Rank4
TypeUniform
Notation
Bowers style acronymProh
Coxeter diagramx4x3o3x ()
Elements
Cells32 triangular prisms, 24 octagonal prisms, 16 cuboctahedra, 8 truncated cubes
Faces64+64 triangles, 96+96 squares, 48 octagons
Edges96+192+192
Vertices192
Vertex figureSkewed rectangular pyramid, base edge lengths 1, 2, 1, 2; lateral edge lengths 2, 2, 2+2, 2+2
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {4+2{\sqrt {2}}}}\approx 2.61313}$
Hypervolume${\displaystyle 8{\frac {27+20{\sqrt {2}}}{3}}\approx 147.42472}$
Dichoral anglesCo–3–trip: 150°
Op–4–trip: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
Co–4–op: 135°
Tic–8–op: 135°
Tic–3–co: 120°
Central density1
Number of external pieces80
Level of complexity16
Related polytopes
ArmyProh
RegimentProh
Abstract & topological properties
Flag count6144
Euler characteristic0
OrientableYes
Properties
SymmetryB4, order 384
ConvexYes
NatureTame

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 edge-inscribed into a small rhombated icositetrachoron. Blending 3 prismatorhombated hexadecachora results in the small rhombic disicositetrachoron.

## Vertex coordinates

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

• ${\displaystyle \left(\pm {\frac {1+2{\sqrt {2}}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}}\right)}$.

## Representations

A prismatorhombated hexadecachoron has the following Coxeter diagrams:

• x4x3o3x () (full symmetry)
• xxwwxx4xxooxx3oxxxxo&#xt (B3 axial, truncated cube-first)
• wx3oo3xw *b3xx&#zx (D4 symmetry)
• Xwx xxw4xxo3oxx&#zx (B3×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 ${\displaystyle {\sqrt {\frac {2a^{2}+3b^{2}+c^{2}+2bc+(3ab+ac){\sqrt {2}}}{2}}}}$.

It has coordinates given by all permutations of:

• ${\displaystyle \left(\pm {\frac {a+(b+c){\sqrt {2}}}{2}},\,\pm {\frac {a+(b+c){\sqrt {2}}}{2}},\,\pm {\frac {a+b{\sqrt {2}}}{2}},\,\pm {\frac {a}{2}}\right)}$.