Prismatorhombated hexadecachoron

Prismatorhombated hexadecachoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Proh
Coxeter diagram
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	$\sqrt{4+2\sqrt2} ≈ 2.61313$
Hypervolume	$8\frac{27+20\sqrt2}{3} ≈ 147.42472$
Dichoral angles	Co–3–trip: 150°
	Op–4–trip: $\arccos\left(-\frac{\sqrt6}{3}\right) ≈ 144.73561°$
	Co–4–op: 135°
	Tic–8–op: 135°
	Tic–3–co: 120°
Central density	1
Number of pieces	80
Level of complexity	16
Related polytopes
Army	Proh
Regiment	Proh
Dual	Rhombipyramidal hecatonenneacontadichoron
Conjugate	Quasiprismatorhombated hexadecachoron
Abstract properties
Euler characteristic	0
Topological properties
Orientable	Yes
Properties
Symmetry	B4, 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 edge-inscribed into a small rhombated icositetrachoron. Blending 3 prismatorhombated hexadecachora results in the small rhombic disicositetrachoron.

Gallery

• 

  B4 Coxeter plane projection

• 

  Stereographic projection

• 

  Net

• 

  Cross-sections

Vertex coordinates

The vertices of a prismatorhombated hexadecachoron of edge length 1 are given by all permutations of:

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

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

It has coordinates given by all permutations of:

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

Related polychora

The 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

• Bowers, Jonathan. "Category 10: Prismatorhombates" (#358).

• Bowers, Jonathan. "Tessic Isogonals".

• Klitzing, Richard. "proh".

• Quickfur. "The Runcitruncated Tesseract".

• Wikipedia Contributors. "Runcitruncated tesseract".