Regular icosahedron
A convex Platonic solid with 20 congruent equilateral-triangle faces, 30 edges, and 12 vertices, with five faces meeting at each vertex.
Platonic solid: A convex regular polyhedron whose faces are congruent regular polygons and whose vertices are equivalent. The icosahedron is one of the five solids in this precisely defined family.
Dihedral angle: The angle between two intersecting planes, measured within a solid along their shared edge. This angle fixes how adjacent triangular faces fold into the convex solid.
Regular dodecahedron: A Platonic solid with 12 regular pentagonal faces, 30 edges, and 20 vertices. It is the icosahedron's dual, exchanging the counts of faces and vertices.
Geodesic dome: A lightweight lattice shell made from interconnected struts arranged along geodesic paths. Geodesic domes often approximate a sphere by subdividing icosahedral faces.
Equilateral triangle: A triangle with three equal sides and three equal angles of 60 degrees. Every face of a regular icosahedron is an equilateral triangle.
Golden ratio: The irrational number (1 + √5)/2, which occurs in many proportions and geometric constructions. The coordinates of a regular icosahedron involve golden-ratio proportions.
Regular octahedron: A Platonic solid with eight equilateral-triangle faces, 12 edges, and six vertices. It shares triangular faces but has four faces, rather than five, meeting at each vertex.
Icosahedral virus: A virus whose protein capsid has icosahedral symmetry, often assembled from repeated protein subunits. The icosahedron describes a common architecture for enclosing viral genetic material.
Icosahedral symmetry: The rotational and reflectional symmetries of an icosahedron, described by the symmetry group of a regular icosahedron. Its symmetry explains why all vertices, edges, and faces are equivalent.
Icosahedral group: The finite group of rotations preserving a regular icosahedron, containing 60 elements. Its 60 rotations encode the solid's rotational symmetry.