Regular icosahedron in the context of "Dual polyhedron"

In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges congruent), and the same number of faces meet at each vertex. There are only five such polyhedra: a tetrahedron (four faces), a cube (six faces), an octahedron (eight faces), a dodecahedron (twelve faces), and an icosahedron (twenty faces).

Geometers have studied the Platonic solids for thousands of years. They are named for the ancient Greek philosopher Plato, who hypothesized in one of his dialogues, the Timaeus, that the classical elements were made of these regular solids.

In geometry, an icosahedron (/ˌaɪkɒsəˈhiːdrən, -kə-, -koʊ-/ or /aɪˌkɒsəˈhiːdrən/) is a polyhedron with 20 faces. The name comes from Ancient Greek εἴκοσι (eíkosi) 'twenty' and ἕδρα (hédra) 'seat'. The plural can be either "icosahedra" (/-drə/) or "icosahedrons".

There are infinitely many non-similar shapes of icosahedra, some of them being more symmetrical than others. The best known is the (convex, non-stellated) regular icosahedron—one of the Platonic solids—whose faces are 20 equilateral triangles.

A regular dodecahedron or pentagonal dodecahedron is a dodecahedron (a polyhedron with 12 faces) composed of regular pentagonal faces, three meeting at each vertex. It is one of the Platonic solids, described in Plato's dialogues as the shape of the universe itself. Johannes Kepler used the dodecahedron in his 1596 model of the Solar System. However, the dodecahedron and other Platonic solids had already been described by other philosophers since antiquity.

The regular dodecahedron is a truncated trapezohedron because it is the result of truncating axial vertices of a pentagonal trapezohedron. It is also a Goldberg polyhedron because it is the initial polyhedron to construct new polyhedra by the process of chamfering. It has a relation with other Platonic solids, one of them is the regular icosahedron as its dual polyhedron. Other new polyhedra can be constructed by using a regular dodecahedron.

In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the icosahedron) and the rhombic triacontahedron.

Every polyhedron with icosahedral symmetry has 60 rotational (or orientation-preserving) symmetries and 60 orientation-reversing symmetries (that combine a rotation and a reflection), for a total symmetry order of 120. The full symmetry group is the Coxeter group of type H₃. It may be represented by Coxeter notation [5,3] and Coxeter diagram . The set of rotational symmetries forms a subgroup that is isomorphic to the alternating group A₅ on 5 letters.

In geometry, the truncated icosahedron is a polyhedron that can be constructed by truncating all of the regular icosahedron's vertices. Intuitively, it may be regarded as footballs (or soccer balls) that are typically patterned with white hexagons and black pentagons. Geodesic dome structures such as those whose architecture Buckminster Fuller pioneered are often based on this structure. It is an example of an Archimedean solid, as well as a Goldberg polyhedron.

A deltahedron is a polyhedron whose faces are all equilateral triangles. The deltahedron was named by Martyn Cundy, after the Greek capital letter delta resembling a triangular shape Δ.

Deltahedra can be categorized by the property of convexity. The simplest convex deltahedron is the regular tetrahedron, a pyramid with four equilateral triangles. There are eight convex deltahedra, which can be used in the applications of chemistry as in the polyhedral skeletal electron pair theory and chemical compounds. There are infinitely many concave deltahedra.