# Small disprismatohexacosihecatonicosachoron

The small disprismatohexacosihecatonicosachoron, or sidpixhi, also commonly called the runcinated 120-cell, is a convex uniform polychoron that consists of 600 regular tetrahedra, 120 regular dodecahedra, 1200 triangular prisms, and 720 pentagonal prisms. 1 tetrahedron, 1 dodecahedron, 3 triangular prisms, and 3 pentagonal prisms join at each vertex. It is the result of expanding the cells of either a hecatonicosachoron or a hexacosichoron outwards, and thus could also be called the runcinated 600-cell.

Small disprismatohexacosihecatonicosachoron
Rank4
TypeUniform
Notation
Bowers style acronymSidpixhi
Coxeter diagramx5o3o3x ()
Elements
Cells600 tetrahedra, 1200 triangular prisms, 720 pentagonal prisms, 120 dodecahedra
Faces2400 triangles, 3600 squares, 1440 pentagons
Edges3600+3600
Vertices2400
Vertex figureTriangular antipodium, edge lengths 1 (small base), (1+5)/2 (large base), and 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle 3+{\sqrt {5}}\approx 5.23607}$
Hypervolume${\displaystyle 25{\frac {281+126{\sqrt {5}}}{4}}\approx 3517.15353}$
Dichoral anglesTet–3–trip: ${\displaystyle \arccos \left(-{\frac {{\sqrt {6}}+{\sqrt {30}}}{8}}\right)\approx 172.23876^{\circ }}$
Pip–4–trip: ${\displaystyle \arccos \left(-{\sqrt {\frac {10+2{\sqrt {5}}}{15}}}\right)\approx 169.18768^{\circ }}$
Doe–5–pip: 162°
Central density1
Number of external pieces2640
Level of complexity8
Related polytopes
ArmySidpixhi
RegimentSidpixhi
DualTriangular-antitegmatic dischiliatetracosichoron
ConjugateQuasidisprismatohexacosihecatonicosachoron
Abstract & topological properties
Flag count115200
Euler characteristic0
OrientableYes
Properties
SymmetryH4, order 14400
Flag orbits8
ConvexYes
NatureTame

## Vertex coordinates

The vertices of a small disprismatohexacosihecatonicosachoron of edge length 1 are all permutations of:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {5+2{\sqrt {5}}}{2}}\right)}$ ,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {9+5{\sqrt {5}}}{4}}\right)}$ ,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right)}$ ,
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}}\right)}$ ,
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}}\right)}$ ,
• ${\displaystyle \left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}}\right)}$ ,

along with the even permutations of:

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

## Semi-uniform variant

The small disprismatohexacosihecatonicosachoron has a semi-uniform variant of the form x5o3o3y that maintains its full symmetry. This variant uses 120 dodecahedra of size x, 600 tetrahedra of size y, 1200 semi-uniform triangular prisms of form x y3o, and 720 semi-uniform pentagonal prisms of form y x5o as cells, with 2 edge lengths.

With edges of length a (of dodecahedra) and b (of tetrahedra), its circumradius is given by ${\displaystyle {\sqrt {\frac {14a^{2}+3b^{2}+11ab+(6a^{2}+b^{2}+5ab){\sqrt {5}}}{2}}}}$ .

## Related polychora

The small disprismatohexacosihecatonicosachoron is the colonel of a 7-member regiment.

Sidpixhi regiment

Index Name OBSA Company Nature
454 Small disprismatohexacosihecatonicosachoron Sidpixhi Sidpixhi Tame
455 Small hecatonicosafaceted prismatohecatonicosihexacosichoron Shif Phix Sidpixhi Tame
456 Small hexacosifaceted prismatodishecatonicosachoron Six Fipady Sidpixhi Tame
457 Small hexacosihecatonicosihecatonicosachoron Sixhihy Sixhihy Tame
458 Prismatoprismatohecatonicosachoron Paphi Wild
459 Small spinoprismatohexacosihecatonicosachoron Snapaxhi Wild