Tetrakis hexahedron

Tetrakis hexahedron
(Click here for rotating model)
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 (also known as a tetrahexahedron) 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 tetrahexahedra 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.

Gallery

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  As tiling by reflection domain of (2,3,3) triangle group

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  As a solid

See also

• Disdyakis triacontahedron
• Disdyakis dodecahedron
• Bisected hexagonal tiling
• Compound of three octahedra

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)

External links

• Eric W. Weisstein, Tetrakis hexahedron (Catalan solid) at MathWorld.
• Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
  • VRML model
  • Conway Notation for Polyhedra Try: "dtO"
• Tetrakis Hexahedron – Interactive Polyhedron model
• The Uniform Polyhedra