The cuboctahedron, also called the heptaparallelohedron or dymaxion (the latter according to Buckminster Fuller; Rawles 1997), is the Archimedean
solid with faces 
.
It is illustrated above together with a wireframe version and a net
that can be used for its construction.
The cuboctahedron is one of the two convex quasiregular polyhedra. It is also the uniform polyhedron
with Maeder index 7 (Maeder 1997), Wenninger index 11 (Wenninger 1989a), Coxeter
index 19 (Coxeter et al. 1954), and Har'El index 12 (Har'El 1993). It has
Schläfli symbol 
and Wythoff symbol 
.
It is shown above in a number of symmetric projections.
The cuboctahedron is implemented in the Wolfram Language as PolyhedronData["Cuboctahedron"] or UniformPolyhedron["Cuboctahedron"]. Precomputed properties are available as PolyhedronData["Cuboctahedron", prop].
A cuboctahedron appears in the lower left as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43), as well is in the mezzotint "Crystal" (Bool et al. 1982, p. 293).
The cuboctahedron is the convex hull of Escher's solid (together with the first rhombic dodecahedron stellation and square dipyramid 3-compound which shares its outer hull) as well as the cubohemioctahedron and octahemioctahedron uniform polyhedra.
The dual polyhedron of the cuboctahedron is the rhombic dodecahedron, both of which are illustrated above together with their common midsphere.
The cuboctahedron has the 
octahedral group of
symmetries. According to Heron, Archimedes ascribed the cuboctahedron to Plato (Heath
1981; Coxeter 1973, p. 30). The polyhedron
vertices of a cuboctahedron with polyhedron edge
length of 
are 
,
, and 
.
The mineral argentite (
)
forms cuboctahedral crystals (Steinhaus 1999, p. 203).
In the Season 2 Star Trek episode "By Any Other Name" (1968), aliens known as Kelvins reduce crewmembers Shae and Yeoman Thompson to two small gray cuboctahedron which are purported to contain their essences. Rojan, the Kelvin leader, then crushes Thompson's polyhedron as a warning to Captain Kirk (William Shatner), thus killing her, but restores Shae to human form.
The inradius 
of the dual, midradius 
of the solid and dual, and circumradius 
of the solid for 
are
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(1)
|
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(2)
|
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(3)
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The distances from the center of the solid to the centroids of the triangular and square faces are
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(4)
|
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(5)
|
The dihedral angle between triangular and square faces is
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(6)
|
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(7)
|
The surface area and volume are
 |  |  |
(8)
|
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(9)
|
The cuboctahedron has Dehn invariant
 |  |  |
(10)
|
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(11)
|
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(12)
|
(OEIS A377296), where the first expression uses the basis of Conway et al. (1999). It can be dissected into the triangular orthobicupola, from which it differs only by the relative rotation of the top and bottom cupolas.
Faceted versions of the cuboctahedron include the cubohemioctahedron and octahemioctahedron.
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The solid common to both the cube and octahedron (left figure) in a cube-octahedron compound is a cuboctahedron (right figure; Ball and Coxeter 1987).
The Minkowski sum of two dual regular tetrahedra is a cuboctahedron.
The cuboctahedron can be inscribed in the rhombic dodecahedron (left figure; Steinhaus 1999, p. 206). The centers of the square faces determine an octahedron (right figure; Ball and Coxeter 1987, p. 143).
Wenninger (1989b) lists four of the possible cuboctahedron stellations, namely the cube-octahedron compound, a truncated form of the stella octangula, a sort of compound of six intersecting square pyramids, and an attractive concave solid formed of rhombi meeting four at a time.
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If a cuboctahedron is oriented with triangles on top and bottom, the two halves may be rotated one sixth of a turn with respect to each other to obtain Johnson
solid 
,
the triangular orthobicupola.
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In cubic close packing, each sphere is surrounded by 12 other spheres. Taking a collection of 13 such spheres gives the cluster illustrated above. Connecting the centers of the external 12 spheres gives a cuboctahedron (Steinhaus 1999, pp. 203-207).
." §3.7.2 in 