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Concept
Cuboctahedron
Summary
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. It is radially equilateral.
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.
Vector Equilibrium (Buckminster Fuller) because its center-to-vertex radius equals its edge length (it has radial equilateral symmetry). Fuller also called a cuboctahedron built of rigid struts and flexible vertices a jitterbug; this object can be progressively transformed into an icosahedron, octahedron, and tetrahedron by folding along the diagonals of its square sides.
With Oh symmetry, order 48, it is a rectified cube or rectified octahedron (Norman Johnson)
With Td symmetry, order 24, it is a cantellated tetrahedron or rhombitetratetrahedron.
With D3d symmetry, order 12, it is a triangular gyrobicupola.
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 A2 Coxeter 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 ) centered at the origin are:
(±1,±1,0)
(±1,0,±1)
(0,±1,±1)
An alternate set of coordinates can be made in 4-space, as 12 permutations of:
(0,1,1,2)
This construction exists as one of 16 orthant facets of the cantellated 16-cell.
Official source
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