Octahedron 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.

Kaolinite (/ˈkeɪ.ələˌnaɪt, -lɪ-/ KAY-ə-lə-nyte, -⁠lih-; also called kaolin) is a clay mineral, with the chemical composition Al₂Si₂O₅(OH)₄. It is a layered silicate mineral, with one "tetrahedral" sheet of silicate tetrahedra (SiO₄) linked to one "octahedral" sheet of aluminate octahedrons (AlO₂(OH)₄) through oxygen atoms on one side, and another such sheet through hydrogen bonds on the other side.

Kaolinite is a soft, earthy, usually white, mineral (dioctahedral phyllosilicate clay), produced by the chemical weathering of aluminium silicate minerals like feldspar. It has a low shrink–swell capacity and a low cation-exchange capacity (1–15 meq/100 g).

In mineralogy, argentite (from Latin argentum 'silver') is cubic silver sulfide (Ag₂S), which can only exist at temperatures above 173 °C (343 °F), 177 °C (351 °F), or 179 °C (354 °F). When it cools to ordinary temperatures it turns into its monoclinic polymorph, acanthite. The International Mineralogical Association has decided to reject argentite as a proper mineral.

The name "argentite" sometimes also refers to pseudomorphs of argentite: specimens of acanthite which still display some of the outward signs of the cubic crystal form, even though their actual crystal structure is monoclinic due to the lower temperature. This form of acanthite is occasionally found as uneven cubes and octahedra, but more often as dendritic or earthy masses, with a blackish lead-grey color and metallic luster.

Cleavage, in mineralogy and materials science, is the tendency of crystalline materials to split along definite crystallographic structural planes. These planes of relative weakness are a result of the regular locations of atoms and ions in the crystal, which create smooth repeating surfaces that are visible both in the microscope and to the naked eye. If bonds in certain directions are weaker than others, the crystal will tend to split along the weakly bonded planes. These flat breaks are termed "cleavage". The classic example of cleavage is mica, which cleaves in a single direction along the basal pinacoid, making the layers seem like pages in a book. In fact, mineralogists often refer to "books of mica".

Diamond and graphite provide examples of cleavage. Each is composed solely of a single element, carbon. In diamond, each carbon atom is bonded to four others in a tetrahedral pattern with short covalent bonds. The planes of weakness (cleavage planes) in a diamond are in four directions, following the faces of the octahedron. In graphite, carbon atoms are contained in layers in a hexagonal pattern where the covalent bonds are shorter (and thus even stronger) than those of diamond. However, each layer is connected to the other with a longer and much weaker van der Waals bond. This gives graphite a single direction of cleavage, parallel to the basal pinacoid. So weak is this bond that it is broken with little force, giving graphite a slippery feel as layers shear apart. As a result, graphite makes an excellent dry lubricant.

A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedron that is dual to an octahedron.

The group of orientation-preserving symmetries is S₄, the symmetric group or the group of permutations of four objects, since there is exactly one such symmetry for each permutation of the four diagonals of the cube.

In geometry, a regular octahedron is a highly symmetrical type of octahedron (eight-sided polyhedron) with eight equilateral triangles as its faces, four of which meet at each vertex. It is a type of square bipyramid or triangular antiprism with equal-length edges. Regular octahedra occur in nature as crystal structures. Other types of octahedra also exist, with various amounts of symmetry.

A regular octahedron is the three-dimensional case of the more general concept of a cross-polytope.

In geometry, every polyhedron is associated with a second dual structure, wherein the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all can also be constructed as geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron.

Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a corresponding symmetry class. For example, the regular polyhedra – the (convex) Platonic solids and (star) Kepler–Poinsot polyhedra – form dual pairs, where the regular tetrahedron is self-dual. The dual of an isogonal polyhedron (one in which any two vertices are equivalent under symmetries of the polyhedron) is an isohedral polyhedron (one in which any two faces are equivalent [...]), and vice versa. The dual of an isotoxal polyhedron (one in which any two edges are equivalent [...]) is also isotoxal.