# Rectified great faceted hexacosichoron

(Redirected from Rigfix)
Rectified great faceted hexacosichoron
Rank4
TypeUniform
Notation
Bowers style acronymRigfix
Coxeter diagramo5o5/2x3o ()
Elements
Cells
Faces
Edges3600
Vertices720
Vertex figureSemi-uniform pentagonal prism, edge lengths (5–1)/2 (base) and 1 (side)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {5-2{\sqrt {5}}}}\approx 0.72654}$
Hypervolume${\displaystyle 5{\frac {545-237{\sqrt {5}}}{4}}\approx 18.81486}$
Dichoral angleGid–3–gid: 120°
Sissid–5/2–gid: 108°
Central density76
Number of external pieces5040
Level of complexity24
Related polytopes
ArmyRox, edge length ${\displaystyle {\sqrt {5}}-2}$
RegimentRigfix
ConjugateRectified faceted hexacosichoron
Convex coreHecatonicosachoron
Abstract & topological properties
Flag count43200
Euler characteristic–480
OrientableYes
Properties
SymmetryH4, order 14400
Flag orbits3
ConvexNo
NatureTame

The rectified great faceted hexacosichoron, or rigfix, is a nonconvex uniform polychoron that consists of 120 small stellated dodecahedra and 120 great icosidodecahedra. Two small stellated dodecahedra and five great icosidodecahedra join at each pentagonal prismatic vertex. As the name suggests, it can be obtained by rectifying the great faceted hexacosichoron.

## Vertex coordinates

The vertices of a rectified great faceted hexacosichoron of edge length 1 are given by all permutations of:

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

along with even permutations of:

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

## Related polychora

The rectified great faceted hexacosichoron is the colonel of a regiment with 15 members. Of these, one other besideds the colonel itself is Wythoffian (the rectified grand hexacosichoron), two are hemi-Wythoffian (the great pentagrammal antiprismatoverted dishecatonicosachoron and quasiprismatohecatonicosachoron), and one is noble (the grand retropental hecatonicosachoron).

Uniform polychoron compounds composed of rectified great faceted hexacosichora include: