# Small ditetrahedronary hexacosihecatonicosachoron

Small ditetrahedronary hexacosihecatonicosachoron
Rank4
TypeUniform
SpaceSpherical
Notation
Bowers style acronymSidtaxhi
Coxeter diagram
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${\displaystyle \frac{\sqrt2+\sqrt{10}}{2} ≈ 2.28825}$
Hypervolume${\displaystyle 5\frac{100+37\sqrt5}{4} ≈ 228.41814}$
Dichoral anglesSidtid–3–tet: ${\displaystyle \arccos\left(-\frac{\sqrt{7+3\sqrt5}}{4}\right) ≈ 157.76124^\circ}$
Sidtid–5/2–sidtid: 144°
Central density2
Number of external pieces2520
Level of complexity9
Related polytopes
ArmyHi
RegimentSidtaxhi
ConjugateGrand ditetrahedronary hexacosihecatonicosachoron
Abstract & topological properties
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:

• ${\displaystyle \left(±\frac{1+\sqrt5}{2},\,±\frac{1+\sqrt5}{2},\,0,\,0\right),}$
• ${\displaystyle \left(±\frac{5+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,±\frac{3+\sqrt5}{4},\,±\frac{3+\sqrt5}{4},\,±\frac{\sqrt5-1}{4}\right),}$
• ${\displaystyle \left(±\frac{2+\sqrt5}{2},\,±\frac12,\,±\frac12,\,±\frac12\right),}$

together with all the even permutations of:

• ${\displaystyle \left(±\frac{2+\sqrt5}{2},\,±\frac{1+\sqrt5}{4},\,±\frac{\sqrt5-1}{4},\,0\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,±\frac{5+\sqrt5}{4},\,0,\,±\frac12\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{2},\,±\frac12\right).}$

## 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.