Snub dodecadodecahedron
Type	Uniform star polyhedron
Elements	F = 84, E = 150 V = 60 (χ = −6)
Faces by sides	60{3}+12{5}+12{5/2}
Coxeter diagram
Wythoff symbol	| 2 5/2 5
Symmetry group	I, [5,3]+, 532
Index references	U40, C49, W111
Dual polyhedron	Medial pentagonal hexecontahedron
Vertex figure	3.3.5/2.3.5
Bowers acronym	Siddid

In geometry, the snub dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U₄₀. It has 84 faces (60 triangles, 12 pentagons, and 12 pentagrams), 150 edges, and 60 vertices.^[1] It is given a Schläfli symbol sr{5⁄2,5}, as a snub great dodecahedron.

Cartesian coordinates

Cartesian coordinates for the vertices of an inverted snub dodecadodecahedron are all the even permutations of

$${\displaystyle {\begin{array}{crrrc}{\Bigl (}&\pm \,2\alpha \ ,&\pm \,2\ ,&\pm \,2\beta \ &{\Bigr )},\\[2pt]{\Bigl (}&\pm {\bigl [}\alpha +{\frac {\beta }{\varphi ))+\varphi {\bigr ]},&\pm {\bigl [}-\alpha \varphi +\beta +{\frac {1}{\varphi )){\bigr ]},&\pm {\bigl [}{\frac {\alpha }{\varphi ))+\beta \varphi -1{\bigr ]}&{\Bigr )},\\[2pt]{\Bigl (}&\pm {\bigl [}-{\frac {\alpha }{\varphi ))+\beta \varphi +1{\bigr ]},&\pm {\bigl [}-\alpha +{\frac {\beta }{\varphi ))-\varphi {\bigr ]},&\pm {\bigl [}\alpha \varphi +\beta -{\frac {1}{\varphi )){\bigr ]}&{\Bigr )},\\[2pt]{\Bigl (}&\pm {\bigl [}-{\frac {\alpha }{\varphi ))+\beta \varphi -1{\bigr ]},&\pm {\bigl [}\alpha -{\frac {\beta }{\varphi ))-\varphi {\bigr ]},&\pm {\bigl [}\alpha \varphi +\beta +{\frac {1}{\varphi )){\bigr ]}&{\Bigr )},\\[2pt]{\Bigl (}&\pm {\bigl [}\alpha +{\frac {\beta }{\varphi ))-\varphi {\bigr ]},&\pm {\bigl [}\alpha \varphi -\beta +{\frac {1}{\varphi )){\bigr ]},&\pm {\bigl [}{\frac {\alpha }{\varphi ))+\beta \varphi +1{\bigr ]}&{\Bigr )},\end{array))}$$

with an even number of plus signs, where

$${\displaystyle \beta ={\frac {\ \ {\frac {\alpha ^{2)){\varphi ))+\varphi \ \ }{\ \alpha \varphi -{\frac {1}{\varphi ))))\ ,}$$

${\displaystyle \varphi ={\tfrac {1+{\sqrt {5))}{2))}$

$${\displaystyle \varphi \alpha ^{4}-\alpha ^{3}+2\alpha ^{2}-\alpha -{\frac {1}{\varphi ))\quad \implies \quad \alpha \approx 0.7964421.}$$

Related polyhedra

Medial pentagonal hexecontahedron

Medial pentagonal hexecontahedron
Type	Star polyhedron
Face
Elements	F = 60, E = 150 V = 84 (χ = −6)
Symmetry group	I, [5,3]+, 532
Index references	DU40
dual polyhedron	Snub dodecadodecahedron

The medial pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the snub dodecadodecahedron. It has 60 intersecting irregular pentagonal faces.