Platonic solid Totally Explained

Everything about Platonic Solid totally explained

In geometry, a Platonic solid is a convex regular polyhedron. These are the three-dimensional analogs of the convex regular polygons. There are precisely five such figures (shown below). They are unique in that the faces, edges and angles are all congruent.

The Five Convex Regular Polyhedra (Platonic solids)
Tetrahedron	Hexahedron or Cube	Octahedron	Dodecahedron	Icosahedron

The name of each figure is derived from the number of its faces: respectively 4, 6, 8, 12, and 20.
The aesthetic beauty and symmetry of the Platonic solids have made them a favorite subject of geometers for thousands of years. They are named for the ancient Greek philosopher Plato who theorized the classical elements were constructed from the regular solids.

History

The Platonic solids have been known since antiquity. Ornamented models of them can be found among the carved stone balls created by the late neolithic people of Scotland at least 1000 years before Plato (Atiyah and Sutcliffe 2003).
   The ancient Greeks studied the Platonic solids extensively. Some sources (such as Proclus) credit Pythagoras with their discovery. Other evidence suggests he may have only been familiar with the tetrahedron, cube, and dodecahedron, and that the discovery of the octahedron and icosahedron belong to Theaetetus, a contemporary of Plato. In any case, Theaetetus gave a mathematical description of all five and may have been responsible for the first known proof that there are no other convex regular polyhedra.
   The Platonic solids feature prominently in the philosophy of Plato for whom they're named. Plato wrote about them in the dialogue Timaeus c.360 B.C. in which he associated each of the four classical elements (earth, air, water, and fire) with a regular solid. Earth was associated with the cube, air with the octahedron, water with the icosahedron, and fire with the tetrahedron. There was intuitive justification for these associations: the heat of fire feels sharp and stabbing (like little tetrahedra). Air is made of the octahedron; its minuscule components are so smooth that one can barely feel it. Water, the icosahedron, flows out of one's hand when picked up, as if it's made of tiny little balls. By contrast, a highly un-spherical solid, the hexahedron (cube) represents earth. These clumsy little solids cause dirt to crumble and break when picked up, in stark difference to the smooth flow of water. The fifth Platonic solid, the dodecahedron, Plato obscurely remarks, "...the god used for arranging the constellations on the whole heaven". Aristotle added a fifth element, aithêr (aether in Latin, "ether" in English) and postulated that the heavens were made of this element, but he'd no interest in matching it with Plato's fifth solid. Euclid gave a complete mathematical description of the Platonic solids in the Elements; the last book (Book XIII) of which is devoted to their properties. Propositions 13–17 in Book XIII describe the construction of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that order. For each solid Euclid finds the ratio of the diameter of the circumscribed sphere to the edge length. In Proposition 18 he argues that there are no further convex regular polyhedra. Much of the information in Book XIII is probably derived from the work of Theaetetus.
   In the 16th century, the German astronomer Johannes Kepler attempted to find a relation between the five known planets at that time (excluding the Earth) and the five Platonic solids. In Mysterium Cosmographicum, published in 1596, Kepler laid out a model of the solar system in which the five solids were set inside one another and separated by a series of inscribed and circumscribed spheres. The six spheres each corresponded to one of the planets (Mercury, Venus, Earth, Mars, Jupiter, and Saturn). The solids were ordered with the innermost being the octahedron, followed by the icosahedron, dodecahedron, tetrahedron, and finally the cube. In this way the structure of the solar system and the distance relationships between the planets was dictated by the Platonic solids. In the end, Kepler's original idea had to be abandoned, but out of his research came the discovery of the Kepler solids, the realization that the orbits of planets are not circles, and Kepler's laws of planetary motion for which he's now famous.

Combinatorial properties

A convex polyhedron is a Platonic solid if and only if

all its faces are congruent convex regular polygons,
none of its faces intersect except at their edges, and
the same number of faces meet at each of its vertices. Each Platonic solid can therefore be denoted by a symbol which is a hexagonal tiling (dual to the triangular tiling). In a similar manner one can consider regular tessellations of the hyperbolic plane. These are characterized the condition 1/p + 1/q < 1/2. There is an infinite number of such tessellations.

Higher dimensions

In more than three dimensions, polyhedra generalize to polytopes, with higher-dimensional convex regular polytopes being the equivalents of the three-dimensional Platonic solids.
In the mid-19th century the Swiss mathematician Ludwig Schläfli discovered the four-dimensional analogues of the Platonic solids, called convex regular 4-polytopes. There are exactly six of these figures; five are analogous to the Platonic solids, while the sixth one, the 24-cell, has no lower-dimensional analogue.
In all dimensions higher than four, there are only three convex regular polytopes: the simplex, the hypercube, and the cross-polytope. In three dimensions, these coincide with the tetrahedron, the cube, and the octahedron.

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