Cuboctahedron
| Cuboctahedral graph | |
|---|---|
4-fold symmetry
|
|
| Vertices | 12 |
| Edges | 24 |
| Automorphisms | 48 |
| Properties | Quartic graph, Hamiltonian, regular |
In geometry, a cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive.
Its dual polyhedron is the rhombic dodecahedron.
The cuboctahedron was probably known to Plato: Heron's Definitiones quotes Archimedes as saying that Plato knew of a solid made of 8 triangles and 6 squares.
The area A and the volume V of the cuboctahedron of edge length a are:
The cuboctahedron has four special orthogonal projections, centered on a vertex, an edge, and the two types of faces, triangular and square. The last two correspond to the B2 and A2Coxeter planes. The skew projections show a square and hexagon passing through the center of the cuboctahedron.
The cuboctahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
The Cartesian coordinates for the vertices of a cuboctahedron (of edge length √2) centered at the origin are:
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