The deltoidal icositetrahedron is the 24-faced dual polyhedron of the small rhombicuboctahedron. It is also called the trapezoidal icositetrahedron (Holden 1971, p. 55). It is illustrated above together with a wireframe version and a net that can be used for its construction.
It is Wenninger dual 
A deltoidal icositetrahedron appears in the middle right as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).
A stella octangula, attractive octahedron 4-compound (whose dual is an attractive cube 4-compound), and cube can all be inscribed in a deltoidal icositetrahedron (E. Weisstein, Dec. 24, 2009). Superposing all these solids gives the beautiful compound illustrated above.
For a small rhombicuboctahedron with unit edge length, the deltoidal icositetrahedron has edge lengths
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(1)
|
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(2)
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and inradius
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(3)
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Normalizing so the smallest edge has unit edge length 
gives a deltoidal icositetrahedron with surface
area and volume
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(4)
|
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(5)
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