|Bowers style acronym||Proh|
|Cells||32 triangular prisms, 24 octagonal prisms, 16 cuboctahedra, 8 truncated cubes|
|Faces||64+64 triangles, 96+96 squares, 48 octagons|
|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)|
|Dichoral angles||Co–3–trip: 150°|
|Number of pieces||80|
|Level of complexity||16|
|Symmetry||B4, order 384|
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.
Gallery[edit | edit source]
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 (BC3 axial, truncated cube-first)
- wx3oo3xw *b3xx&#zx (D4 symmetry)
- Xwx xxw4xxo3oxx&#zx (BC3×A1 symmetry)
Semi-uniform variant[edit | edit source]
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 .
It has coordinates given by all permutations of:
Related polychora[edit | edit source]
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.
|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|
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".