Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymIsdid
Coxeter diagrams5/3s5s ()
Elements
Faces60 triangles, 12 pentagons, 12 pentagrams
Edges60+60+30
Vertices60
Vertex figureIrregular pentagon, edge lengths 1, 1, (5–1)/2, 1, (1+5)/2
Measures (edge length 1)
Volume≈ 4.61431
Dihedral angles3–3: ≈ 130.49074°
5–3: ≈ 68.64088°
5/2–3: ≈ 11.12448°
Central density9
Number of pieces372
Level of complexity39
Related polytopes
ArmyNon-uniform snid
RegimentIsdid
DualMedial inverted pentagonal hexecontahedron
Convex coreDodecahedron
Abstract properties
Euler characteristic-6
Topological properties
OrientableYes
Properties
SymmetryH3+, order 60
ConvexNo
NatureTame
Discovered by{{{discoverer}}}

The inverted snub dodecadodecahedron or isdid, is a uniform polyhedron. It consists of 60 snub triangles, 12 pentagrams, and 12 pentagons. Three triangles, 1 pentagon, and one pentagram meeting at each vertex. It can be constructed by alternation of the quasitruncated dodecadodecahedron and then setting all edge lengths to be equal.

## Measures

The circumradius R ≈ 0.85163 of the inverted snub dodecadodecahedron with unit edge length is the smallest positive real root of:

${\displaystyle 64x^8-192x^6+180x^4-65x^2+8.}$

Its volume V ≈ 4.61431 is given by the smallest positive real root of:

${\displaystyle 64x^8-21440x^6+18100x^4+5895625x^2+60062500.}$

These same polynomials define the circumradius and volume of the snub dodecadodecahedron.

## Related polyhedra

The inverted disnub dodecadodecahedron is a uniform polyhedron compound composed of the 2 opposite chiral forms of the inverted snub dodecadodecahedron.

o5/3o5o truncations
Name OBSA Schläfli symbol CD diagram Picture
Small stellated dodecahedron sissid {5/3,5} x5/3o5o ()
Quasitruncated small stellated dodecahedron quit sissid t{5/3,5} x5/3x5o ()