# Prismatorhombated hexacosichoron

Prismatorhombated hexacosichoron
Rank4
TypeUniform
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$\displaystyle \sqrt{48+21\sqrt5} \approx 9.74461$
Hypervolume$\displaystyle 25\frac{3367+1475\sqrt5}{4} \approx 41657.5017$
Dichoral anglesCo–3–trip: $\displaystyle \arccos\left(-\frac{\sqrt6+\sqrt{30}}{8}\right) \approx 172.23876^\circ$
Dip–4–trip: $\displaystyle \arccos\left(-\sqrt{\frac{10+2\sqrt5}{15}}\right) \approx 169.18768^\circ$
Co–4–dip: $\displaystyle \arccos\left(-\sqrt{\frac{5+2\sqrt5}{10}}\right) \approx 166.71747^\circ$
Tid–10–dip: 162°
Tid–3–co: $\displaystyle \arccos\left(-\frac{\sqrt{7+3\sqrt5}}{4}\right) \approx 157.76124^\circ$
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
Flag orbits16
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:

• $\displaystyle \left(0,\,\pm1,\,\pm\frac{7+3\sqrt5}{2},\,\pm\frac{7+3\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm\frac12,\,\pm\frac12,\,\pm3\frac{2+\sqrt5}{2},\,\pm\frac{8+3\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm\frac12,\,\pm\frac12,\,\pm\frac{3+2\sqrt5}{2},\,\pm\frac{9+4\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm\frac{2+\sqrt5}{2},\,\pm\frac{5+2\sqrt5}{2},\,\pm3\frac{2+\sqrt5}{2},\,\pm3\frac{2+\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm\frac{3+\sqrt5}{2},\,\pm\frac{3+\sqrt5}{2},\,\pm\frac{5+3\sqrt5}{2},\,\pm\frac{7+3\sqrt5}{2}\right)$ ,

Plus all even permutations of:

• $\displaystyle \left(0,\,\pm\frac12,\,\pm\frac{17+9\sqrt5}{4},\,\pm\frac{5+3\sqrt5}{4}\right)$ ,
• $\displaystyle \left(0,\,\pm\frac{5+\sqrt5}{4},\,\pm\frac{13+7\sqrt5}{4},\,\pm3\frac{2+\sqrt5}{2}\right)$ ,
• $\displaystyle \left(0,\,\pm\frac{2+\sqrt5}{2},\,\pm\frac{19+7\sqrt5}{4},\,\pm3\frac{3+\sqrt5}{4}\right)$ ,
• $\displaystyle \left(0,\,\pm3\frac{1+\sqrt5}{4},\,\pm\frac{17+7\sqrt5}{4},\,\pm\frac{5+2\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm\frac12,\,\pm(3+\sqrt5),\,\pm\frac{11+5\sqrt5}{4},\,\pm\frac{13+5\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm\frac12,\,\pm\frac{3+\sqrt5}{4},\,\pm\frac{15+7\sqrt5}{4},\,\pm\frac{5+3\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm\frac12,\,\pm\frac{3+\sqrt5}{4},\,\pm\frac{19+7\sqrt5}{4},\,\pm(2+\sqrt5)\right)$ ,
• $\displaystyle \left(\pm\frac12,\,\pm\frac{3+\sqrt5}{4},\,\pm\frac{17+9\sqrt5}{4},\,\pm\frac{3+\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm\frac12,\,\pm\frac{1+\sqrt5}{2},\,\pm\frac{17+7\sqrt5}{4},\,\pm\frac{9+5\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm\frac12,\,\pm\frac{2+\sqrt5}{2},\,\pm\frac{9+4\sqrt5}{2},\,\pm\frac{5+2\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm\frac12,\,\pm\frac{3+\sqrt5}{2},\,\pm\frac{13+7\sqrt5}{4},\,\pm\frac{13+5\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm\frac12,\,\pm\frac{5+3\sqrt5}{4},\,\pm\frac{15+7\sqrt5}{4},\,\pm(3+\sqrt5)\right)$ ,
• $\displaystyle \left(\pm1,\,\pm\frac{3+\sqrt5}{4},\,\pm\frac{9+4\sqrt5}{2},\,\pm\frac{7+3\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm1,\,\pm\frac{2+\sqrt5}{2},\,\pm\frac{15+7\sqrt5}{4},\,\pm\frac{11+5\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm\frac{3+\sqrt5}{4},\,\pm\frac{9+5\sqrt5}{4},\,\pm3\frac{2+\sqrt5}{2},\,\pm(3+\sqrt5)\right)$ ,
• $\displaystyle \left(\pm\frac{3+\sqrt5}{4},\,\pm\frac{1+\sqrt5}{2},\,\pm\frac{17+9\sqrt5}{4},\,\pm\frac{2+\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm\frac{3+\sqrt5}{4},\,\pm3\frac{1+\sqrt5}{4},\,\pm\frac{9+4\sqrt5}{2},\,\pm\frac{3+\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm\frac{3+\sqrt5}{4},\,\pm\frac{5+2\sqrt5}{2},\,\pm\frac{7+3\sqrt5}{2},\,\pm\frac{13+5\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm\frac{3+\sqrt5}{4},\,\pm\frac{7+3\sqrt5}{4},\,\pm\frac{8+3\sqrt5}{2},\,\pm(3+\sqrt5)\right)$ ,
• $\displaystyle \left(\pm\frac{3+\sqrt5}{4},\,\pm\frac{5+2\sqrt5}{2},\,\pm\frac{5+3\sqrt5}{2},\,\pm\frac{13+5\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm\frac{1+\sqrt5}{2},\,\pm\frac{5+\sqrt5}{4},\,\pm\frac{5+3\sqrt5}{4},\,\pm\frac{9+4\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm\frac{1+\sqrt5}{2},\,\pm3\frac{3+\sqrt5}{4},\,\pm3\frac{2+\sqrt5}{2},\,\pm\frac{13+5\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm\frac{1+\sqrt5}{2},\,\pm(2+\sqrt5),\,\pm\frac{7+3\sqrt5}{2},\,\pm(3+\sqrt5)\right)$ ,
• $\displaystyle \left(\pm\frac{5+\sqrt5}{4},\,\pm\frac{3+\sqrt5}{2},\,\pm\frac{8+3\sqrt5}{2},\,\pm\frac{11+5\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm\frac{2+\sqrt5}{2},\,\pm\frac{9+5\sqrt5}{4},\,\pm\frac{5+3\sqrt5}{2},\,\pm\frac{11+5\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm\frac{2+\sqrt5}{2},\,\pm\frac{3+\sqrt5}{2},\,\pm\frac{5+3\sqrt5}{4},\,\pm\frac{19+7\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm\frac{2+\sqrt5}{2},\,\pm\frac{3+2\sqrt5}{2},\,\pm\frac{8+3\sqrt5}{2},\,\pm\frac{5+2\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm3\frac{1+\sqrt5}{4},\,\pm(2+\sqrt5),\,\pm3\frac{2+\sqrt5}{2},\,\pm\frac{11+5\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm\frac{3+\sqrt5}{2},\,\pm\frac{5+3\sqrt5}{4},\,\pm\frac{8+3\sqrt5}{2},\,\pm\frac{9+5\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm\frac{3+\sqrt5}{2},\,\pm\frac{5+2\sqrt5}{2},\,\pm\frac{7+3\sqrt5}{4},\,\pm\frac{17+7\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm\frac{3+\sqrt5}{2},\,\pm\frac{3+2\sqrt5}{2},\,\pm\frac{15+7\sqrt5}{4},\,\pm3\frac{3+\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm\frac{5+3\sqrt5}{4},\,\pm\frac{5+2\sqrt5}{2},\,\pm(2+\sqrt5),\,\pm\frac{15+7\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm\frac{5+3\sqrt5}{4},\,\pm\frac{7+3\sqrt5}{4},\,\pm3\frac{2+\sqrt5}{2},\,\pm\frac{5+3\sqrt5}{2}\right)$ ,
• $\displaystyle \left(\pm\frac{5+3\sqrt5}{4},\,\pm\frac{3+2\sqrt5}{2},\,\pm\frac{7+3\sqrt5}{2},\,\pm\frac{9+5\sqrt5}{4}\right)$ ,
• $\displaystyle \left(\pm\frac{7+3\sqrt5}{4},\,\pm\frac{3+2\sqrt5}{2},\,\pm\frac{13+7\sqrt5}{4},\,\pm(2+\sqrt5)\right)$ .

## 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.