# Deltoidal icositetrahedron: Wikis

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# Encyclopedia

Deltoidal icositetrahedron

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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"
$\arccos ( -\frac{7 + 4\sqrt{2}}{17} )$
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.

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 $6\sqrt{29-2\sqrt{2}}$ and its volume is $\sqrt{122+71\sqrt{2}}$.

## Related polyhedra

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.

## 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 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.

## 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)
• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, MR730208, ISBN 978-0-521-54325-5   (The thirteen semiregular convex polyhedra and their duals, Page 23, Deltoidal icositetrahedron)
• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 [1] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 286, tetragonal icosikaitetrahedron)