Prismatorhombated hexadecachoron

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Prismatorhombated hexadecachoron
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
Hypervolume
Dichoral anglesCo–3–trip: 150°
 Op–4–trip:
 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
DualRhombipyramidal hecatonenneacontadichoron
ConjugateQuasiprismatorhombated hexadecachoron
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.

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 (B3 axial, truncated cube-first)
  • wx3oo3xw *b3xx&#zx (D4 symmetry)
  • Xwx xxw4xxo3oxx&#zx (B3×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.

External links[edit | edit source]