Cuboctahedron

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Template:Semireg polyhedra db A cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasi-regular polyhedron, i.e. an Archimedean solid (vertex-uniform) with in addition edge-uniformity.

1 Cartesian coordinates
2 Geometric relations
3 Cuboctahedra in the world
4 Related polyhedra
5 See also
6 External links

Image:Cuboctahedron flat.png

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Cartesian coordinates

The Cartesian coordinates for the vertices of a cuboctahedron centered at the origin are

(±1,±1,0)

(±1,0,±1)

(0,±1,±1)

Its dual polyhedron is the rhombic dodecahedron.

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Geometric relations

Image:Kuboctaeder-Animation.gif

A cuboctahedron can be obtained by taking an appropriate cross section of a four-dimensional cross-polytope.

A cuboctahedron has octahedral symmetry. Its first stellation is the compound of a cube and its dual octahedron, with the vertices of the cuboctahedron located at the midpoints of the edges of either.

The cuboctahedron is a rectified cube and also a rectified octahedron.

It is also a runcinated tetrahedron. With this construction it is given the Wythoff Symbol: 3 3 | 2.

A skew runcination of the tetrahedron produces a solid with faces parallel to those of the cuboctahedron, namely eight triangles of two sizes, and six rectangles. While its edges are unequal, this solid remains vertex-uniform: the solid has the full tetrahedral symmetry group and its vertices are equivalent under that group.

The edges of a cuboctahedron form four regular hexagons. If the cuboctahedron is cut in the plane of one of these hexagons, each half is a triangular cupola, one of the Johnson solids; the cuboctahedron itself thus can also be called a triangular gyrobicupola, the simplest of a series. If the halves are put back together with a twist, so that triangles meet triangles and squares meet squares, the result is another Johnson solid, the triangular orthobicupola.

Both triangular bicupolae are important in sphere packing. Each sphere can have up to twelve neighbors, and in a face-centered cubic lattice these take the positions of a cuboctahedron's vertices. In a hexagonal close-packed lattice they correspond to the corners of the triangular orthobicupola.

Cuboctahedra and octahedra are the cells of the rectified cubic tessellation, one of the Andreini tessellations.

The volume of the cuboctahedron is 5/6 of that of the enclosing cube and 5/8 of that of the enclosing octahedron; it is 5/3 √2 times the cube of the length of an edge.

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