Icosaedri

Icosaedri is the plural term for the icosahedron in several European languages; in English the corresponding plural is icosahedra. The icosahedron is a regular polyhedron, one of the five Platonic solids. It has 20 equilateral triangular faces, 12 vertices, and 30 edges. At each vertex, five faces meet, and the dihedral angle between neighboring faces is approximately 138.19 degrees. The solid exhibits icosahedral symmetry, with a total of 120 symmetries including reflections, and 60 orientation-preserving rotations isomorphic to the alternating group A5.

The dual polyhedron of the icosahedron is the dodecahedron, which has 12 pentagonal faces. In geometry, the icosahedron serves as a key example of high symmetry and is often used to illustrate regular polyhedra, Euler characteristic properties, and spherical tiling relationships.

Coordinates and construction models: A common coordinate realization uses the golden ratio φ = (1+√5)/2. One standard set of vertex coordinates is (0, ±1, ±φ) and all permutations, scaled so that edges have unit length. This embedding highlights the close connection between the icosahedron’s geometry and the golden ratio.

Applications and appearances: The icosahedron appears in various domains. In chemistry and materials science, Buckminsterfullerene (C60) exhibits icosahedral symmetry. In virology, many virus capsids are modeled with icosahedral symmetry. In architecture and design, it informs geodesic domes and ornamental structures. In mathematics, it is central to studies of symmetry groups, tessellations, and polyhedral combinatorics.