Prismatorhombated tesseract

Prismatorhombated tesseract
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Prit
Coxeter diagram
Elements
Cells	24 cubes, 32 hexagonal prisms, 16 truncated tetrahedra, 8 small rhombicuboctahedra
Faces	64 triangles, 48+96+96 squares, 64 hexagons
Edges	96+192+192
Vertices	192
Vertex figure	Isosceles trapezoidal pyramid, base edge lengths 1, √2, √2, √2; lateral edge lengths √2, √2, √3, √3
Measures (edge length 1)
Circumradius	$\sqrt{\frac{7+3\sqrt2}{2}} ≈ 2.37093$
Hypervolume	$5\frac{67+48\sqrt2}{6} ≈ 112.40188$
Dichoral angles	Tut–6–hip: 150°
	Hip–4–cube: $\arccos\left(-\frac{\sqrt6}{3}\right) ≈ 144.73561°$
	Sirco–4–cube: 135°
	Sirco–4–hip: $\arccos\left(-\frac{\sqrt3}{3}\right) ≈ 125.26439°$
	Sirco–3–tut: 120°
Central density	1
Number of pieces	80
Level of complexity	16
Related polytopes
Army	Prit
Regiment	Prit
Dual	Deltopyramidal hecatonenneacontadichoron
Conjugate	Prismatoquasirhombated tesseract
Abstract properties
Euler characteristic	0
Topological properties
Orientable	Yes
Properties
Symmetry	B4, order 384
Convex	Yes
Nature	Tame

The prismatorhombated tesseract, or prit, also commonly called the runcitruncated 16-cell, is a convex uniform polychoron that consists of 24 cubes, 32 hexagonal prisms, 16 truncated tetrahedra, and 8 small rhombicuboctahedra. 1 cube, 2 hexagonal prisms, 1 truncated tetrahedron, and 1 small rhombicuboctahedron join at each vertex. As one of its names suggests, it can be obtained by runcitruncating the hexadecachoron.

Gallery

• 

  B4 Coxeter plane projection

• 

  Net

• 

  Cross-sections

Vertex coordinates

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

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

Representations

A prismatorhombated tesseract has the following Coxeter diagrams:

• x4o3x3x (full symmetry)
• xxxxxx4ooxxoo3xuxxux&#xt (BC₃ axial, small rhombicuboctahedron-first)
• qo3xx3oq *b3xx&#zx (D₄ symmetry)
• Xwx xxx4oox3xux&#zx (BC₃×A₁ symmetry)

Semi-uniform variant

The prismatorhombated tesseract has a semi-uniform variant of the form x4o3y3z that maintains its full symmetry. This variant uses 8 small rhombicuboctahedra of form x4o3y, 16 truncated tetrahedra of form z3y3o, 24 square prisms of form z x4o, and 32 ditrigonal prisms of form x y3z as cells, with 3 edge lengths.

With edges of length a, b, and c (such that it forms a4o3b3c), its circumradius is given by $\sqrt{\frac{2a^2+2b^2+c^2+2bc+(2ab+ac)\sqrt2}{2} }$.

It has coordinates given by all permutations of:

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

Related polychora

The prismatorhombated tesseract is the colonel of a 3-member regiment that also includes the small cubihexadecadisoctachoron and the small rhombiprismic disoctachoron.

The great rhombicuboctahedral prism can be obtained as the central segment of the prismatorhombated tesseract in small rhombicuboctahedron-first orientation.

External links

• Bowers, Jonathan. "Category 10: Prismatorhombates" (#355).

• Bowers, Jonathan. "Tessic Isogonals".

• Klitzing, Richard. "prit".

• Quickfur. "The Runcitruncated 16-cell".

• Wikipedia Contributors. "Runcitruncated 16-cell".