Prismatorhombated hexadecachoron

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Prismatorhombated hexadecachoron
Schlegel half-solid runcitruncated 8-cell.png
Bowers style acronymProh
Coxeter diagramCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Cells32 triangular prisms, 24 octagonal prisms, 16 cuboctahedra, 8 truncated cubes
Faces64+64 triangles, 96+96 squares, 48 octagons
Vertex figureSkewed rectangular pyramid, base edge lengths 1, 2, 1, 2; lateral edge lengths 2, 2, 2+2, 2+2 Runcitruncated 8-cell verf.png
Measures (edge length 1)
Dichoral anglesCo–3–trip: 150°
 Co–4–op: 135°
 Tic–8–op: 135°
 Tic–3–co: 120°
Central density1
Number of pieces80
Level of complexity16
Related polytopes
DualRhombipyramidal hecatonenneacontadichoron
ConjugateQuasiprismatorhombated hexadecachoron
Abstract properties
Euler characteristic0
Topological properties
SymmetryB4, 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.

o4o3o3o truncations
Name OBSA CD diagram Picture
Tesseract tes x4o3o3o
Schlegel wireframe 8-cell.png
Truncated tesseract tat x4x3o3o
Schlegel half-solid truncated tesseract.png
Rectified tesseract rit o4x3o3o
Schlegel half-solid rectified 8-cell.png
Tesseractihexadecachoron tah o4x3x3o
Schlegel half-solid bitruncated 8-cell.png
Rectified hexadecachoron = Icositetrachoron ico o4o3x3o
Schlegel half-solid rectified 16-cell.png
Truncated hexadecachoron thex o4o3x3x
Schlegel half-solid truncated 16-cell.png
Hexadecachoron hex o4o3o3x
Schlegel wireframe 16-cell.png
Small rhombated tesseract srit x4o3x3o
Schlegel half-solid cantellated 8-cell.png
Great rhombated tesseract grit x4x3x3o
Schlegel half-solid cantitruncated 8-cell.png
Small rhombated hexadecachoron = Rectified icositetrachoron rico o4x3o3x
Schlegel half-solid cantellated 16-cell.png
Great rhombated hexadecachoron = Truncated icositetrachoron tico o4x3x3x
Schlegel half-solid cantitruncated 16-cell.png
Small disprismatotesseractihexadecachoron sidpith x4o3o3x
Schlegel half-solid runcinated 8-cell.png
Prismatorhombated hexadecachoron proh x4x3o3x
Schlegel half-solid runcitruncated 8-cell.png
Prismatorhombated tesseract prit x4o3x3x
Schlegel half-solid runcitruncated 16-cell.png
Great disprismatotesseractihexadecachoron gidpith x4x3x3x
Schlegel half-solid omnitruncated 8-cell.png

External links[edit | edit source]