# Dishecatonicosachoron

Dishecatonicosachoron
Rank4
TypeUniform
Notation
Bowers style acronymDahi
Coxeter diagramo5/2x3x5o ()
Elements
Cells
Faces
Edges3600+3600
Vertices3600
Vertex figureDigonal disphenoid, edge lengths (1+5)/2 (base 1), (5–1)/2 (base 2) and 3 (sides)
Measures (edge length 1)
Hypervolume${\displaystyle {\frac {6825}{2}}=3412.5}$
Dichoral angleTiggy–5/2–tiggy: 144°
Tiggy–6–ti: 120°
Ti–5–ti: 72°
Central density20
Number of external pieces21720
Level of complexity75
Related polytopes
ArmySemi-uniform Srahi, edge lengths ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$ (pentagons), ${\displaystyle {\sqrt {5}}-2}$ (octahedra)
RegimentDahi
ConjugateDishecatonicosachoron
Convex coreHecatonicosachoron
Abstract & topological properties
Flag count86400
Euler characteristic0
OrientableYes
Properties
SymmetryH4, order 14400
Flag orbits6
ConvexNo
NatureTame

The dishecatonicosachoron, or dahi, is a nonconvex uniform polychoron that consists of 120 truncated icosahedra and 120 truncated great icosahedra. 2 of each join at each vertex.

It is the medial stage of the truncation series between a grand hecatonicosachoron and its dual great stellated hecatonicosachoron, which makes it the bitruncation of both of these polychora.

## Vertex coordinates

Coordinates for the vertices of a dishecatonicosachoron of edge length 1 are given by all permutations of:

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

together with all even permutations of:

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