Prismatorhombated hexadecachoron
Prismatorhombated hexadecachoron  

Rank  4 
Type  Uniform 
Notation  
Bowers style acronym  Proh 
Coxeter diagram  x4x3o3x () 
Elements  
Cells  32 triangular prisms, 24 octagonal prisms, 16 cuboctahedra, 8 truncated cubes 
Faces  64+64 triangles, 96+96 squares, 48 octagons 
Edges  96+192+192 
Vertices  192 
Vertex figure  Skewed rectangular pyramid, base edge lengths 1, √2, 1, √2; lateral edge lengths √2, √2, √2+√2, √2+√2 
Measures (edge length 1)  
Circumradius  
Hypervolume  
Dichoral angles  Co–3–trip: 150° 
Op–4–trip:  
Co–4–op: 135°  
Tic–8–op: 135°  
Tic–3–co: 120°  
Central density  1 
Number of external pieces  80 
Level of complexity  16 
Related polytopes  
Army  Proh 
Regiment  Proh 
Dual  Rhombipyramidal hecatonenneacontadichoron 
Conjugate  Quasiprismatorhombated hexadecachoron 
Abstract & topological properties  
Flag count  6144 
Euler characteristic  0 
Orientable  Yes 
Properties  
Symmetry  B_{4}, order 384 
Convex  Yes 
Nature  Tame 
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 edgeinscribed into a small rhombated icositetrachoron. Blending 3 prismatorhombated hexadecachora results in the small rhombic disicositetrachoron.
Gallery[edit  edit source]

B4 Coxeter plane projection

Stereographic projection

Net
Vertex coordinates[edit  edit source]
The vertices of a prismatorhombated hexadecachoron of edge length 1 are given by all permutations of:
 .
Representations[edit  edit source]
A prismatorhombated hexadecachoron has the following Coxeter diagrams:
 x4x3o3x () (full symmetry)
 xxwwxx4xxooxx3oxxxxo&#xt (B_{3} axial, truncated cubefirst)
 wx3oo3xw *b3xx&#zx (D_{4} symmetry)
 Xwx xxw4xxo3oxx&#zx (B_{3}×A_{1} symmetry)
Semiuniform variant[edit  edit source]
The prismatorhombated hexadecachoron has a semiuniform 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 .
It has coordinates given by all permutations of:
 .
Related polychora[edit  edit source]
The prismatorhombated hexadecachoron is the colonel of a 3member 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.
External links[edit  edit source]
 Bowers, Jonathan. "Category 10: Prismatorhombates" (#358).
 Bowers, Jonathan. "Tessic Isogonals".
 Klitzing, Richard. "proh".
 Quickfur. "The Runcitruncated Tesseract".
 Wikipedia contributors. "Runcitruncated tesseract".