Prismatorhombated hexadecachoron

The prismatorhombated hexadecachoron, or proh, also commonly called the runcitruncated tesseract, is a convex uniform polychoron that consists of 32 triangular prisms, 24 octagonal prisms, 16 cuboctahedra, and 8 truncated cubes. 1 triangular prism, 2 octagonal prisms, 1 cuboctahedron, and 1 truncated cube join at each vertex. As one of its names suggests, it can be obtained by runcitruncating the tesseract.

The prismatorhombated hexadecachoron can be vertex-inscribed into a small rhombated icositetrachoron.

Vertex coordinates
The vertices of a prismatorhombated hexadecachoron of edge length 1 are given by all permutations of:


 * $$\left(±\frac{1+2\sqrt2}{2},\,±\frac{1+\sqrt2}{2},\,±\frac{1+\sqrt2}{2},\,±\frac12\right).$$

Representations
A prismatorhombated hexadecachoron has the following Coxeter diagrams:


 * x4x3o3x (full symmetry)
 * xxwwxx4xxooxx3oxxxxo&#xt (BC3 axial, truncated cube-first)
 * wx3oo3xw *b3xx&#zx (D4 symmetry)
 * Xwx xxw4xxo3oxx&#zx (BC3×A1 symmetry)

Semi-uniform variant
The prismatorhombated hexadecachoron has a semi-uniform variant of the form x4y3o3z that maintains its full symmetry. This variant uses 8 truncated cubes of form x4y3o, 16 rhombitetratetrahedra of form y3o3z, 32 triangular prisms of form x z3o, and 24 ditetragonal prisms of form z x4y as cells, with 3 edge lengths.

With edges of length a, b, and c (such that it forms a4b3o3c), its circumradius is given by $$\sqrt{\frac{2a^2+3b^2+c^2+2bc+(3ab+ac)\sqrt2}{2}}$$.

It has coordinates given by all permutations of:


 * $$\left(±\frac{a+(b+c)\sqrt2}{2},\,±\frac{a+(b+c)\sqrt2}{2},\,±\frac{a+b\sqrt2}{2},\,±\frac{a}{2}\right).$$

Related polychora
The prismatorhombated hexadecachoron is the colonel of a 3-member regiment that also includes the small prismatohexadecadisoctachoron and small rhombiprismic tesseractihexadecachoron.

The segmentochoron truncated cube atop great rhombicuboctahedron can be obtained as a cap of the prismatorhombated hexadecachoron.

The truncated cubes of the prismatorhombated hexadecachoron can be augmented by cuboctahedron atop truncated cube segmentochora. If all 8 truncated cubes are augmented, the result is the small rhombated icositetrachoron, as square cupolas from the segmentochora merge with octagonal prisms to form small rhombicuboctahedra.