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Tetrakis cuboctahedron

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Tetrakis cuboctahedron
TypeGeodesic polyhedron (2,0)
Faces32 triangles (2 types)
Edges48 (2 types)
Vertices18 (2 types)
Vertex configuration(6) 35
(12) 36
Conway notationk4aC
Symmetry groupOctahedral (Oh)
Dual polyhedronchamfered cube
Propertiesconvex
Net
A cat toy in the shape of a tetrakis cuboctahedron projected onto a sphere
Tetrakis cuboctahedrons usefully represent carbon atoms in a 3D ball-and-stick model of a diamond lattice as the normals to alternate yellow-shaded faces in the top image correspond exactly to the tetrahedral bond angles
3D model of a tetrakis cuboctahedron

In geometry, the tetrakis cuboctahedron is a convex polyhedron with 32 triangular faces, 48 edges, and 18 vertices. It is a dual of the truncated rhombic dodecahedron.

Its name comes from a topological construction from the cuboctahedron with the kis operator applied to the square faces. In this construction, all the vertices are assumed to be the same distance from the center, while in general octahedral symmetry can be maintain even with the 6 order-4 vertices at a different distance from the center as the other 12.

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It can also be topologically constructed from the octahedron, dividing each triangular face into 4 triangles by adding mid-edge vertices (an ortho operation). From this construction, all 32 triangles will be equilateral.

This polyhedron can be confused with a slightly smaller Catalan solid, the tetrakis hexahedron, which has only 24 triangles, 32 edges, and 14 vertices.

See also

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References

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  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5
    • Chapter 21: Naming the Archimedean and Catalan polyhedra and Tilings (p284)
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