The Deltoidal Icositetrahedron
The deltoidal icositetrahedron is a 3D Catalan solid bounded by 24 kites, 48 edges (12 short, 12 long), and 26 vertices. It is the dual of the rhombicuboctahedron.
The kite faces are transitive, with two short edges and two long edges with a (4+√2) : 7 length ratio.
The 26 vertices are of three kinds: 6 apical vertices corresponding to an inscribed octahedron, 8 vertices corresponding to an inscribed cube, and 12 vertices corresponding to an inscribed cuboctahedron.
The deltoidal icositetrahedron is the projection envelope of its direct 4D analogue, the triangular antitegmatic hexacontatetrachoron.
Projections
The following are images of the deltoidal icositetrahedron from various viewpoints:
Projection  Description 

Front view, centered on an axial vertex. 

Side view, centered on a cuboctahedron vertex. 

Octant view, centered on a cube vertex. 
Animation
Here's an animation of a deltoidal icositetrahedron rotating around the vertical axis:
Coordinates
The Cartesian coordinates for the deltoidal icositetrahedron are all permutations of coordinate and changes of sign of:
 (0, 0, (2√2−1))
 (0, (4−√2)/2, (4−√2)/2)
 (1, 1, 1)
These coordinates can be obtained by inverting a rhombicuboctahedron of edge length (3−√2)/7.