# Rectified hecatonicosachoron

Rectified hecatonicosachoron
Rank4
TypeUniform
Notation
Bowers style acronymRahi
Coxeter diagramo5x3o3o ()
Elements
Cells600 tetrahedra, 120 icosidodecahedra
Faces2400 triangles, 720 pentagons
Edges3600
Vertices1200
Vertex figureSemi-uniform triangular prism, edge lengths 1 (base) and (1+5)/2 (side)
Edge figuretet 3 id 5 id 3
Measures (edge length 1)
Circumradius${\displaystyle {\frac {3{\sqrt {3}}+{\sqrt {15}}}{2}}\approx 4.53457}$
Hypervolume${\displaystyle 5{\frac {730+331{\sqrt {5}}}{4}}\approx 1837.67313}$
Dichoral anglesId–3–tet: ${\displaystyle \arccos \left(-{\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 157.76124^{\circ }}$
Id–5–id: 144°
Central density1
Number of external pieces720
Level of complexity3
Related polytopes
ArmyRahi
RegimentRahi
DualJoined hexacosichoron
ConjugateRectified great grand stellated hecatonicosachoron
Abstract & topological properties
Flag count43200
Euler characteristic0
OrientableYes
Properties
SymmetryH4, order 14400
Flag orbits3
ConvexYes
NatureTame

The rectified hecatonicosachoron, or rahi, also commonly called the rectified 120-cell, is a convex uniform polychoron that consists of 600 regular tetrahedra and 120 icosidodecahedra. Two tetrahedra and three icosidodecahedra join at each triangular prismatic vertex. As the name suggests, it can be obtained by rectifying the hecatonicosachoron.

## Vertex coordinates

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

• ${\displaystyle \left(0,\,0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm (2+{\sqrt {5}})\right)}$,
• ${\displaystyle \left(0,\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}}\right)}$,

along with all even permutations of:

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

## Representations

A rectified hecatonicosachoron has the following Coxeter diagrams:

• o5x3o3o (full symmetry)
• ofxoxooxFf(oV)fFxooxoxfo5xoxfofFxoo(xo)ooxFfofxox3oooxFfofxF(Vo)FxfofFxooo&#xt (H3 axial, icosidodecahedron-first)

## Related polychora

The rectified hecatonicosachoron is the colonel of a 3-member regiment that also includes the facetorectified hecatonicosachoron and small hecatonicosintercepted hecatonicosachoron.

### Isogonal derivatives

Substitution by vertices of these following elements will produce these convex isogonal polychora: