahedron (dual polyhedron) |
 Net |
The truncated cuboctahedron is an Archimedean solid. It has 12 regular square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices and 72 edges. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated cuboctahedron is a zonohedron.
Contents |
[edit] Other names
Alternate interchangeable names are:
- Rhombitruncated cuboctahedron
- Great rhombicuboctahedron
- Omnitruncated cuboctahedron
The name truncated cuboctahedron, given originally by Johannes Kepler, is a little misleading. If you truncate a cuboctahedron by cutting the corners off, you do not get this uniform figure: some of the faces will be rectangles. However, the resulting figure is topologically equivalent to a truncated cuboctahedron and can always be deformed until the faces are regular.
The alternative name great rhombicuboctahedron refers to the fact that the 12 square faces lie in the same planes as the 12 faces of the rhombic dodecahedron which is dual to the cuboctahedron. Compare to small rhombicuboctahedron.
One unfortunate point of confusion: There is a nonconvex uniform polyhedron by the same name. See uniform great rhombicuboctahedron.
[edit] Area and volume
The area A and the volume V of the truncated cuboctahedron of edge length a are:
[edit] Cartesian coordinates
Cartesian coordinates for the vertices of a truncated cuboctahedron centered at the origin are all permutations of
- (±1, ±(1+√2), ±(1+√8)). (edge length = 2)
[edit] See also
[edit] References
- Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)
[edit] External links
- Eric W. Weisstein, Great rhombicuboctahedron (Archimedean solid) at MathWorld.
- The Uniform Polyhedra
- Virtual Reality Polyhedra The Encyclopedia of Polyhedra
- great Rhombicuboctahedron: paper strips for plaiting
