icosohedron

An icosahedron is a polyhedron with 20 faces, each of which is an equilateral triangle. It is one of the five Platonic solids, which are convex polyhedra where each face is a congruent regular polygon and the same number of faces meet at each vertex. The icosahedron is dual to the dodecahedron, meaning that the vertices of one correspond to the faces of the other, and vice versa.

The icosahedron has 20 faces, 30 edges, and 12 vertices. Each vertex is the meeting point of five faces, and each edge is shared by two faces. The surface area and volume of a regular icosahedron with edge length a are given by the formulas 5√3a^2 and (5(3+√5)/12)a^3, respectively.

In geometry, the icosahedron is notable for its symmetrical properties. It is a highly symmetrical object, with 60 rotational symmetries and 120 reflections. The icosahedron is also the only Platonic solid that cannot be tiled to cover a plane without gaps or overlaps.

The icosahedron has numerous applications in various fields, including mathematics, physics, and chemistry. In mathematics, it is studied for its symmetrical properties and as a fundamental shape in three-dimensional space. In physics, it is used to model the structure of certain crystals and molecules. In chemistry, it is used to represent the arrangement of atoms in some molecules, such as boron clusters.