Deltoidal icositetrahedron  

Click on picture for large version. Click here for spinning version. 

Type  Catalan 
Face polygon  kite 
Faces  24 
Edges  48 
Vertices  26 = 6 + 8 + 12 
Face configuration  V3.4.4.4 
Symmetry group  octahedral (O_{h}) or *432 
Dihedral angle  138° 6' 34" 
Dual polyhedron  rhombicuboctahedron 
Properties  convex, facetransitive 
Net 
In geometry, a deltoidal icositetrahedron (also a trapezoidal icositetrahedron and tetragonal icosikaitetrahedron) is a Catalan solid which looks a bit like an overinflated cube. Its dual polyhedron is the rhombicuboctahedron.
The 24 faces are deltoids or kites, not trapezoids; the trapezohedron is similarly misnamed. The short and long edges of each kite are in the ratio 1.00:1.29.
If its smallest edges have length 1, its surface area is and its volume is .
Contents 
The deltoidal icositetrahedron is topologically identical to a cube which has all of its edges bisected:
The great triakis octahedron is a stellation of the deltoidal icositetrahedron.
The deltoidal icositetrahedron is a crystal habit often formed by the mineral analcime and occasionally garnet. The shape is often called a trapezohedron in mineral contexts, although in solid geometry that name refers to a particular class of polyhedron.
The Shining Trapezohedron of the fictional Lovecraft Mythos was probably intended to refer to a crystal of this shape.

