# Tetrakis hexahedron

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Tetrakis hexahedron
Tetrakis hexahedron

Type Catalan solid
Face type isosceles triangle
Faces 24
Edges 36
Vertices 14
Vertices by type 6{4}+8{6}
Face configuration V4.6.6
Symmetry group Oh, [4,3], *432
Dihedral angle 143°7'48"
$\arccos ( -\frac{4}{5} )$
Properties convex, face-transitive

Truncated octahedron
(dual polyhedron)

Net

In geometry, a tetrakis hexahedron is a Catalan solid. Its dual is the truncated octahedron, an Archimedean solid. It can be seen as a cube with square pyramids covering each square face; that is, it is the Kleetope of the cube.

It also can be called a disdyakis hexahedron as the dual of an omnitruncated tetrahedron.

## Uses

Naturally occurring (crystal) formations of tetrahexahedrons are observed in copper and fluorite systems.

Polyhedral dice shaped like the tetrakis hexahedron are occasionally used by gamers.

A 24-cell viewed under a vertex-first perspective projection has a surface topology of a tetrakis hexahedron and the geometric proportions of the rhombic dodecahedron, with the rhombic faces divided into two triangles.

## 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 14, Tetrakishexahedron)
• 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 284, Tetrakis hexahedron )