정다면체: 두 판 사이의 차이
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46번째 줄:
#* 정사각형: 내각의 크기가 90°이므로, 하나의 꼭지점에 모일 수 있는 사각형면의 개수는 3개이다. 이것은 정육면체에 해당한다.
#* 정오각형: 내각의 크기가 108°이므로, 하나의 꼭지점에 모일 수 있는 오각형면의 개수는 3개이다. 이것은 정십이면체에 해당한다.
==History==
[[Image:Kepler-solar-system-1.png|right|250px|thumb|Kepler's Platonic solid model of the solar system from ''[[Mysterium Cosmographicum]]'' (1596)]]
The Platonic solids have been known since antiquity. Ornamented models of them can be found among the [[Carved Stone Balls|carved stone balls]] created by the late [[neolithic]] people of [[Scotland]] at least 1000 years before Plato (Atiyah and Sutcliffe 2003). Dice go back to the dawn of civilization with shapes that augured formal charting of Platonic solids.
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 (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 there are no other convex regular polyhedra.
The Platonic solids feature prominently in the philosophy of [[Plato]] for whom they are named. Plato wrote about them in the dialogue [[Timaeus (dialogue)|''Timaeus'']] ''c''.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. 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 is 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. Moreover, the solidity of the Earth was believed to be due to the fact that the cube is the only regular solid that [[tesselation|tesselates]] [[Euclidean space]]. 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, [[aether (classical element)|aithêr]] (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.
[[Euclid]] gave a complete mathematical description of 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 5 regular solids is the chief goal of the deductive system canonized in "Elements".<ref>{{cite book | author1 = Weyl H. | title = Symmetry | publisher = Princeton | year = 1952 | page = 74 }}</ref> Much of the information in Book XIII is probably derived from the work of Theaetetus.
In the 16th century, the [[Germans|German]] [[astronomer]] [[Johannes Kepler]] attempted to find a relation between the five extraterrestrial [[planet]]s known at that time 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 (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. 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 recognition that [[Kepler's laws of planetary motion#First law|the orbits of planets are ellipses]] rather than circles, as well as [[Kepler's laws of planetary motion#Second law|his two laws of orbital dynamics]], changing the courses of physics and astronomy, plus the discovery of the [[Kepler-Poinsot polyhedron|Kepler solids]].
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