# Great dipentary trishecatonicosachoron

(Redirected from Gidipthi)
Great dipentary trishecatonicosachoron
Rank4
TypeUniform
Notation
Bowers style acronymGidipthi
Coxeter diagramo5o3x5/3x5/2*b ()
Elements
Cells120 icosahedra, 120 small stellated dodecahedra, 120 great dodecicosidodecahedra
Faces2400 triangles, 1440 pentagrams, 720 decagrams
Edges3600+3600
Vertices1440
Vertex figurePentagonal frustum, edge lengths 1 (large base), (5–1)/2 (small base), and (5–5)/2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {4-{\sqrt {5}}}}\approx 1.32813}$
Hypervolume${\displaystyle 5{\frac {207{\sqrt {5}}-370}{2}}\approx 232.16518}$
Number of external pieces6600
Related polytopes
ArmySemi-uniform Tex, edge lengths ${\displaystyle {\sqrt {5}}-2}$ (icosahedra), ${\displaystyle {\frac {3-{\sqrt {5}}}{2}}}$ (surrounded by truncated tetrahedra)
RegimentGidipthi
ConjugateSmall dipentary trishecatonicosachoron
Abstract & topological properties
Flag count86400
Euler characteristic–1560
OrientableYes
Properties
SymmetryH4, order 14400
ConvexNo
NatureTame

The great dipentary trishecatonicosachoron, or gidipthi, is a nonconvex uniform polychoron that consists of 120 icosahedra, 120 small stellated dodecahedra, and 120 great dodecicosidodecahedra. 1 icosahedron, 1 small stellated dodecahedron, and 5 great dodecicosidodecahedra join at each vertex.

## Vertex coordinates

The vertices of a great dipentary trishecatonicosachoron of edge length 1 are given by all permutations of:

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

plus all even permutations of:

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

## Related polychora

The great dipentary trishecatonicosachoron is the colonel of a regiment that includes 81 uniform members, as well as 78 fissary uniforms, 18 normal and 59 fissary scaliforms, and 1 scaliform compound.