# Small ditrigonary icosidodecahedral antiprism

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Small ditrigonary icosidodecahedral antiprism
Rank4
TypeUniform
SpaceSpherical
Notation
Bowers style acronymSidtidap
Coxeter diagramβ2β5o3o ()
Elements
Cells40 tetrahedra, 12 pentagrammic antiprisms, 2 small ditrigonary icosidodecahedra
Faces40+120 triangles, 24 pentagrams
Edges60+120
Vertices40
Vertex figureTriangular cupola, edge lengths (5–1)/2 (3 base edges), 1 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle \sqrt{\frac{5+\sqrt5}{8}} ≈ 0.95106}$
Hypervolume${\displaystyle \frac{\sqrt{110+85\sqrt5}}{8} ≈ 2.16530}$
Dichoral anglesTet–3–stap: ${\displaystyle \arccos\left(-\sqrt{\frac{5+3\sqrt5}{20}}\right) ≈ 139.91776^\circ}$
Sidtid–3–tet: ${\displaystyle \arccos\left(-\frac{\sqrt{7-3\sqrt5}}{2}\right) ≈ 105.66967^\circ}$
Sidtid–5/2–stap: 90°
Height${\displaystyle \sqrt{\frac{\sqrt5-1}{2}} ≈ 0.78615}$
Number of external pieces174
Level of complexity32
Related polytopes
ArmySemi-uniform Dope
RegimentSidtidap
DualSmall triambic icosahedral antitegum
ConjugateNone
Abstract & topological properties
Euler characteristic–10
OrientableYes
Properties
SymmetryH3×A1, order 240
ConvexNo
NatureTame

The small ditrigonary icosidodecahedral antiprism or sidtidap, is a nonconvex uniform polychoron that consists of 2 small ditrigonary icosidodecahedra, 12 pentagrammic antiprisms, and 40 tetrahedra. Each vertex joins 1 small ditrigonary icosidodecahedron, 3 pentagrammic antiprisms, and 4 tetrahedra.

It can be obtained as a holosnub dodecahedral prism.

## Vertex coordinates

The vertices of a small ditrigonary icosidodecahedral antiprism of edge length 1 are given by:

• ${\displaystyle \left(±\frac12,\,±\frac12,\,±\frac12,\,±\sqrt{\frac{\sqrt5-1}{8}}\right),}$

along with all even permutations of the first three coordinates of:

• ${\displaystyle \left(±\frac{1+\sqrt5}{4},\,±\frac{\sqrt5-1}{4},\,0,\,±\sqrt{\frac{\sqrt5-1}{8}}\right).}$

## Related polychora

The regiment of the small ditrigonary icosidodecahedral antiprism also includes the ditrigonary dodecadodecahedral antiprism and the great ditrigonary icosidodecahedral antiprism.

The small icosicosidodecahedral alterprism is a partial Stott expansion of this polychoron.