Tetrakis cube
In geometry, a tetrakis hexahedron (also known as a tetrahexahedron, hextetrahedron and kiscube) is a Catalan solid. Its dual is the truncated octahedron, an Archimedean solid.
It also can be called a disdyakis hexahedron as the dual of an omnitruncated tetrahedron.
The tetrakis hexahedron, dual of the truncated octahedron has 3 symmetry positions, two located on vertices and one mid-edge.
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.
The tetrakis hexahedron appears as one of the simplest examples in building theory. Consider the Riemannian symmetric space associated to the group SL4(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.
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