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. It can be called a disdyakis hexahedron or hexakis tetrahedron as the dual of an omnitruncated tetrahedron, and as the barycentric subdivision of a tetrahedron. Cartesian coordinates Cartesian coordinates for the 14 vertices of a tetrakis hexahedron centered at the origin, are the points (±3/2, 0, 0), (0, ±3/2, 0), (0, 0, ±3/2) and (±1, ±1, ±1). The length of the shorter edges of this tetrakis hexahedron equals 3/2 and that of the longer edges equals 2. The faces are acute isosceles triangles. The larger angle of these equals and the two smaller ones equal . Orthogonal projections The tetrakis hexahedron, dual of the truncated octahedron has 3 symmetry positions, two located on vertices and one mid-edge. 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. 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. Symmetry With Td, [3,3] (*332) tetrahedral symmetry, the triangular faces represent the 24 fundamental domains of tetrahedral symmetry. This polyhedron can be constructed from 6 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. The edges of the spherical tetrakis hexahedron belong to six great circles, which correspond to mirror planes in tetrahedral symmetry. They can be grouped into three pairs of orthogonal circles (which typically intersect on one coordinate axis each). In the images below these square hosohedra are colored red, green and blue. Dimensions If we denote the edge length of the base cube by a, the height of each pyramid summit above the cube is .