The Full Wiki

Deltoidal icositetrahedron: Wikis


Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.


From Wikipedia, the free encyclopedia

Deltoidal icositetrahedron
Deltoidal icositetrahedron
Click on picture for large version.
Click here 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"
 \arccos ( -\frac{7 + 4\sqrt{2}}{17} )
Dual polyhedron rhombicuboctahedron
Properties convex, face-transitive
Deltoidal icositetrahedron

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:

Partial cubic honeycomb.png

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.

See also


  • 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)

External links



Got something to say? Make a comment.
Your name
Your email address