Rank4
TypeUniform
SpaceSpherical
Notation
Bowers style acronymProh
Coxeter diagram
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${\displaystyle \sqrt{4+2\sqrt2} ≈ 2.61313}$
Hypervolume${\displaystyle 8\frac{27+20\sqrt2}{3} ≈ 147.42472}$
Dichoral anglesCo–3–trip: 150°
Op–4–trip: ${\displaystyle \arccos\left(-\frac{\sqrt6}{3}\right) ≈ 144.73561°}$
Co–4–op: 135°
Tic–8–op: 135°
Tic–3–co: 120°
Central density1
Number of pieces80
Level of complexity16
Related polytopes
ArmyProh
RegimentProh
Abstract properties
Euler characteristic0
Topological properties
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:

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

## 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 ${\displaystyle \sqrt{\frac{2a^2+3b^2+c^2+2bc+(3ab+ac)\sqrt2}{2}}}$.

It has coordinates given by all permutations of:

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

## Related polychora

The 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.

o4o3o3o truncations
Name OBSA CD diagram Picture
Tesseract tes x4o3o3o
Truncated tesseract tat x4x3o3o
Rectified tesseract rit o4x3o3o
Rectified hexadecachoron = Icositetrachoron ico o4o3x3o
Small rhombated tesseract srit x4o3x3o
Great rhombated tesseract grit x4x3x3o
Small rhombated hexadecachoron = Rectified icositetrachoron rico o4x3o3x
Great rhombated hexadecachoron = Truncated icositetrachoron tico o4x3x3x