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The Platonic solids are the tetrahedron, the cube (or hexahedron regular), the octahedron, the dodecahedron and the icosahedron. Also known as Platonic bodies, bodies cosmic Pythagorean solids, solid perfect, Plato's polyhedra or, more accurately, convex regular polyhedra. Are characterized by convex polyhedra whose faces are equal regular polygons and whose vertices are attached to the same number of faces. Given these names after the Greek philosopher Plato (ca . AdC/428 427 BC - 347 BC), who is credited if any studied in the first instance. Properties

Regularity

As has been stated for define these polyhedra:

• All faces of a Platonic solid are equal regular polygons.
• In all the vertices of a Platonic solid attend the same number of faces and edges.
• All edges of a Platonic solid are the same length.
• All dihedral angles formed by the faces of a Platonic solid with each other are equal.
• All its vertices are convex to the icosahedron.

Symmetry

The Platonic solids are highly symmetrical:

• All of them are central symmetry about a point space (center of symmetry) which is equidistant from their faces, vertices and edges.
• They also have axial symmetry on a number of axes of symmetry passing through the center of symmetry above.
• They also have mirror symmetry about a series of planes of symmetry (or principal planes), which divided into two equal parts.

geometric Following the above, can be traced all Platonic solid three particular areas, all focused in the center of symmetry of the polyhedron:

• A sphere inscribed tangent to all sides in the middle.
• A second sphere tangent to all edges in the center.
• A circumscribed sphere, passing through all vertices of the polyhedron.

projecting the centers of the edges of a Platonic polyhedron circumscribed about the sphere from the center of symmetry of the polyhedron is obtained a regular spherical network, composed of equal-circle arcs, which are regular spherical polygons.

polyhedra whose faces are equal regular polygons are called regular polyhedra . There are five regular polyhedrons:

hexahedron (6 square):

A cube or regular hexahedron is a polyhedron with six congruent square faces, one of the so-called Platonic solids.
A cube, and is a hexahedron, may also be classified as a parallelepiped rectangle straight, because all sides are parallel sides and four pairs, and even as a prism with square base and height equal to the side of the base.

tetrahedron (4 triangles equilateral) :

A tetrahedron is a polyhedron with four faces. With this number of heads is bound to be a convex polyhedron, and triangular faces, meeting three in each corner. If the four faces of the tetrahedron are equilateral triangles, necessarily equal to each other, the tetrahedron is called regular. The tetrahedron is the three-dimensional simplex.

Dodecahedron (12 regular pentagons)

A dodecahedron is a twelve-sided polyhedron, convex or concave. Their faces are to be polygon of eleven sides or less. If the twelve faces of the dodecahedron are regular pentagons necessarily equal to each other, the dodecahedron is convex and is called regular, then being a so-called Platonic solids.

Icosahedron (20 equilateral triangles)

An icosahedron is a twenty-sided polyhedron, convex or concave. Their faces are to be nineteen-sided polygons or less. If the twenty faces of the icosahedron are equilateral triangles, necessarily equal to each other, the icosahedron is convex and is called regular , being then a so-called Platonic solids. The conjugate polyhedron of the icosahedron is the dodecahedron.

Octahedron (8 triangles equilateral)

An octahedron is a polyhedron with eight faces. With this number of faces can be a convex polyhedron or a concave polyhedron. Their faces are to be seven-sided polygons or less. If the eight faces of the octahedron are equilateral triangles, necessarily equal to each other, the octahedron is convex and is called regular , being then a so-called Platonic solids.