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Cuboctahedron
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{{Semireg polyhedra db|Semireg polyhedron stat table|CO}}
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.
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Image:cuboctahedron flat.png
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.
Geometric relations
Image:Kuboctaeder-Animation.gif left
A cuboctahedron can be obtained by taking an appropriate
cross section (geometry) cross section of a four-dimensional
cross-polytope.
A cuboctahedron has octahedral symmetry. Its first
stellation is the
polyhedral compound compound of a
cube (geometry) cube and its dual
octahedron, with the vertices of the cuboctahedron located at the midpoints of the edges of either.
The cuboctahedron is a
Rectification (geometry) rectified cube and also a rectified
octahedron.
It is also a
runcination (geometry) 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
bicupola (geometry) 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.
Cuboctahedra in the world
Buckminster Fuller applied the name "
Dymaxion" to this shape.
Related polyhedra
Compare:
image:hexahedron.jpg thumb|left|100px|[[Cube]]
image:truncatedhexahedron.jpg thumb|left|100px|[[Truncated cube]]
image:cuboctahedron.jpg thumb|left|100px|cuboctahedron
image:truncatedoctahedron.jpg thumb|left|100px|[[Truncated octahedron]]
image:octahedron.jpg thumb|left|100px|[[Octahedron]]
See also
*
:Image:Kuboctaeder-Animation.gif Animation of rotating cuboctahedron
*
Cube
*
Icosidodecahedron
*
Octahedron
*
Rhombicuboctahedron
*
Truncated cuboctahedron
External links
-
The Uniform Polyhedra
-
Virtual Reality Polyhedra The Encyclopedia of Polyhedra
Category:Archimedean solids
Category:Quasiregular polyhedra
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es:Cuboctaedro
nl:Kuboctaëder
ja:立方八�体
pt:Cuboctaedro
zh:截�立方體
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