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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 with identical faces and identical vertices. The icosahedron has 12 vertices and 30 edges. It is dual to the dodecahedron, meaning that the vertices of one correspond to the faces of the other, and vice versa.

The icosahedron can be found in various contexts, including mathematics, chemistry, and architecture. In mathematics, it is studied for its symmetrical properties and its role in various geometric constructions. In chemistry, the icosahedral structure is found in some molecules, such as borane (B12H12) and some fullerenes. In architecture, the icosahedron has been used as a design element in various buildings and structures.

The icosahedron can be inscribed in a sphere, with each vertex touching the sphere. This sphere is known as the circumsphere of the icosahedron. The radius of the circumsphere is equal to the edge length of the icosahedron. The icosahedron can also be circumscribed by a sphere, with the sphere touching each face of the icosahedron. This sphere is known as the insphere of the icosahedron.

The icosahedron has several interesting properties, including its symmetrical arrangement of faces and vertices, its role in various geometric constructions, and its appearance in various natural and man-made structures. It is a fascinating object of study in the fields of mathematics, chemistry, and architecture.