Disnub icosidodecahedron

Disnub icosidodecahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Dissid
Coxeter diagram
Elements
Components	2 snub dodecahedra
Faces	120 triangles, 40 triangles as 20 hexagrams, 24 pentagons as 12 stellated decagons
Edges	60+120+120
Vertices	120
Vertex figure	Floret pentagon, edge lengths 1, 1, 1, 1, (1+√5)/2
Measures (edge length 1)
Circumradius	≈ 2.15584
Volume	≈ 75.23330
Dihedral angles	3–3: ≈ 164.17537°
	5–3: ≈ 152.92992°
Central density	2
Number of external pieces	392
Level of complexity	26
Related polytopes
Army	Semi-uniform Grid
Regiment	Dissid
Dual	Compound of two pentagonal hexecontahedra
Conjugates	Great disnub icosidodecahedron, great inverted disnub icosidodecahedron, great diretrosnub icosidodecahedron
Convex core	Icosidodecatruncated disdyakis triacontahedron
Abstract & topological properties
Flag count	1200
Orientable	Yes
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

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.

Its quotient prismatic equivalent is the snub dodecahedral antiprism, which is four-dimensional.

Measures

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

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

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

$\begin{align}&531441x^{12}-3120444405x^{10}+633684343500x^8+16101978406875x^6\\ {} &+384557737134375x^4-45531087887643750x^2+187445810737515625.\end{align}$

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

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

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

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

External links

• Bowers, Jonathan. "Polyhedron Category C10: Disnubs" (#69).

• Klitzing, Richard. "dissid".

• Wikipedia Contributors. "Compound of two snub dodecahedra".