# Deltoidal icositetrahedron

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Deltoidal icositetrahedron
Deltoidal icositetrahedron

Click on picture for large version.
Type Catalan
Face polygon kite
Faces 24
Edges 48
Vertices 26 = 6 + 8 + 12
Face configuration V3.4.4.4
Symmetry group Oh, [4,3], *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, 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 $6\sqrt{29-2\sqrt{2}}$ and its volume is $\sqrt{122+71\sqrt{2}}$.

## 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.

## 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.

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, ISBN 978-0-521-54325-5, MR730208  (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)

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