In geometry, a tetrakis hexahedron (also known as a tetrahexahedron, hextetrahedron, tetrakis cube, and kiscube) is a Catalan solid. Its dual is the truncated octahedron, an Archimedean solid

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.

The tetrakis hexahedron appears as one of the simplest examples in building theory. Consider the Riemannian symmetric space associated to the group SL<sub>4</sub>(R). Its Tits boundary has the structure of a spherical building whose apartments are 2-dimensional spheres. The partition of this sphere into spherical simplices (chambers) can be obtained by taking the radial projection of a tetrakis hexahedron.

Symmetry

With tetrahedral symmetry, the triangular faces represent the 24 fundamental domains of tetrahedral symmetry. This polyhedron can be constructed from six great circles on a sphere. It can also be seen by a cube with its square faces triangulated by their vertices and face centers, and a tetrahedron with its faces divided by vertices, mid-edges, and a central point.

See also

  • Disdyakis triacontahedron
  • Disdyakis dodecahedron
  • Kisrhombille tiling
  • Compound of three octahedra
  • Deltoidal icositetrahedron, another 24-face Catalan solid.

References

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  • (The thirteen semiregular convex polyhedra and their duals, Page 14, Tetrakishexahedron)
  • The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, [https://web.archive.org/web/20100919143320/https://akpeters.com/product.asp?ProdCode=2205] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Tetrakis hexahedron)
  • Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
  • VRML model
  • Conway Notation for Polyhedra Try: "dtO" or "kC"
  • Tetrakis Hexahedron – Interactive Polyhedron model
  • The Uniform Polyhedra