# Small ditetrahedronary hexacosihecatonicosachoron

Small ditetrahedronary hexacosihecatonicosachoron
Rank4
TypeUniform
Notation
Bowers style acronymSidtaxhi
Coxeter diagramo3o3x5/2o3*b ()
Elements
Cells600 tetrahedra, 120 small ditrigonary icosidodecahedra
Faces2400 triangles, 720 pentagrams
Edges3600
Vertices600
Vertex figureSemi-uniform Truncated tetrahedron, edge lengths 1 (triangle edges) and (5–1)/2 (other edges)
Edge figure(tet 3 sidtid 5/2 sidtid 3)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {{\sqrt {2}}+{\sqrt {10}}}{2}}\approx 2.28825}$
Hypervolume${\displaystyle 5{\frac {100+37{\sqrt {5}}}{4}}\approx 228.41814}$
Dichoral anglesSidtid–3–tet: ${\displaystyle \arccos \left(-{\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 157.76124^{\circ }}$
Sidtid–5/2–sidtid: 144°
Central density2
Number of external pieces2520
Level of complexity9
Related polytopes
ArmyHi, edge length ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$
RegimentSidtaxhi
ConjugateGrand ditetrahedronary hexacosihecatonicosachoron
Abstract & topological properties
Flag count43200
Euler characteristic—600
OrientableYes
Properties
SymmetryH4, order 14400
ConvexNo
NatureTame

The small ditetrahedronary hexacosihecatonicosachoron, or sidtaxhi, is a nonconvex uniform polychoron that consists of 600 regular tetrahedra and 120 small ditrigonary icosidodecahedra. 4 small ditrigonary icosidodecahedra and 4 tetrahedra join at each vertex, with a variant of the truncated tetrahedron as the vertex figure.

The small ditetrahedronary hexacosihecatonicosachoron contains the vertices of a small rhombicosidodecahedral prism and decagonal duoprism.

It can be formed as a holosnub hecatonicosachoron.

## Vertex coordinates

The vertices of a small ditetrahedronary hexacosihecatonicosachoron of edge length 1, centered at the origin, are given by all permutations of:

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

together with all the even permutations of:

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

## Related polychora

The small ditetrahedronary hexacosihecatonicosachoron is the colonel of a regiment with 37 members, plus 5 fissaries and three compounds (two are subsymmetric), as well as 11 scaliform members plus 54 scaliform fissaries and 2 scaliform compounds.