Editing Platonic solid (section)

== History ==
The Platonic solids have been known since antiquity. It has been suggested that certain [[carved stone balls]] created by the [[late Neolithic]] people of [[Scotland]] represent these shapes; however, these balls have rounded knobs rather than being polyhedral, the numbers of knobs frequently differed from the numbers of vertices of the Platonic solids, there is no ball whose knobs match the 20 vertices of the dodecahedron, and the arrangement of the knobs was not always symmetrical.{{sfn|Lloyd|2012}}

The [[ancient Greeks]] studied the Platonic solids extensively. Some sources (such as [[Proclus]]) credit [[Pythagoras]] with their discovery. Other evidence suggests that he may have only been familiar with the tetrahedron, cube, and dodecahedron and that the discovery of the octahedron and icosahedron belong to [[Theaetetus (mathematician)|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 no other convex regular polyhedra exist.

{{multiple image
 | align = right |total_width=500
 | image1 = Kepler Hexahedron Earth.jpg |width1=290|height1=304
 | image2 = Kepler Icosahedron Water.jpg |width2=306|height2=328
 | image3 = Kepler Octahedron Air.jpg |width3=328|height3=334
 | image4 = Kepler Tetrahedron Fire.jpg |width4=367|height4=328
 | image5 = Kepler Dodecahedron Universe.jpg |width5=330|height5=332
 | footer = Assignment to the elements in Kepler's ''Harmonices Mundi''
}}
The Platonic solids are prominent in the philosophy of [[Plato]], their namesake. Plato wrote about them in the dialogue [[Timaeus (dialogue)|''Timaeus'']] {{circa}} 360 B.C. in which he associated each of the four [[classical element]]s ([[earth (classical element)|earth]], [[air (classical element)|air]], [[water (classical element)|water]], and [[fire (classical element)|fire]]) with a regular solid. Earth was associated with the cube, air with the octahedron, water with the icosahedron, and fire with the tetrahedron. Of the fifth Platonic solid, the dodecahedron, Plato obscurely remarked, "...the god used [it] for arranging the constellations on the whole heaven". [[Aristotle]] added a fifth element, [[aether (classical element)|aither]] (aether in Latin, "ether" in English) and postulated that the heavens were made of this element, but he had no interest in matching it with Plato's fifth solid.<ref>Wildberg (1988): Wildberg discusses the correspondence of the Platonic solids with elements in ''Timaeus'' but notes that this correspondence appears to have been forgotten in ''[[Epinomis]]'', which he calls "a long step towards Aristotle's theory", and he points out that Aristotle's ether is above the other four elements rather than on an equal footing with them, making the correspondence less apposite.</ref>

[[Euclid]] completely mathematically described the Platonic solids in the [[Euclid's Elements|''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. [[Andreas Speiser]] has advocated the view that the construction of the five regular solids is the chief goal of the deductive system canonized in the ''Elements''.{{sfn|Weyl|1952|p=74}} Much of the information in Book XIII is probably derived from the work of Theaetetus.

[[File:Mysterium Cosmographicum solar system model.jpg|upright=1|thumb|[[Johannes Kepler|Kepler's]] Platonic solid model of the [[Solar System]] from ''[[Mysterium Cosmographicum]]'' (1596)]]
In the 16th century, the German [[astronomer]] [[Johannes Kepler]] attempted to relate the five extraterrestrial [[planet]]s known at that time to the five Platonic solids. In ''[[Mysterium Cosmographicum]]'', published in 1596, Kepler proposed 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. Kepler proposed that the distance relationships between the six planets known at that time could be understood in terms of the five Platonic solids enclosed within a sphere that represented the orbit of [[Saturn]]. The six spheres each corresponded to one of the planets ([[Mercury (planet)|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, thereby dictating the structure of the solar system and the distance relationships between the planets by the Platonic solids. In the end, Kepler's original idea had to be abandoned, but out of his research came his [[Kepler's laws of planetary motion|three laws of orbital dynamics]], the first of which was that [[Kepler's laws of planetary motion#First law|the orbits of planets are ellipses]] rather than circles, changing the course of physics and astronomy.<ref>{{cite book |last1=Olenick |first1=R. P. |title=[[The Mechanical Universe|The Mechanical Universe: Introduction to Mechanics and Heat]] |last2=Apostol |first2=T. M. |author-link2=Tom M. Apostol |last3=Goodstein |first3=D. L. |author-link3=David Goodstein |publisher=Cambridge University Press |year=1986 |isbn=0-521-30429-6 |pages=434–436}}</ref> He also discovered the [[Kepler–Poinsot polyhedron|Kepler solids]], which are two ''nonconvex'' regular polyhedra.