Cuboctahedron

Cuboctahedron
(Click here for rotating model)
Type	Archimedean solid Uniform polyhedron
Elements	F = 14, E = 24, V = 12 (χ = 2)
Faces by sides	8{3}+6{4}
Conway notation	aC aaT
Schläfli symbols	r{4,3} or ${\begin{Bmatrix}4\\3\end{Bmatrix}}$ rr{3,3} or $r{\begin{Bmatrix}3\\3\end{Bmatrix}}$
Schläfli symbols	t₁{4,3} or t_0,2{3,3}
Wythoff symbol	2 \| 3 4 3 3 \| 2
Coxeter diagram	or or
Symmetry group	O_h, B₃, [4,3], (432), order 48 T_d, [3,3], (332), order 24
Rotation group	O, [4,3]⁺, (432), order 24
Dihedral angle	$\operatorname {arcsec} \left(-{\sqrt {3}}\right)\approx 125.26^{\circ }$
References	U₀₇, C₁₉, W₁₁
Properties	Semiregular convex quasiregular
Colored faces	3.4.3.4 (Vertex figure)
Rhombic dodecahedron (dual polyhedron)	Net

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.^[1] 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.^[2]

^ Coxeter 1973, pp. 18–19, §2.3 Quasi-regular polyhedra.
^ Heath, Thomas L. (1931), "A manual of Greek mathematics", Nature, 128 (3235), Clarendon: 739–740, Bibcode:1931Natur.128..739T, doi:10.1038/128739a0, S2CID 3994109

[FOOTNOTECoxeter197318–19§2.3_Quasi-regular_polyhedra-1] Coxeter 1973, pp. 18–19, §2.3 Quasi-regular polyhedra.

[2] Heath, Thomas L. (1931), "A manual of Greek mathematics", Nature, 128 (3235), Clarendon: 739–740, Bibcode:1931Natur.128..739T, doi:10.1038/128739a0, S2CID 3994109

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