Deltoidal icositetrahedron
Click on picture for large version. Click deltoidalicositetrahedron.gif 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 (Oh) or *432
Dihedral angle	138° 6' 34"
Dual polyhedron	rhombicuboctahedron
Properties	convex, face-transitive
Net

A deltoidal icositetrahedron (or trapezoidal icositetrahedron) 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.

Related polyhedra

The deltoidal icositetrahedron is topologically identical to a cube which has all of its edges bisected:

Occurances 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 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.

See also

• Deltoidal hexecontahedron
• The Haunter of the Dark H.P. Lovecraft's book includes this figure in the plot.

References

• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)

External links

• Deltoidal (Trapezoidal) Icositetrahedron -- Interactive Polyhedron model
• Printable Geometric Net of a Deltoidal Icositetrahedron Life is a Story Problem.org