# Small ditrigonary icosidodecahedron

(Redirected from Sidtid)
Small ditrigonary icosidodecahedron Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymSidtid
Coxeter diagramx5/2o3o3*a (   )
Elements
Faces20 triangles, 12 pentagrams
Edges60
Vertices20
Vertex figureDitrigon, edge lengths 1 and (5–1)/2 Measures (edge length 1)
Circumradius$\frac{\sqrt3}{2} \approx 0.86603$ Volume$\frac{4\sqrt5}{3} \approx 2.98142$ Dihedral angle$\arccos\left(-\sqrt{\frac{5+2\sqrt5}{15}}\right) \approx 142.62263^\circ$ Central density2
Number of external pieces72
Level of complexity5
Related polytopes
ArmyDoe, edge length $\frac{\sqrt5-1}{2}$ RegimentSidtid
DualSmall triambic icosahedron
ConjugateGreat ditrigonary icosidodecahedron
Convex coreTruncated icosahedron
Abstract & topological properties
Flag count240
Euler characteristic-8
OrientableYes
Genus5
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The small ditrigonary icosidodecahedron, or sidtid, is a quasiregular uniform polyhedron. It consists of 20 equilateral triangles and 12 pentagrams, with three of each joining at a vertex.

It can be constructed as a holosnub dodecahedron. The pentagrammic faces lie in the same planes as the pentagons of the convex hull dodecahedron, and the triangles are the dodecahedron's vertex figures.

This polyhedron is the vertex figure of the small ditrigonary hexacosihecatonicosachoron.

## Vertex coordinates

A small ditrigonary icosidodecahedron of side length 1 has vertex coordinates given by all permutations of

• $\left(±\frac12,\,±\frac12,\,±\frac12\right),$ and even permutations of

• $\left(±\frac{1+\sqrt5}{4},\,±\frac{\sqrt5-1}{4},\,0\right).$ The first set of vertices correspond to those of an inscribed unit cube. This relates to the fact that a uniform compound of 5 cubes has the same vertices and edges as this polyhedron.

## Representations

A small ditrigonary icosidodecahedron has the following Coxeter diagrams:

• x5/2o3o3*a
• ß5o3o (as holosnub)

## Related polyhedra

The small ditrigonary icosidodecahedron is the colonel of a three-member regiment that also includes the ditrigonary dodecadodecahedron and the great ditrigonary icosidodecahedron. This regiment also contains the rhombihedron, the uniform compound of 5 cubes. The pentagrammic cuploid and pentagonal cuploid are contained within the edge structure.

o5/2o3o3*a truncations
Name OBSA CD diagram Picture
Small ditrigonary icosidodecahedron sidtid x5/2o3o3*a (   )
(degenerate, double cover of id) x5/2x3o3*a (   )
(degenerate, double cover of ike) o5/2o3x3*a (   )
Small icosicosidodecahedron siid x5/2o3x3*a (   )
(degenerate, double cover of ti) x5/2x3x3*a (   )
Small snub icosicosidodecahedron seside s5/2s3s3*a (   )