Prismatorhombated hecatonicosachoron

Prismatorhombated hecatonicosachoron
Rank4
TypeUniform
Notation
Bowers style acronymPrahi
Coxeter diagramx5o3x3x ()
Elements
Cells720 pentagonal prisms, 1200 hexagonal prisms, 600 truncated tetrahedra, 120 small rhombicosidodecahedra
Faces2400 triangles, 3600+3600 squares, 1440 pentagons, 2400 hexagons
Edges3600+7200+7200
Vertices7200
Vertex figureIsosceles trapezoidal pyramid, base edge lengths 1, 2, (1+5)/2, 2; lateral edge lengths 2, 2, 3, 3
Measures (edge length 1)
Circumradius${\displaystyle {\frac {5{\sqrt {2}}+3{\sqrt {10}}}{2}}\approx 8.27895}$
Hypervolume${\displaystyle 25{\frac {1814+779{\sqrt {5}}}{4}}\approx 22224.3560}$
Dichoral anglesTut–6–hip: ${\displaystyle \arccos \left(-{\frac {{\sqrt {6}}+{\sqrt {30}}}{8}}\right)\approx 172.23876^{\circ }}$
Pip–4–hip: ${\displaystyle \arccos \left(-{\sqrt {\frac {10+2{\sqrt {5}}}{15}}}\right)\approx 169.18768^{\circ }}$
Srid–5–pip: 162°
Srid–4–hip: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}$
Srid–3–tut: ${\displaystyle \arccos \left(-{\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 157.76124^{\circ }}$
Central density1
Number of external pieces2640
Level of complexity16
Related polytopes
ArmyPrahi
RegimentPrahi
ConjugatePrismatoquasirhombated great grand stellated hecatonicosachoron
Abstract & topological properties
Flag count230400
Euler characteristic0
OrientableYes
Properties
SymmetryH4, order 14400
ConvexYes
NatureTame

The prismatorhombated hecatonicosachoron, or prahi, also commonly called the runcitruncated 600-cell, is a convex uniform polychoron that consists of 720 pentagonal prisms, 1200 hexagonal prisms, 600 truncated tetrahedra, and 120 small rhombicosidodecahedra. 1 pentagonal prism, 2 hexagonal prisms, 1 truncated tetrahedron, and 1 small rhombicosidodecahedron join at each vertex. As one of its names suggests, it can be obtained by runcitruncating the hexacosichoron.

Vertex coordinates

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

• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {7+4{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm {\frac {8+3{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {7+{\sqrt {5}}}{4}},\,\pm {\frac {11+5{\sqrt {5}}}{4}},\,\pm {\frac {11+5{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {9+5{\sqrt {5}}}{4}},\,\pm {\frac {13+5{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {9+5{\sqrt {5}}}{4}},\,\pm {\frac {9+5{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {7+5{\sqrt {5}}}{4}},\,\pm {\frac {11+5{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(0,\,\pm {\frac {1}{2}},\,\pm {\frac {9+5{\sqrt {5}}}{4}},\,\pm 5{\frac {3+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(0,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {7+4{\sqrt {5}}}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(0,\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {15+7{\sqrt {5}}}{4}},\,\pm {\frac {7+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(0,\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{2}},\,\pm (3+{\sqrt {5}})\right)}$,
• ${\displaystyle \left(0,\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm 3{\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {11+3{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{2}},\,\pm {\frac {7+5{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm 1,\,\pm {\frac {11+5{\sqrt {5}}}{4}},\,\pm {\frac {13+5{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm 1,\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {15+7{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{2}},\,\pm {\frac {11+5{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {8+3{\sqrt {5}}}{2}},\,\pm {\frac {4+{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{2}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {9+5{\sqrt {5}}}{4}},\,\pm (3+{\sqrt {5}})\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm (2+{\sqrt {5}}),\,\pm {\frac {11+5{\sqrt {5}}}{4}},\,\pm {\frac {11+3{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {7+4{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {15+7{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {4+{\sqrt {5}}}{2}},\,\pm {\frac {11+5{\sqrt {5}}}{4}},\,\pm (3+{\sqrt {5}})\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {13+5{\sqrt {5}}}{4}},\,\pm {\frac {11+3{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm 1,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm 3{\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {9+5{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm 1,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {8+3{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm 1,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{2}},\,\pm (2+{\sqrt {5}})\right)}$,
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {8+3{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm 5{\frac {3+{\sqrt {5}}}{4}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {7+5{\sqrt {5}}}{4}},\,\pm {\frac {11+5{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {8+3{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {4+{\sqrt {5}}}{2}},\,\pm {\frac {9+5{\sqrt {5}}}{4}},\,\pm {\frac {11+5{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm {\frac {13+5{\sqrt {5}}}{4}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm 5{\frac {3+{\sqrt {5}}}{4}},\,\pm (2+{\sqrt {5}})\right)}$,
• ${\displaystyle \left(\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{2}},\,\pm {\frac {4+{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {7+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {7+5{\sqrt {5}}}{4}},\,\pm {\frac {13+5{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {4+{\sqrt {5}}}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm 3{\frac {2+{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm (2+{\sqrt {5}}),\,\pm {\frac {7+5{\sqrt {5}}}{4}},\,\pm {\frac {9+5{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm {\frac {11+5{\sqrt {5}}}{4}},\,\pm (2+{\sqrt {5}})\right)}$,
• ${\displaystyle \left(\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{2}}\right)}$.

Semi-uniform variant

The prismatorhombated hecatonicosachoron has a semi-uniform variant of the form x5o3y3z that maintains its full symmetry. This variant uses 120 small rhombicosidodecahedra of form x5o3y, 600 truncated tetrahedra of form z3y3o, 720 pentagonal prisms of form z x5o, and 1200 ditrigonal prisms of form x y3z as cells, with 3 edge lengths.

With edges of length a, b, and c (such that it forms a5o3b3c), its circumradius is given by ${\displaystyle {\sqrt {\frac {14a^{2}+10b^{2}+3c^{2}+22ab+11ac+10bc+(6a^{2}+4b^{2}+c^{2}+10ab+5ac+4bc){\sqrt {5}}}{2}}}}$.

Related polychora

The prismatorhombated hecatonicosachoron is the colonel of a 3-member regiment that also includes the small prismatohexacosidishecatonicosachoron and the small rhombiprismic dishecatonicosachoron.