Deltoidal icositetrahedron

In geometry, the deltoidal icositetrahedron (or trapezoidal icositetrahedron, tetragonal icosikaitetrahedron, tetragonal trisoctahedron, strombic icositetrahedron) is a Catalan solid. Its 24 faces are congruent kites. The deltoidal icositetrahedron, whose dual is the (uniform) rhombicuboctahedron, is tightly related to the pseudo-deltoidal icositetrahedron, whose dual is the pseudorhombicuboctahedron; but the actual and pseudo-d.i. are not to be confused with each other.

Deltoidal icositetrahedron
(rotating and 3D model)
Type	Catalan
Conway notation	oC or deC
Coxeter diagram
Face polygon	Kite with 3 equal acute angles & 1 obtuse angle
Faces	24, congruent
Edges	24 short + 24 long = 48
Vertices	8 (connecting 3 short edges) + 6 (connecting 4 long edges) + 12 (connecting 4 alternate short & long edges) = 26
Face configuration	V3.4.4.4
Symmetry group	O_h, BC₃, [4,3], *432
Rotation group	O, [4,3]⁺, (432)
Dihedral angle	same value for short & long edges: $\arccos \left(-{\frac {7+4{\sqrt {2}}}{17}}\right)$ $\approx 138^{\circ }07'05''$
Dual polyhedron	Rhombicuboctahedron
Properties	convex, face-transitive
Net

D.i. as artwork and die

D.i. projected onto cube and octahedron in Perspectiva Corporum Regularium

Dyakis dodecahedron crystal model and projection onto octahedron

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