# Disnub icosidodecahedron

Disnub icosidodecahedron
Rank3
TypeUniform
SpaceSpherical
Bowers style acronymDissid
Info
SymmetryH3, order 120
ArmySemi-uniform Grid
RegimentDissid
Elements
Vertex figureFloret pentagon, edge lengths 1, 1, 1, 1, (1+5)/2
Components2 snub dodecahedra
Faces120 triangles, 20 hexagrams, 12 stellated decagons
Edges60+120+120
Vertices120
Measures (edge length 1)
Volume≈ 75.23330
Dihedral angles3–3: ≈ 164.17537°
5–3: ≈ 152.92992°
Central density2
Related polytopes
DualCompound of two pentagonal hexecontahedra
Properties
ConvexNo
OrientableYes
NatureTame

The disnub icosidodecahedron, dissid, or compound of two snub dodecahedra is a uniform polyhedron compound. It consists of 120 snub triangles, 40 further triangles, and 24 pentagons (the latter two can combine in pairs due to faces in the same plane). Four triangles and one pentagon join at each vertex.

## Measures

The circumradius R ≈ 2.15584 of the disnub icosidodecahedron with unit edge length is the largest real root of

${\displaystyle 4096x^{12}-27648x^{10}+47104x^8-35776x^6+13872x^4-2696x^2+209.}$

Its volume V ≈ 75.23330 is given by twice the largest real root of

{\displaystyle \begin{align}&2176782336x^{12}-3195335070720x^{10}+162223191936000x^8+1030526618040000x^6\\ {} &+6152923794150000x^4-182124351550575000x^2+187445810737515625.\end{align}}

Its dihedral angles may be given as acos(α) for the angle between two triangles, and acos(β) for the angle between a pentagon and a triangle, where α ≈ –0.96210 is the smallest real root of

${\displaystyle 729x^6-486x^5-729x^4+756x^3+63x^2-270x+1,}$

and β ≈ –0.89045 is the second to smallest root of

${\displaystyle 91125x^{12}-668250x^{10}+2006775x^8-2735100x^6+1768275x^4-502410x^2+43681.}$[1]