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In geometry, a deltoidal icositetrahedron (also a trapezoidal icositetrahedron, tetragonal icosikaitetrahedron,[1] and strombic icositetrahedron) is a Catalan solid which looks a bit like an overinflated cube. Its dual polyhedron is the rhombicuboctahedron.
The 24 faces are kites. The short and long edges of each kite are in the ratio \scriptstyle1:(2-\frac{1}{\sqrt{2}})\approx 1:1.292893\dots
If its smallest edges have length 1, its surface area is \scriptstyle{6\sqrt{29-2\sqrt{2}}} and its volume is \scriptstyle{\sqrt{122+71\sqrt{2}}}.
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 has another meaning.
The deltoidal icositetrahedron has three symmetry positions, all centered on vertices:
The great triakis octahedron is a stellation of the deltoidal icositetrahedron.
The deltoidal icositetrahedron is topologically equivalent to a cube whose faces are divided in quadrants. It can also be projected onto a regular octahedron, with kite faces, or more general quadrilaterals with pyritohedral symmetry. In Conway polyhedron notation, they represent an ortho operation to a cube or octahedron.
In crystallography a rotational variation is called a dyakis dodecahedron[2][3] or diploid.[4]
The deltoidal icositetrahedron is one of a family of duals to the uniform polyhedra related to the cube and regular octahedron.
This polyhedron is topologically related as a part of sequence of deltoidal polyhedra with face figure (V3.4.n.4), and continues as tilings of the hyperbolic plane. These face-transitive figures have (*n32) reflectional symmetry.
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