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|Conway notation||oC or deC|
|Vertices||26 = 6 + 8 + 12|
|Symmetry group||Oh, BC3, [4,3], *432|
|Rotation group||O, [4,3]+, (432)|
|Dihedral angle||138° 7' 5"
In geometry, a deltoidal icositetrahedron (also a trapezoidal icositetrahedron, tetragonal icosikaitetrahedron, and strombic icositetrahedron) is a Catalan solid which looks a bit like an overinflated cube. Its dual polyhedron is the rhombicuboctahedron.
The 24 faces are kites. The short and long edges of each kite are in the ratio
If its smallest edges have length 1, its surface area is and its volume is .
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 deltoidal icositetrahedron has three symmetry positions, all centered on vertices:
The great triakis octahedron is a stellation of the deltoidal icositetrahedron.
The deltoidal icositetrahedron is topologically equivalent to a cube whose faces are divided in quadrants. It can also be projected onto a regular octahedron, with kite faces, or more general quadrilaterals with pyritohedral symmetry. In Conway polyhedron notation, they represent an ortho operation to a cube or octahedron.
|Octahedral, Oh, order 24||Pyritohedral, Th, order 12|
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.
|Symmetry: [4,3], (*432)||[4,3]+, (432)||[3+,4], (3*2)|
|Duals to uniform polyhedra|
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.
- Deltoidal hexecontahedron
- "The Haunter of the Dark", a story by H.P. Lovecraft, whose plot involves this figure
- 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, MR 730208 (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  (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 286, tetragonal icosikaitetrahedron)
- Eric W. Weisstein, Deltoidal icositetrahedron (Catalan solid) at MathWorld
- Deltoidal (Trapezoidal) Icositetrahedron – Interactive Polyhedron model