Editing Regular icosahedron (section)

=== Symmetry ===
[[File:Sphere symmetry group ih.png|thumb|Illustration of a icosahedral symmetry. The five-fold, three-fold, and two-fold are labeled in blue, red, and magenta respectively. The mirror planes are the cyan [[great circle]].]]
The regular icosahedron has six five-fold rotation axes passing through two opposite vertices, ten three-fold axes rotating a triangular face, and fifteen two-fold axes passing through any of its edges. It has fifteen mirror planes as in a cyan [[great circle]] on the sphere meeting at order <math>\pi/5, \pi/3, \pi/2</math> angles, dividing a sphere into 120 triangles [[fundamental domain]]s. The full symmetry group of the icosahedron (including reflections) is known as the [[full icosahedral symmetry]] <math> \mathrm{I}_\mathrm{h} </math>.<ref>{{multiref
 |{{harvnb|Cann|2012|p=[https://books.google.com/books?id=mDbsN9LvE8gC&pg=PA34 34]}}
 |{{harvnb|Benz|Neumann|2014|p=[https://books.google.com/books?id=YagtBAAAQBAJ&pg=RA1-SA1-PA98 1-98]}}
}}</ref> It is isomorphic to the product of the rotational symmetry group and the [[cyclic group]] of size two, generated by the reflection through the center of the regular icosahedron.{{sfn|Seidel|1991|p=[https://books.google.com/books?id=brziBQAAQBAJ&pg=PA364 364]}} It shares the [[dual polyhedron]] of a regular icosahedron, the regular dodecahedron: a regular icosahedron can be inscribed in a regular dodecahedron by placing its vertices at the face centers of the dodecahedron, and vice versa.<ref>{{multiref
|{{harvnb|Erickson|2011|p=[https://books.google.com/books?id=LgeP62-ZxikC&pg=PA62 62]}}
|{{harvnb|Herrmann|Sally|2013|p=[https://books.google.com/books?id=b2fjR81h6yEC&pg=PA257 257]}}
}}</ref>

The rotational [[symmetry group]] of the regular icosahedron is [[isomorphic]] to the [[alternating group]] on five letters. This non-[[abelian group|abelian]] [[simple group]] is the only non-trivial [[normal subgroup]] of the [[symmetric group]] on five letters.{{sfn|Gray|2018|p=[https://books.google.com/books?id=gl1oDwAAQBAJ&pg=PA371 371]}} Since the [[Galois group]] of the general [[quintic equation]] is isomorphic to the symmetric group on five letters, and this normal subgroup is simple and non-abelian, the general quintic equation does not have a solution in radicals. The proof of the [[Abel–Ruffini theorem]] uses this simple fact,{{sfn|Rotman|1998|p=[https://books.google.com/books?id=0kHhBwAAQBAJ&pg=PA75 74&ndash;75]}} and [[Felix Klein]] wrote a book that made use of the theory of icosahedral symmetries to derive an analytical solution to the general quintic equation.<ref>{{multiref
|{{harvnb|Klein|1884}}. See [[Icosahedral symmetry#Related geometries|related geometries of that symmetry group]] for further history and related symmetries on seven and eleven letters.
|{{harvnb|Klein|1888}}
}}</ref>

The regular icosahedron, as one of the Platonic solids, is a [[regular polyhedron]]. It is [[Isogonal figure|isogonal]], [[isohedral]], and [[isotoxal]]: any two vertices, two faces, and two edges of a regular icosahedron respectively can be transformed by rotations and reflections under its symmetry orbit, which preserves the appearance. Each regular polyhedron has a [[convex hull]] on its edge midpoints; [[icosidodecahedron]] is the convex hull of a regular icosahedron.{{sfn|Senechal|1989|p=[https://books.google.com/books?id=OToVjZW9CKMC&pg=PA12 12]}} Each vertex is surrounded by five equilateral triangles, so the regular icosahedron denotes <math> 3.3.3.3.3 </math> in [[vertex configuration]] or <math> \{3,5\} </math> in [[Schläfli symbol]].{{sfn|Walter|Deloudi|2009|p=[https://books.google.com/books?id=nVx-tu596twC&pg=PA50 50]}}