Tetrahemihexahedron

The tetrahemihexahedron, tetrahemicube, or thah, is a quasiregular polyhedron and one of 10 uniform hemipolyhedra. It consists of 4 equilateral triangles and 3 "hemi" squares passing through the center, with two of each joining at a vertex. It also has 4 triangular pseudofaces. It can be derived as a facetorectified tetrahedron. Its triangular faces are parallel to those of a tetrahedron, and its hemi square faces are parallel to those of a cube or hexahedron: hence the name. It is also the 3D demicross. It can be derived as a rectified petrial tetrahedron.

It is the only non-convex, non-skew uniform polyhedron that only has A3 symmetry in its highest symmetry form. It's also the only uniform polyhedron, other than prisms whose bases have an odd amount of sides, with an odd amount of faces. It is also the only uniform polyhedron with an odd Euler characteristic. Further, it is the only uniform polyhedron with an odd number of faces aside from prisms of an odd-sided polygon.

The visible portion of this solid resembles an octahedron with four triangular pyramids carved out. In fact the four triangular faces are a tetrahedral subset of those of an octahedron, while the squares are the 3 equatorial planes of the octahedron. It also shares its vertices and edges with the octahedron.

It also happens to be a 3/2-gonal cuploid (retrograde triangular cuploid), as it can be formed from a retrograde triangular cupola by removing the bottom degenerate face.

The tetrahemihexahedron has the curious property that it can be blended with an octahedron to produce another tetrahemihexahedron using the other half of the octahedron's faces. These two orientations of the tetrahemihexahedron are sometimes important to distinguish when it appears in higher polytopes.

Hemicuboctahedron
The hemicuboctahedron is a quotient of the cuboctahedron, where antipodal elements are identified. It is a tiling of the real projective plane and is abstractly equivalent to the tetrahemihexahedron. This means that the surface of a tetrahemihexahedron is homeomorphic to a real projective plane, and thus it is non-orientable and has Euler characteristic 1. Its surface closely resembles the Roman surface embedding of the real projective plane, with both having A3 symmetry.

Since the double-cover of the hemicuboctahedron is the cuboctahedron, the double-cover of the tetrahemihexahedron is abstractly equivalent to the cuboctahedron as well. Conversely, quasicantellating a tetrahedron produces a doubly-covered tetrahemihexahedron.

The square faces of the hemicuboctahedron can be subdivided into triangles to form the hemiicosahedron. Likewise certain faces of the hemiicosahedron can be combined into squares to form the hemicuboctahedron.

Vertex coordinates
Its vertices are the same as those of its regiment colonel, the octahedron.

In vertex figures
The tetrahemihexahedron appears as a vertex figure in one uniform polychoron, that being the tesseractihemioctachoron. It has an edge length of 1.

Irregular tetrahemihexahedra appear as vertex figures of several uniform polychora in Bowers' category 11, namely piphid, stefacoth, shafipto, shif phix, six fipady, hi fipady, mohi fipady, six fixady, shi fixady, gohi fohixhi, and gaxifthi. These all have the symmetry of triangular cuploids, with the squares becoming trapezoids..

Polyhedra isomorphic to the tetrahemihexahedron appear as vertex figures of several uniform polychora in Bowers' category 12, namely sto, gittifcoth, gahfipto, gix fixady, gohi fixady, ghif phix, gix fipady, shi fohixhi, and saxifthi. These resemble triangular cuploids except that the side trapezoids have been turned into crossed trapezoids.

Related polyhedra
Two uniform polyhedron compounds are composed of tetrahemihexahedra, both of which share edges with compounds of octahedra:


 * Hemirhombichiricosahedron (5)
 * Snub pseudosnub rhombicosahedron (20)