Icosahedral symmetry

Icosahedral symmetry is invariance under the rotations of a regular icosahedron. Its rotational group has 60 elements and is isomorphic to the alternating group A₅.

Icosahedral rotation group: The 60-element group of orientation-preserving rotations of a regular icosahedron. It is the precise mathematical object describing the rotations that preserve icosahedral symmetry.
Icosahedron: A regular polyhedron with 20 triangular faces, 30 edges, and 12 vertices. Its rotational invariance is the geometric source of the symmetry.
Icosahedral virus: A virus whose protein capsid has icosahedral symmetry. Viral capsids use this symmetry to assemble many subunits into a closed shell.
Tetrahedral symmetry: The rotational symmetry of a regular tetrahedron, comprising 12 rotations. It has fewer rotations and lacks the fivefold axes characteristic of icosahedral symmetry.
Plato: An ancient Greek philosopher whose dialogues discuss the regular solids and their cosmological associations. The icosahedron appears in the Platonic account of the elements and regular solids.
Alternating group A₅: The group of even permutations of five objects, containing 60 elements. It is abstractly isomorphic to the icosahedral rotation group.
Dodecahedron: A regular polyhedron with 12 pentagonal faces, 30 edges, and 20 vertices. It shares the icosahedron's rotational symmetry group as its dual solid.
Caspar–Klug theory: A geometric theory classifying icosahedral virus capsids by triangulation number. It connects icosahedral symmetry to the repeated arrangement of capsid proteins.
Octahedral symmetry: The rotational symmetry of a cube or octahedron, comprising 24 rotations. Its threefold axes coexist with fourfold axes, unlike the icosahedron's fivefold axes.
Euclid's Elements: A Greek mathematical treatise that systematically presents geometry and number theory. Its final book constructs the regular icosahedron and establishes classical results about it.

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