# Small dishexacosidishecatonicosachoron

The small dishexacosidishecatonicosachoron, or sadixady, is a nonconvex uniform polychoron that consists of 600 regular tetrahedra, 120 great stellated dodecahedra, 600 truncated tetrahedra, and 120 truncated great icosahedra. 1 tetrahedron, 1 great stellated dodecahedron, 3 truncated tetrahedra, and 3 truncated great icosahedra join at each vertex.

Small dishexacosidishecatonicosachoron
Rank4
TypeUniform
Notation
Coxeter diagramx3x3o3o5/2*a ()
Elements
Cells600 tetrahedra, 120 great stellated dodecahedra, 600 truncated tetrahedra, 120 truncated great icosahedra
Faces2400 triangles, 1440 pentagrams, 2400 hexagons
Edges3600+3600
Vertices2400
Vertex figureTriangular antipodium, edge lengths 1 (large base), (5–1)/2 (small base), and 3 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {13-3{\sqrt {5}}}{2}}}\approx 1.77367}$
Hypervolume${\displaystyle 5{\frac {322{\sqrt {5}}-365}{2}}\approx 887.53471}$
Dichoral anglesTet–3–tut: ${\displaystyle \arccos \left(-{\frac {3{\sqrt {5}}-1}{8}}\right)\approx 164.47751^{\circ }}$
Tiggy–6–tut: ${\displaystyle \arccos \left(-{\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 157.76124^{\circ }}$
Gissid–5/2–tiggy: 144°
Number of external pieces57120
Level of complexity293
Related polytopes
ArmySemi-uniform Sidpixhi, edge lengths ${\displaystyle {\frac {3-{\sqrt {5}}}{2}}}$ (dodecahedra), ${\displaystyle {\sqrt {5}}-2}$ (tetrahedra)
ConjugateGreat dishexacosidishecatonicosachoron
Abstract & topological properties
Flag count115200
Euler characteristic0
OrientableYes
Properties
SymmetryH4, order 14400
ConvexNo
NatureTame

## Vertex coordinates

The vertices of a small dishexacosidishecatonicosachoron of edge length 1 are given by all permutations of:

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

along with all even permutations of:

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

## Related polychora

The small dishexacosidishecatonicosachoron is the colonel of a regiment of 7 members. Its other members include the small hexacosifaceted hexacosidishecatonicosachoron, small hecatonicosafaceted hexacosidishecatonicosachoron, small ditrigonal hecatonicosihexacosihecatonicosachoron, hecatonicosihecatonicosihexacosichoron, small spinotrishecatonicosachoron, and small spinohecatonicosidishexacosichoron.