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.

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 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
Genus4
Properties
SymmetryH3+, order 60
ConvexNo
NatureTame

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 (       )
Dodecadodecahedron did r{5,5/3} o5/3x5o (       )
Truncated great dodecahedron tigid t{5,5/3} o5/3x5x (       )
Great dodecahedron gad {5,5/3} o5/3o5x (       )
Complex ditrigonal rhombidodecadodecahedron (degenerate, ditdid+rhom) cadditradid rr{5,5/3} x5/3o5x (       )
Quasitruncated dodecadodecahedron quitdid tr{5,5/3} x5/3x5x (       )
Inverted snub dodecadodecahedron isdid sr{5,5/3} s5/3s5s (       )