Abstract simplicial complex

Deltoidal icositetrahedron
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Type	Catalan
Coxeter diagram	Template:CDD
Face polygon	kite
Faces	24
Edges	48
Vertices	26 = 6 + 8 + 12
Face configuration	V3.4.4.4
Symmetry group	Oh, BC3, [4,3], *432
Rotation group	O, [4,3]+, (432)
Dihedral angle	138° 7' 5" $\arccos(-{\frac {7+4{\sqrt {2}}}{17}})$
Dual polyhedron	rhombicuboctahedron
Properties	convex, face-transitive
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.

Dimensions

The 24 faces are deltoids or kites, also called trapezia in the US and trapezoids in Britain. The short and long edges of each kite are in the ratio 1:1.292893...

If its smallest edges have length 1, its surface area is $\scriptstyle {6{\sqrt {29-2{\sqrt {2}}}}}$ and its volume is $\scriptstyle {\sqrt {122+71{\sqrt {2}}}}$.

Occurrences in nature and culture

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.

Related polyhedra

The deltoidal icositetrahedron is topologically equivalent to a cube whose faces are divided in quadrants.

The great triakis octahedron is a stellation of the deltoidal icositetrahedron.

Related polyhedra and tilings

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.

See also

• Deltoidal hexecontahedron
• "The Haunter of the Dark", a story by H.P. Lovecraft, whose plot involves this figure

References

• Template:The Geometrical Foundation of Natural Structure (book) (Section 3-9)
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External links

• Template:Mathworld2
• Deltoidal (Trapezoidal) Icositetrahedron – Interactive Polyhedron model