The (small) rhombicuboctahedron (Cundy and Rowlett 1989, p. 105), sometimes simply called the rhombicuboctahedron (Wenninger 1989, p. 27; Maeder 1997, Conway
et al. 1999), is the 26-faced Archimedean
solid consisting of faces 
. Although this solid is sometimes also called the
truncated icosidodecahedron, this name is inappropriate since true truncation
would yield rectangular instead of square faces.
It is also the uniform polyhedron with Maeder index 10 (Maeder 1997), Wenninger index 13 (Wenninger 1989), Coxeter index 22 (Coxeter
et al. 1954), and Har'El index 15 (Har'El 1993). It has Schläfli
symbol r
and Wythoff symbol 
.
Some symmetric projections of the small rhombicuboctahedron are illustrated above.
The solid is an expanded (or cantellated) cube or octahedron since it may be constructed from either of these solids by the process of expansion.
The small rhombicuboctahedron is implemented in the Wolfram Language as UniformPolyhedron["Rhombicuboctahedron"]. Precomputed properties are available as PolyhedronData["SmallRhombicuboctahedron", prop].
A small rhombicuboctahedron appears in the middle right as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).
Its dual polyhedron of the small rhombicuboctahedron is the deltoidal icositetrahedron,
illustrated above together with their common midsphere.
The inradius 
of the dual, midradius 
of the solid and dual, and circumradius 
of the solid for 
are
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
The distances between the solid center and centroids of the triangular and square faces are
 |  |  |
(4)
|
 |  |  |
(5)
|
The surface area and volume are
 |  |  |
(6)
|
 |  |  |
(7)
|
The Dehn invariant of the unit small rhombicuboctahedron is
 |  |  |
(8)
|
 |  |  |
(9)
|
where the first expression uses the basis of Conway et al. (1999). It can be dissected into the elongated square gyrobicupola, which differs only by the relative rotation of the top and bottom cupolas.
The small rhombicuboctahedron can be constructed as the convex hull of the 24 vertices given by 
and the 16 distinct permutations of these
values.
The small rhombicuboctahedron is the convex hull of the small cubicuboctahedron, small rhombihexahedron, and stellated truncated hexahedron. Since the convex hull of the small cubicuboctahedron is the small rhombicuboctahedron, whose dual is the deltoidal icositetrahedron, the dual of the small cubicuboctahedron (i.e., the small hexacronic icositetrahedron) is one of the stellations of the deltoidal icositetrahedron (Wenninger 1989, p. 57).
The Minkowski sum of a unit cube and unit regular octahdron in dual position is a small rhombicuboctahedron.
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§3.7.5 in 