# Prismatorhombated tesseract

Prismatorhombated tesseract
Rank4
TypeUniform
Notation
Bowers style acronymPrit
Coxeter diagramx4o3x3x ()
Elements
Cells24 cubes, 32 hexagonal prisms, 16 truncated tetrahedra, 8 small rhombicuboctahedra
Faces64 triangles, 48+96+96 squares, 64 hexagons
Edges96+192+192
Vertices192
Vertex figureIsosceles trapezoidal pyramid, base edge lengths 1, 2, 2, 2; lateral edge lengths 2, 2, 3, 3
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {7+3{\sqrt {2}}}{2}}}\approx 2.37093}$
Hypervolume${\displaystyle 5{\frac {67+48{\sqrt {2}}}{6}}\approx 112.40188}$
Dichoral anglesTut–6–hip: 150°
Hip–4–cube: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
Sirco–4–cube: 135°
Sirco–4–hip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Sirco–3–tut: 120°
Central density1
Number of external pieces80
Level of complexity16
Related polytopes
ArmyPrit
RegimentPrit
ConjugatePrismatoquasirhombated tesseract
Abstract & topological properties
Flag count6144
Euler characteristic0
OrientableYes
Properties
SymmetryB4, order 384
Flag orbits16
ConvexYes
NatureTame

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.

## Vertex coordinates

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

• ${\displaystyle \left(\pm {\frac {1+2{\sqrt {2}}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right)}$.

## Representations

A prismatorhombated tesseract has the following Coxeter diagrams:

• x4o3x3x () (full symmetry)
• xxxxxx4ooxxoo3xuxxux&#xt (B3 axial, small rhombicuboctahedron-first)
• qo3xx3oq *b3xx&#zx (D4 symmetry)
• Xwx xxx4oox3xux&#zx (B3×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 ${\displaystyle {\sqrt {\frac {2a^{2}+2b^{2}+c^{2}+2bc+(2ab+ac){\sqrt {2}}}{2}}}}$.

It has coordinates given by all permutations of:

• ${\displaystyle \left(\pm {\frac {a+(b+c){\sqrt {2}}}{2}},\,\pm {\frac {a+b{\sqrt {2}}}{2}},\,\pm {\frac {a}{2}},\,\pm {\frac {a}{2}}\right)}$.

## 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.

Uniform polychoron compounds composed of prismatorhombated tesseracts include: