# Prismatorhombated hexacosichoron

Prismatorhombated hexacosichoron
Rank4
TypeUniform
SpaceSpherical
Notation
Bowers style acronymPrix
Coxeter diagramx5x3o3x ()
Elements
Cells1200 triangular prisms, 720 decagonal prisms, 600 cuboctahedra, 120 truncated dodecahedra
Faces2400+2400 triangles, 3600+3600 squares, 1440 decagons
Edges3600+7200+7200
Vertices7200
Vertex figureSkewed rectangular pyramid, base edge lengths 1, 2, 1, 2; lateral edge lengths 2, 2, (5+5)/2, (5+5)/2
Measures (edge length 1)
Circumradius${\displaystyle \sqrt{48+21\sqrt5} ≈ 9.74461}$
Hypervolume${\displaystyle 25\frac{3367+1475\sqrt5}{4} ≈ 41657.5017}$
Dichoral anglesCo–3–trip: ${\displaystyle \arccos\left(-\frac{\sqrt6+\sqrt{30}}{8}\right) ≈ 172.23876^\circ}$
Dip–4–trip: ${\displaystyle \arccos\left(-\sqrt{\frac{10+2\sqrt5}{15}}\right) ≈ 169.18768^\circ}$
Co–4–dip: ${\displaystyle \arccos\left(-\sqrt{\frac{5+2\sqrt5}{10}}\right) ≈ 166.71747^\circ}$
Tid–10–dip: 162°
Tid–3–co: ${\displaystyle \arccos\left(-\frac{\sqrt{7+3\sqrt5}}{4}\right) ≈ 157.76124^\circ}$
Central density1
Number of external pieces2640
Level of complexity16
Related polytopes
ArmyPrix
RegimentPrix
ConjugateQuasiprismatorhombated grand hexacosichoron
Abstract & topological properties
Flag count230400
Euler characteristic0
OrientableYes
Properties
SymmetryH4, order 14400
ConvexYes
NatureTame

The prismatorhombated hexacosichoron, or prix, also commonly called the runcitruncated 120-cell, is a convex uniform polychoron that consists of 1200 triangular prisms, 720 decagonal prisms, 600 cuboctahedra, and 120 truncated dodecahedra. 1 triangular prism, 2 decagonal prisms, 1 cuboctahedron, and 1 truncated dodecahedron join at each vertex. As one of its names suggests, it can be obtained by runcitruncating the hecatonicosachoron.

## Vertex coordinates

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

• ${\displaystyle \left(0,\,±1,\,±\frac{7+3\sqrt5}{2},\,±\frac{7+3\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac12,\,±3\frac{2+\sqrt5}{2},\,±\frac{8+3\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac12,\,±\frac{3+2\sqrt5}{2},\,±\frac{9+4\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac{2+\sqrt5}{2},\,±\frac{5+2\sqrt5}{2},\,±3\frac{2+\sqrt5}{2},\,±3\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{2},\,±\frac{3+\sqrt5}{2},\,±\frac{5+3\sqrt5}{2},\,±\frac{7+3\sqrt5}{2}\right),}$

Plus all even permutations of:

• ${\displaystyle \left(0,\,±\frac12,\,±\frac{17+9\sqrt5}{4},\,±\frac{5+3\sqrt5}{4}\right),}$
• ${\displaystyle \left(0,\,±\frac{5+\sqrt5}{4},\,±\frac{13+7\sqrt5}{4},\,±3\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(0,\,±\frac{2+\sqrt5}{2},\,±\frac{19+7\sqrt5}{4},\,±3\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(0,\,±3\frac{1+\sqrt5}{4},\,±\frac{17+7\sqrt5}{4},\,±\frac{5+2\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac12,\,±(3+\sqrt5),\,±\frac{11+5\sqrt5}{4},\,±\frac{13+5\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{3+\sqrt5}{4},\,±\frac{15+7\sqrt5}{4},\,±\frac{5+3\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{3+\sqrt5}{4},\,±\frac{19+7\sqrt5}{4},\,±(2+\sqrt5)\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{3+\sqrt5}{4},\,±\frac{17+9\sqrt5}{4},\,±\frac{3+\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{1+\sqrt5}{2},\,±\frac{17+7\sqrt5}{4},\,±\frac{9+5\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{2+\sqrt5}{2},\,±\frac{9+4\sqrt5}{2},\,±\frac{5+2\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{3+\sqrt5}{2},\,±\frac{13+7\sqrt5}{4},\,±\frac{13+5\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{5+3\sqrt5}{4},\,±\frac{15+7\sqrt5}{4},\,±(3+\sqrt5)\right),}$
• ${\displaystyle \left(±1,\,±\frac{3+\sqrt5}{4},\,±\frac{9+4\sqrt5}{2},\,±\frac{7+3\sqrt5}{4}\right),}$
• ${\displaystyle \left(±1,\,±\frac{2+\sqrt5}{2},\,±\frac{15+7\sqrt5}{4},\,±\frac{11+5\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,±\frac{9+5\sqrt5}{4},\,±3\frac{2+\sqrt5}{2},\,±(3+\sqrt5)\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{2},\,±\frac{17+9\sqrt5}{4},\,±\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,±3\frac{1+\sqrt5}{4},\,±\frac{9+4\sqrt5}{2},\,±\frac{3+\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,±\frac{5+2\sqrt5}{2},\,±\frac{7+3\sqrt5}{2},\,±\frac{13+5\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,±\frac{7+3\sqrt5}{4},\,±\frac{8+3\sqrt5}{2},\,±(3+\sqrt5)\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,±\frac{5+2\sqrt5}{2},\,±\frac{5+3\sqrt5}{2},\,±\frac{13+5\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{2},\,±\frac{5+\sqrt5}{4},\,±\frac{5+3\sqrt5}{4},\,±\frac{9+4\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{2},\,±3\frac{3+\sqrt5}{4},\,±3\frac{2+\sqrt5}{2},\,±\frac{13+5\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{2},\,±(2+\sqrt5),\,±\frac{7+3\sqrt5}{2},\,±(3+\sqrt5)\right),}$
• ${\displaystyle \left(±\frac{5+\sqrt5}{4},\,±\frac{3+\sqrt5}{2},\,±\frac{8+3\sqrt5}{2},\,±\frac{11+5\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{2+\sqrt5}{2},\,±\frac{9+5\sqrt5}{4},\,±\frac{5+3\sqrt5}{2},\,±\frac{11+5\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{2+\sqrt5}{2},\,±\frac{3+\sqrt5}{2},\,±\frac{5+3\sqrt5}{4},\,±\frac{19+7\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{2+\sqrt5}{2},\,±\frac{3+2\sqrt5}{2},\,±\frac{8+3\sqrt5}{2},\,±\frac{5+2\sqrt5}{2}\right),}$
• ${\displaystyle \left(±3\frac{1+\sqrt5}{4},\,±(2+\sqrt5),\,±3\frac{2+\sqrt5}{2},\,±\frac{11+5\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{2},\,±\frac{5+3\sqrt5}{4},\,±\frac{8+3\sqrt5}{2},\,±\frac{9+5\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{2},\,±\frac{5+2\sqrt5}{2},\,±\frac{7+3\sqrt5}{4},\,±\frac{17+7\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{2},\,±\frac{3+2\sqrt5}{2},\,±\frac{15+7\sqrt5}{4},\,±3\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{5+3\sqrt5}{4},\,±\frac{5+2\sqrt5}{2},\,±(2+\sqrt5),\,±\frac{15+7\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{5+3\sqrt5}{4},\,±\frac{7+3\sqrt5}{4},\,±3\frac{2+\sqrt5}{2},\,±\frac{5+3\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac{5+3\sqrt5}{4},\,±\frac{3+2\sqrt5}{2},\,±\frac{7+3\sqrt5}{2},\,±\frac{9+5\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{7+3\sqrt5}{4},\,±\frac{3+2\sqrt5}{2},\,±\frac{13+7\sqrt5}{4},\,±(2+\sqrt5)\right).}$

## Semi-uniform variant

The prismatorhombated hexacosichoron has a semi-uniform variant of the form x5y3o3z that maintains its full symmetry. This variant uses 120 truncated dodecahedra of form x5y3o, 600 rhombitetratetrahedra of form y3o3z, 1200 triangular prisms of form x z3o, and 720 dipentagonal prisms of form z x5y as cells, with 3 edge lengths.

With edges of length a, b, and c (such that it forms a5b3o3c), its circumradius is given by ${\displaystyle \sqrt{\frac{14a^2+21b^2+3c^2+33ab+11ac+14bc+(6a^2+9b^2+c^2+15ab+5ac+6bc)\sqrt5}{2}}}$.

## Related polychora

The prismatorhombated hexacosichoron is the colonel of a 3-member regiment that also includes the small prismatohexacosihecatonicosihecatonicosachoron and the small rhombiprismic hexacosihecatonicosachoron.

The segmentochoron truncated dodecahedron atop great rhombicosidodecahedron can be obtained as a cap of the prismatorhombated hexacosichoron.

o5o3o3o truncations
Name OBSA CD diagram Picture
Hecatonicosachoron hi x5o3o3o
Truncated hecatonicosachoron thi x5x3o3o
Rectified hecatonicosachoron rahi o5x3o3o
Hexacosihecatonicosachoron xhi o5x3x3o
Rectified hexacosichoron rox o5o3x3o
Truncated hexacosichoron tex o5o3x3x
Hexacosichoron ex o5o3o3x
Small rhombated hecatonicosachoron srahi x5o3x3o
Great rhombated hecatonicosachoron grahi x5x3x3o
Small rhombated hexacosichoron srix o5x3o3x
Great rhombated hexacosichoron grix o5x3x3x
Small disprismatohexacosihecatonicosachoron sidpixhi x5o3o3x
Prismatorhombated hexacosichoron prix x5x3o3x
Prismatorhombated hecatonicosachoron prahi x5o3x3x
Great disprismatohexacosihecatonicosachoron gidpixhi x5x3x3x