Summary
| Great snub icosidodecahedron | |
|---|---|
| |
| Type | Uniform star polyhedron |
| Elements | F = 92, E = 150 V = 60 (χ = 2) |
| Faces by sides | (20+60){3}+12{5/2} |
| Coxeter diagram |     
|
| Wythoff symbol | | 2 5/2 3 |
| Symmetry group | I, [5,3]+, 532 |
| Index references | U57, C88, W113 |
| Dual polyhedron | Great pentagonal hexecontahedron |
| Vertex figure |  34.5/2 |
| Bowers acronym | Gosid |
In geometry, the great snub icosidodecahedron is a nonconvex uniform polyhedron, indexed as U57. It has 92 faces (80 triangles and 12 pentagrams), 150 edges, and 60 vertices.[1] It can be represented by a Schläfli symbol sr{5⁄2,3}, and Coxeter-Dynkin diagram 
.
This polyhedron is the snub member of a family that includes the great icosahedron, the great stellated dodecahedron and the great icosidodecahedron.
In the book Polyhedron Models by Magnus Wenninger, the polyhedron is misnamed great inverted snub icosidodecahedron, and vice versa.
Cartesian coordinates edit
Cartesian coordinates for the vertices of a great retrosnub icosidodecahedron are all the even permutations of
![{\displaystyle {\begin{array}{crrrc}{\Bigl (}&\pm \,2\alpha ,&\pm \,2,&\pm \,2\beta &{\Bigr )},\\{\Bigl (}&\pm {\bigl [}\alpha -\beta \varphi -{\frac {1}{\varphi }}{\bigr ]},&\pm {\bigl [}{\frac {\alpha }{\varphi }}+\beta -\varphi {\bigr ]},&\pm {\bigl [}-\alpha \varphi -{\frac {\beta }{\varphi }}-1{\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm {\bigl [}\alpha \varphi -{\frac {\beta }{\varphi }}+1{\bigr ]},&\pm {\bigl [}-\alpha -\beta \varphi +{\frac {1}{\varphi }}{\bigr ]},&\pm {\bigl [}-{\frac {\alpha }{\varphi }}+\beta +\varphi {\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm {\bigl [}\alpha \varphi -{\frac {\beta }{\varphi }}-1{\bigr ]},&\pm {\bigl [}\alpha +\beta \varphi +{\frac {1}{\varphi }}{\bigr ]},&\pm {\bigl [}-{\frac {\alpha }{\varphi }}+\beta -\varphi {\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm {\bigl [}\alpha -\beta \varphi +{\frac {1}{\varphi }}{\bigr ]},&\pm {\bigl [}-{\frac {\alpha }{\varphi }}-\beta -\varphi {\bigr ]},&\pm {\bigl [}-\alpha \varphi -{\frac {\beta }{\varphi }}+1{\bigr ]}&{\Bigr )},\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f7e1b1ee5692b0155448d8f632b2ffab5088df1)
![{\displaystyle {\begin{aligned}\alpha &=\xi -{\frac {1}{\xi }},\\[4pt]\beta &=-{\frac {\xi }{\varphi }}+{\frac {1}{\varphi ^{2}}}-{\frac {1}{\xi \varphi }},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e6a3e86c3eab5801ad1f161516d1c4927862671)
where is the golden ratio and ξ is the negative real root of:
The circumradius for unit edge length is
The four positive real roots of the sextic in R2,
Related polyhedra edit
Great pentagonal hexecontahedron edit
| Great pentagonal hexecontahedron | |
|---|---|
| |
| Type | Star polyhedron |
| Face | 
|
| Elements | F = 60, E = 150 V = 92 (χ = 2) |
| Symmetry group | I, [5,3]+, 532 |
| Index references | DU57 |
| dual polyhedron | Great snub icosidodecahedron
 The great pentagonal hexecontahedron (or great petaloid ditriacontahedron) is a nonconvex isohedral polyhedron and dual to the uniform great snub icosidodecahedron. It has 60 intersecting irregular pentagonal faces, 120 edges, and 92 vertices. Proportions editDenote the golden ratio by . Let be the negative zero of the polynomial . Then each pentagonal face has four equal angles of and one angle of . Each face has three long and two short edges. The ratio between the lengths of the long and the short edges is given by
The dihedral angle equals . Part of each face lies inside the solid, hence is invisible in solid models. The other two zeroes of the polynomial play a similar role in the description of the great inverted pentagonal hexecontahedron and the great pentagrammic hexecontahedron. See also editReferences edit
External links edit
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