Rank3
TypeUniform
Notation
Bowers style acronymIsdid
Coxeter diagrams5/3s5s ()
Elements
Faces60 triangles, 12 pentagons, 12 pentagrams
Edges60+60+30
Vertices60
Vertex figureIrregular nonconvex 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 external pieces372
Level of complexity39
Related polytopes
ArmyNon-uniform snid
RegimentIsdid
DualMedial inverted pentagonal hexecontahedron
Convex coreDodecahedron
Abstract & topological properties
Flag count600
Euler characteristic–6
OrientableYes
Properties
SymmetryH3+, order 60
Flag orbits10
ChiralYes
ConvexNo
NatureTame

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.

This polyhedron has tiny holes near the pentagrams' "notched" regions, as well as ambiguously-tunneled "arches" opposite the notches. These make the inverted snub dodecadodecahedron the only uniform polyhedron to have holes not caused by filling method (e.g. the great dirhombicosidodecahedron's numerous, very obvious holes when binary-filled).

## 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.