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| In [[Euclidean geometry]], a '''Platonic solid''' is a [[Regular polyhedron|regular]], [[Convex set|convex]] [[polyhedron]] with [[Congruence (geometry)|congruent]] [[Face (geometry)|faces]] of [[Regular polygon|regular]] [[polygon]]s and the same number of faces meeting at each [[Vertex (geometry)|vertex.]] Five solids meet those criteria, and each is named after its number of faces.
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| {| border style="margin: 1em auto; text-align: center; border-collapse: collapse; border: 1pt solid #aaa;"
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| | [[Tetrahedron]]<br /> (four faces) || [[Cube]] or [[hexahedron]]<br />(six faces) || [[Octahedron]]<br />(eight faces) || [[Dodecahedron]]<br />(twelve faces) || [[Icosahedron]]<br />(twenty faces)
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| |- style="vertical-align: bottom;"
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| |width=120| [[Image:Tetrahedron.svg|80px]]<br />
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| <small>([[:image:tetrahedron.gif|Animation]])</small>
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| |width=120 style="padding-top: 4pt;"|[[Image:Hexahedron.svg|80px]]<br />
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| <small>([[:image:hexahedron.gif|Animation]])</small>
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| |width=120|[[Image:Octahedron.svg|80px]]<br />
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| <small>([[:image:octahedron.gif|Animation]])</small>
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| |width=120|[[Image:POV-Ray-Dodecahedron.svg|80px]]<br />
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| <small>([[:image:dodecahedron.gif|Animation]])</small>
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| |width=120|[[Image:Icosahedron.svg|80px]]<br />
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| <small>([[:image:icosahedron.gif|Animation]])</small>
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| |}
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| [[Geometer]]s have studied the [[mathematical beauty]] and [[symmetry]] of the Platonic solids for thousands of years. They are named for the [[Greek philosophy|ancient Greek philosopher]] [[Plato]] who theorized in his dialogue, the [[Timaeus (dialogue)|Timaeus]], that the [[classical element]]s were made of these regular solids.<ref name="The Stanford Encyclopedia of Philosophy">{{cite web|last=Zeyl|first=Donald|title=The Stanford Encyclopedia of Philosophy: Plato's Timaeus|url=http://plato.stanford.edu/entries/plato-timaeus/}}</ref>
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| ==History==
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| [[Image:Kepler-solar-system-1.png|right|250px|thumb| [[Johannes Kepler|Kepler's]] Platonic solid model of the [[solar system]] from ''[[Mysterium Cosmographicum]]'' (1596)]]
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| The Platonic solids have been known since antiquity. [[Carved Stone Balls|Carved stone balls]] created by the late [[neolithic]] people of [[Scotland]] lie near ornamented models resembling them, but the Platonic solids do not appear to have been preferred over less-symmetrical objects, and some of the Platonic solids are even absent.<ref>{{cite web|last=Hart|first=George|title=Neolithic Carved Stone Polyhedra|url=http://www.georgehart.com/virtual-polyhedra/neolithic.html}}; see also Lloyd D. R, (2012), ''How old are the Platonic Solids?'', BSHM Bulletin: Journal of
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| the British Society for the History of Mathematics, 27:3, 131-140</ref> Dice go back to the dawn of civilization with shapes that augured formal charting of Platonic solids.
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| 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.
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| The Platonic solids are prominent in the philosophy of [[Plato]], their namesake. 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 nonspherical 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 cube's being the only regular solid that [[tessellation|tesselates]] [[Euclidean space]] was believed to cause the solidity of the Earth. 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.<ref>See e.g. {{citation|title=John Philoponus' Criticism of Aristotle's Theory of Aether|first=Christian|last=Wildberg|publisher=Walter de Gruyter|year=1988|isbn=9783110104462|pages=11–12|url=http://books.google.com/books?id=af3XzdAvB_cC&pg=PA11}}. 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>
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| [[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.
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| [[Andreas Speiser]] has advocated the view that the construction of the 5 regular solids is the chief goal of the deductive system canonized in the ''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.
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| In the 16th century, the [[Germans|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. He also discovered the [[Kepler-Poinsot polyhedron|Kepler solids]].
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| In the 20th century, attempts to link Platonic solids to the physical world were expanded to the [[electron shell model]] in chemistry by [[Robert James Moon|Robert Moon]] in a theory known as the "[[Robert James Moon#Moon model|Moon model]]".<ref>{{harvnb|Hecht|Stevens|2004}}</ref>
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| <div style="clear: both"></div>
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| ==Combinatorial properties==
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| A convex polyhedron is a Platonic solid if and only if
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| # all its faces are [[Congruence (geometry)|congruent]] convex [[regular polygon]]s,
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| # none of its faces intersect except at their edges, and
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| # the same number of faces meet at each of its [[vertex (geometry)|vertices]].
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| Each Platonic solid can therefore be denoted by a symbol {''p'', ''q''} where
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| :''p'' = the number of edges of each face (or the number of vertices of each face) and
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| :''q'' = the number of faces meeting at each vertex (or the number of edges meeting at each vertex).
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| The symbol {''p'', ''q''}, called the [[Schläfli symbol]], gives a [[combinatorics|combinatorial]] description of the polyhedron. The Schläfli symbols of the five Platonic solids are given in the table below.
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| {| class="wikitable sortable"
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| |-
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| !colspan=2 | Polyhedron
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| ![[Vertex (geometry)|Vertices]]
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| ![[Edge (geometry)|Edges]]
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| ![[Face (geometry)|Faces]]
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| ![[Schläfli symbol]]
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| ![[Vertex configuration|Vertex config.]]
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| |- align=center
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| |[[tetrahedron]]
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| |[[Image:tetrahedron.svg|50px|Tetrahedron]]
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| |4||6||4||{3, 3}||3.3.3
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| |- align=center
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| |[[hexahedron]]<BR>([[cube]])
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| |[[Image:hexahedron.svg|50px|Hexahedron (cube)]]
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| |8||12||6||{4, 3}||4.4.4
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| |- align=center
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| |[[octahedron]]
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| |[[Image:octahedron.svg|50px|Octahedron]]
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| |6||12||8||{3, 4}||3.3.3.3
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| |- align=center
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| |[[dodecahedron]]
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| | [[Image:POV-Ray-Dodecahedron.svg|50px|Dodecahedron]]
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| |20||30||12||{5, 3}||5.5.5
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| |- align=center
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| |[[icosahedron]]
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| | [[Image:icosahedron.svg|50px|Icosahedron]]
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| |12||30||20||{3, 5}||3.3.3.3.3
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| |}
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| All other combinatorial information about these solids, such as total number of vertices (''V''), edges (''E''), and faces (''F''), can be determined from ''p'' and ''q''. Since any edge joins two vertices and has two adjacent faces we must have:
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| :<math>pF = 2E = qV.\,</math>
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| The other relationship between these values is given by [[Euler characteristic|Euler's formula]]:
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| :<math>V - E + F = 2.\,</math>
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| This nontrivial fact can be proved in a great variety of ways (in [[algebraic topology]] it follows from the fact that the Euler characteristic of the [[sphere]] is two). Together these three relationships completely determine ''V'', ''E'', and ''F'':
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| :<math>V = \frac{4p}{4 - (p-2)(q-2)},\quad E = \frac{2pq}{4 - (p-2)(q-2)},\quad F = \frac{4q}{4 - (p-2)(q-2)}.</math>
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| Note that swapping ''p'' and ''q'' interchanges ''F'' and ''V'' while leaving ''E'' unchanged (for a geometric interpretation of this fact, see the section on dual polyhedra below).
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| ==Classification==
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| The classical result is that only five convex regular polyhedra exist. Two common arguments below demonstrate no more than five Platonic solids can exist, but positively demonstrating the existence of any given solid is a separate question – one that an explicit construction cannot easily answer.
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| ===Geometric proof===
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| The following geometric argument is very similar to the one given by [[Euclid]] in the [[Euclid's Elements|''Elements'']]:
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| # Each vertex of the solid must coincide with one vertex each of at least three faces.
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| # At each vertex of the solid, the total, among the adjacent faces, of the angles between their respective adjacent sides must be less than 360°.
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| # The angles at all vertices of all faces of a Platonic solid are identical: each vertex of each face must contribute less than 360°/3 = 120°.
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| # Regular polygons of [[Hexagon|six]] or more sides have only angles of 120° or more, so the common face must be the triangle, square, or pentagon. And for:
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| #* [[Triangle|Triangular]] faces: each vertex of a regular triangle is 60°, so a shape may have 3, 4, or 5 triangles meeting at a vertex; these are the tetrahedron, octahedron, and icosahedron respectively.
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| #* [[Square (geometry)|Square]] faces: each vertex of a square is 90°, so there is only one arrangement possible with three faces at a vertex, the cube.
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| #* [[Pentagon]]al faces: each vertex is 108°; again, only one arrangement, of three faces at a vertex is possible, the dodecahedron.
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| ===Topological proof===
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| A purely [[topology|topological]] proof can be made using only combinatorial information about the solids. The key is [[Euler characteristic|Euler's observation]] that <math>V - E + F = 2</math>, and the fact that <math>pF = 2E = qV</math>, where ''p'' stands for the number of edges of each face and ''q'' for the number of edges meeting at each vertex. Combining these equations one obtains the equation
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| :<math>\frac{2E}{q} - E + \frac{2E}{p} = 2.</math>
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| Simple algebraic manipulation then gives
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| :<math>{1 \over q} + {1 \over p}= {1 \over 2} + {1 \over E}.</math>
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| Since <math>E</math> is strictly positive we must have
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| :<math>\frac{1}{q} + \frac{1}{p} > \frac{1}{2}.</math>
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| Using the fact that ''p'' and ''q'' must both be at least 3, one can easily see that there are only five possibilities for (''p'', ''q''):
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| :<math>(3, 3),\quad (4, 3),\quad (3, 4),\quad (5, 3),\quad (3,5).</math>
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| ==Geometric properties==
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| ===Angles===
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| There are a number of [[angle]]s associated with each Platonic solid. The [[dihedral angle]] is the interior angle between any two face planes. The dihedral angle, θ, of the solid {''p'',''q''} is given by the formula
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| :<math>\sin{\theta\over 2} = \frac{\cos(\pi/q)}{\sin(\pi/p)}.</math>
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| This is sometimes more conveniently expressed in terms of the [[tangent (trigonometric function)|tangent]] by
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| :<math>\tan{\theta\over 2} = \frac{\cos(\pi/q)}{\sin(\pi/h)}.</math>
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| The quantity ''h'' is 4, 6, 6, 10, and 10 for the tetrahedron, cube, octahedron, dodecahedron, and icosahedron respectively.
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| The [[angular deficiency]] at the vertex of a polyhedron is the difference between the sum of the face-angles at that vertex and 2π. The defect, δ, at any vertex of the Platonic solids {''p'',''q''} is
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| :<math>\delta = 2\pi - q\pi\left(1-{2\over p}\right).</math>
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| By a theorem of Descartes, this is equal to 4π divided by the number of vertices (i.e. the total defect at all vertices is 4π).
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| The 3-dimensional analog of a plane angle is a [[solid angle]]. The solid angle, Ω, at the vertex of a Platonic solid is given in terms of the dihedral angle by
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| :<math>\Omega = q\theta - (q-2)\pi.\,</math>
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| This follows from the [[spherical excess]] formula for a [[spherical polygon]] and the fact that the [[vertex figure]] of the polyhedron {''p'',''q''} is a regular ''q''-gon.
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| The solid angle of a face subtended from the center of a platonic solid is equal to the solid angle of a full sphere (4π steradians) divided by the number of faces. Note that this is equal to the angular deficiency of its dual.
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| The various angles associated with the Platonic solids are tabulated below. The numerical values of the solid angles are given in [[steradian]]s. The constant φ = (1+√5)/2 is the [[golden ratio]].
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| {| class="wikitable"
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| !Polyhedron
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| ![[Dihedral angle]]<br><math>\theta</math>
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| !<math>\tan\frac{\theta}{2}</math>
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| ![[Vertex angle]]
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| ![[Defect (geometry)|Defect]] (<math>\delta</math>)
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| !colspan = 2|Vertex [[solid angle]] (<math>\Omega</math>)
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| !Face<br>solid angle
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| |-
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| |[[tetrahedron]] || 70.53° || <math>1\over{\sqrt 2}</math> || 60° || <math>\pi</math>
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| |<math>\cos^{-1}\left(\frac{23}{27}\right)</math>
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| |<math>\approx 0.551286</math>
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| |<math>\pi</math>
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| |-
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| |[[cube]] || 90° || <math>1</math> || 90° || <math>\pi\over 2</math>
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| |<math>\frac{\pi}{2}</math>
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| |<math>\approx 1.57080</math>
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| |<math>2\pi\over 3</math>
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| |-
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| |[[octahedron]] || 109.47° || <math>\sqrt 2</math> || 60°, 90° || <math>{2\pi}\over 3</math>
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| |<math>4\sin^{-1}\left({1\over 3}\right)</math>
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| |<math>\approx 1.35935</math>
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| |<math>\pi\over 2</math>
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| |-
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| |[[dodecahedron]] || 116.57° || <math>\varphi</math> || 108° || <math>\pi\over 5</math>
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| |<math>\pi - \tan^{-1}\left(\frac{2}{11}\right)</math>
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| |<math>\approx 2.96174</math>
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| |<math>\pi\over 3</math>
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| |-
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| |[[icosahedron]] || 138.19° || <math>\varphi^2</math> || 60°, 108° || <math>\pi\over 3</math>
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| |<math>2\pi - 5\sin^{-1}\left({2\over 3}\right)</math>
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| |<math>\approx 2.63455</math>
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| |<math>\pi\over 5</math>
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| |}
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| ===Radii, area, and volume===
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| Another virtue of regularity is that the Platonic solids all possess three concentric spheres:
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| * the [[circumscribed sphere]] that passes through all the vertices,
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| * the [[midsphere]] that is tangent to each edge at the midpoint of the edge, and
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| * the [[inscribed sphere]] that is tangent to each face at the center of the face.
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| The [[radius|radii]] of these spheres are called the ''circumradius'', the ''midradius'', and the ''inradius''. These are the distances from the center of the polyhedron to the vertices, edge midpoints, and face centers respectively. The circumradius ''R'' and the inradius ''r'' of the solid {''p'', ''q''} with edge length ''a'' are given by
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| :<math>R = \left({a\over 2}\right)\tan\frac{\pi}{q}\tan\frac{\theta}{2}</math>
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| :<math>r = \left({a\over 2}\right)\cot\frac{\pi}{p}\tan\frac{\theta}{2}</math>
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| where θ is the dihedral angle. The midradius ρ is given by
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| :<math>\rho = \left({a\over 2}\right)\frac{\cos(\pi/p)}{\sin(\pi/h)}</math>
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| where ''h'' is the quantity used above in the definition of the dihedral angle (''h'' = 4, 6, 6, 10, or 10). Note that the ratio of the circumradius to the inradius is symmetric in ''p'' and ''q'':
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| :<math>{R\over r} = \tan\frac{\pi}{p}\tan\frac{\pi}{q}=\frac{{\sqrt{{sin^{-2}{(\theta/2)}}-{cos^{2}{(\alpha/2)}}}}}{\sin{(\alpha/2)}}. </math>
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| The [[surface area]], ''A'', of a Platonic solid {''p'', ''q''} is easily computed as area of a regular ''p''-gon times the number of faces ''F''. This is:
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| :<math>A = \left({a\over 2}\right)^2 Fp\cot\frac{\pi}{p}.</math>
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| The [[volume]] is computed as ''F'' times the volume of the [[pyramid (geometry)|pyramid]] whose base is a regular ''p''-gon and whose height is the inradius ''r''. That is,
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| :<math>V = {1\over 3}rA.</math>
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| The following table lists the various radii of the Platonic solids together with their surface area and volume. The overall size is fixed by taking the edge length, ''a'', to be equal to 2.
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| {| class="wikitable"
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| |-
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| !Polyhedron<br><small>(''a'' = 2)</small>|| Inradius (''r'') || Midradius (ρ) || Circumradius (''R'') || Surface area (''A'') || Volume (''V'')
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| |-
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| |[[tetrahedron]] || <math>1\over {\sqrt 6}</math> || <math>1\over {\sqrt 2}</math> || <math>\sqrt{3\over 2}</math> || <math>4\sqrt 3</math> || <math>\frac{\sqrt 8}{3}</math>
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| |-
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| |[[cube]] || <math>1\,</math> || <math>\sqrt 2</math> || <math>\sqrt 3</math> || <math>24\,</math> || <math>8\,</math>
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| |-
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| |[[octahedron]] || <math>\sqrt{2\over 3}</math> || <math>1\,</math> || <math>\sqrt 2</math> || <math>8\sqrt 3</math> || <math>\frac{\sqrt {128}}{3}</math>
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| |-
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| |[[dodecahedron]] || <math>\frac{\varphi^2}{\xi}</math> || <math>\varphi^2</math> || <math>\sqrt 3\,\varphi</math> || <math>60\frac{\varphi}{\xi}</math> || <math>20\frac{\varphi^3}{\xi^2}</math>
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| |-
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| |[[icosahedron]] || <math>\frac{\varphi^2}{\sqrt 3}</math> || <math>\varphi</math> || <math>\xi\varphi</math> || <math>20\sqrt 3</math> || <math>\frac{20\varphi^2}{3}</math>
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| |}
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| The constants φ and ξ in the above are given by
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| :<math>\varphi = 2\cos{\pi\over 5} = \frac{1+\sqrt 5}{2}\qquad\xi = 2\sin{\pi\over 5} = \sqrt{\frac{5-\sqrt 5}{2}} = 5^{1/4}\varphi^{-1/2}.</math>
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| Among the Platonic solids, either the dodecahedron or the icosahedron may be seen as the best approximation to the sphere. The icosahedron has the largest number of faces and the largest dihedral angle, it hugs its inscribed sphere the most tightly, and its surface area to volume ratio is closest to that of a sphere of the same size (i.e. either the same surface area or the same volume.) The dodecahedron, on the other hand, has the smallest angular defect, the largest vertex solid angle, and it fills out its circumscribed sphere the most.
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| ==Symmetry==
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| ===Dual polyhedra===
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| [[Image:Dual Cube-Octahedron.svg|thumb|150px|right|A dual pair: cube and octahedron.]]
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| Every polyhedron has a [[dual polyhedron|dual (or "polar") polyhedron]] '''with faces and vertices interchanged'''. The dual of every Platonic solid is another Platonic solid, so that we can arrange the five solids into dual pairs.
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| * The tetrahedron is [[self-dual polyhedron|self-dual]] (i.e. its dual is another tetrahedron).
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| * The cube and the octahedron form a dual pair.
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| * The dodecahedron and the icosahedron form a dual pair.
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| If a polyhedron has Schläfli symbol {''p'', ''q''}, then its dual has the symbol {''q'', ''p''}. Indeed every combinatorial property of one Platonic solid can be interpreted as another combinatorial property of the dual.
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| One can construct the dual polyhedron by taking the vertices of the dual to be the centers of the faces of the original figure. Connecting the centers of adjacent faces in the original forms the edges of the dual and thereby interchanges the number of faces and vertices while maintaining the number of edges.
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| More generally, one can dualize a Platonic solid with respect to a sphere of radius ''d'' concentric with the solid. The radii (''R'', ρ, ''r'') of a solid and those of its dual (''R''*, ρ*, ''r''*) are related by
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| :<math>d^2 = R^\ast r = r^\ast R = \rho^\ast\rho.</math>
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| Dualizing with respect to the midsphere (''d'' = ρ) is often convenient because the midsphere has the same relationship to both polyhedra. Taking ''d''<sup>2</sup> = ''Rr'' yields a dual solid with the same circumradius and inradius (i.e. ''R''* = ''R'' and ''r''* = ''r'').
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| ===Symmetry groups===
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| In mathematics, the concept of [[symmetry]] is studied with the notion of a [[group (mathematics)|mathematical group]]. Every polyhedron has an associated [[symmetry group]], which is the set of all transformations ([[Euclidean isometry|Euclidean isometries]]) which leave the polyhedron invariant. The [[order (group theory)|order]] of the symmetry group is the number of symmetries of the polyhedron. One often distinguishes between the ''full symmetry group'', which includes [[reflection (mathematics)|reflections]], and the ''proper symmetry group'', which includes only [[rotation (mathematics)|rotations]].
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| The symmetry groups of the Platonic solids are known as [[polyhedral group]]s (which are a special class of the [[point groups in three dimensions]]). The high degree of symmetry of the Platonic solids can be interpreted in a number of ways. Most importantly, the vertices of each solid are all equivalent under the [[group action|action]] of the symmetry group, as are the edges and faces. One says the action of the symmetry group is [[transitive action|transitive]] on the vertices, edges, and faces. In fact, this is another way of defining regularity of a polyhedron: a polyhedron is ''regular'' if and only if it is [[vertex-uniform]], [[edge-uniform]], and [[face-uniform]].
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| There are only three symmetry groups associated with the Platonic solids rather than five, since the symmetry group of any polyhedron coincides with that of its dual. This is easily seen by examining the construction of the dual polyhedron. Any symmetry of the original must be a symmetry of the dual and vice-versa. The three polyhedral groups are:
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| * the [[tetrahedral group]] ''T'',
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| * the [[octahedral group]] ''O'' (which is also the symmetry group of the cube), and
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| * the [[icosahedral group]] ''I'' (which is also the symmetry group of the dodecahedron).
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| The orders of the proper (rotation) groups are 12, 24, and 60 respectively – precisely twice the number of edges in the respective polyhedra. The orders of the full symmetry groups are twice as much again (24, 48, and 120). See (Coxeter 1973) for a derivation of these facts. All Platonic solids except the tetrahedron are ''centrally symmetric,'' meaning they are preserved under [[reflection through the origin]].
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| The following table lists the various symmetry properties of the Platonic solids. The symmetry groups listed are the full groups with the rotation subgroups given in parenthesis (likewise for the number of symmetries). [[Wythoff's construction|Wythoff's kaleidoscope construction]] is a method for constructing polyhedra directly from their symmetry groups. They are listed for reference Wythoff's symbol for each of the Platonic solids.
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| {| class="wikitable"
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| |-
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| !rowspan=2|Polyhedron
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| !rowspan=2|[[Schläfli symbol|Schläfli<BR>symbol]]
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| !rowspan=2|[[Wythoff symbol|Wythoff<BR>symbol]]
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| !rowspan=2|[[Dual polyhedron|Dual<BR>polyhedron]]
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| !colspan=5|[[Symmetry group]] (Reflection, rotation)
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| |-
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| ![[Polyhedral group|Polyhedral]]
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| ![[Schönflies notation|Schönflies]]
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| ![[Coxeter notation|Coxeter]]
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| ![[Orbifold notation|Orbifold]]
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| !Order
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| |- align=center
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| |[[tetrahedron]]
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| |{3, 3} || <nowiki>3 | 2 3</nowiki> || tetrahedron
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| | [[tetrahedral symmetry|Tetrahedral]] [[File:Tetrahedral_reflection_domains.png|40px]]
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| |''T''<sub>d</sub>, ''T''
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| | [3,3], [3,3]<sup>+</sup>
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| | *332, 332
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| | 24, 12
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| |- align=center
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| |[[cube]]
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| |{4, 3} || <nowiki>3 | 2 4</nowiki> || octahedron
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| | rowspan=2 | [[octahedral symmetry|Octahedral]] [[File:Octahedral_reflection_domains.png|40px]]
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| | rowspan=2 | ''O''<sub>h</sub>, ''O''
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| | rowspan=2 | [4,3], [4,3]<sup>+</sup>
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| | rowspan=2 | *432, 432
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| | rowspan=2 | 48, 24
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| |- align=center
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| |[[octahedron]]
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| |{3, 4} || <nowiki>4 | 2 3</nowiki> || cube
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| |- align=center
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| |[[dodecahedron]]
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| |{5, 3} || <nowiki>3 | 2 5</nowiki> || icosahedron
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| | rowspan=2 | [[icosahedral symmetry|Icosahedral]] [[File:Icosahedral_reflection_domains.png|40px]]
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| | rowspan=2 | ''I''<sub>h</sub>, ''I''
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| | rowspan=2 | [5,3], [5,3]<sup>+</sup>
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| | rowspan=2 | *532, 532
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| | rowspan=2 | 120, 60
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| |- align=center
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| |[[icosahedron]]
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| |{3, 5} || <nowiki>5 | 2 3</nowiki> || dodecahedron
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| |}
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| ==In nature and technology==
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| The tetrahedron, cube, and octahedron all occur naturally in [[crystal structure]]s. These by no means exhaust the numbers of possible forms of crystals. However, neither the regular icosahedron nor the regular dodecahedron are amongst them. One of the forms, called the [[pyritohedron]] (named for the group of [[pyrite|minerals]] of which it is typical) has twelve pentagonal faces, arranged in the same pattern as the faces of the regular dodecahedron. The faces of the pyritohedron are, however, not regular, so the pyritohedron is also not regular.
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| [[Image:Circogoniaicosahedra ekw.jpg|left|frame|Circogonia icosahedra, a species of [[Radiolaria]], shaped like a regular icosahedron.]]
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| In the early 20th century, [[Ernst Haeckel]] described (Haeckel, 1904) a number of species of [[Radiolaria]], some of whose skeletons are shaped like various regular polyhedra. Examples include ''Circoporus octahedrus'', ''Circogonia icosahedra'', ''Lithocubus geometricus'' and ''Circorrhegma dodecahedra''. The shapes of these creatures should be obvious from their names.
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| Many [[virus]]es, such as the [[herpes]] virus, have the shape of a regular icosahedron. Viral structures are built of repeated identical [[protein]] subunits and the icosahedron is the easiest shape to assemble using these subunits. A regular polyhedron is used because it can be built from a single basic unit protein used over and over again; this saves space in the viral [[genome]].
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| In [[meteorology]] and [[climatology]], global numerical models of atmospheric flow are of increasing interest which employ [[geodesic grid]]s that are based on an icosahedron (refined by [[triangulation]]) instead of the more commonly used [[longitude]]/[[latitude]] grid. This has the advantage of evenly distributed spatial resolution without [[Mathematical singularity|singularities]] (i.e. the [[poles]]) at the expense of somewhat greater numerical difficulty.
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| Geometry of [[space frame]]s is often based on platonic solids. In MERO system, Platonic solids are used for naming convention of various space frame configurations. For example ½O+T refers to a configuration made of one half of octahedron and a tetrahedron.
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| Several [[Platonic hydrocarbons]] have been synthesised, including [[cubane]] and [[dodecahedrane]].
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| Platonic solids are often used to make [[dice]], because dice of these shapes can be made fair ([[fair dice]]). 6-sided dice are very common, but the other numbers are commonly used in [[role-playing game]]s. Such dice are commonly referred to as d''n'' where ''n'' is the number of faces (d8, d20, etc.); see [[dice notation]] for more details.
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| [[Image:BluePlatonicDice.jpg|thumb|500px|center|[[Polyhedral dice]] are often used in [[role-playing games]].]]
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| These shapes frequently show up in other games or puzzles. Puzzles similar to a [[Rubik's Cube]] come in all five shapes – see [[magic polyhedra]].
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| === Liquid crystals with symmetries of Platonic solids ===
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| For the intermediate material phase called [[liquid crystals]], the existence of such symmetries was first proposed in 1981 by [[Hagen Kleinert|H. Kleinert]] and K. Maki and their structure was analyzed in.<ref>
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| {{Cite journal| title = Lattice Textures in Cholesteric Liquid Crystals
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| | author = [[Hagen Kleinert|Kleinert, H.]] and Maki, K.
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| | journal = Fortschritte der Physik
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| | volume = 29
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| | issue = 5
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| | pages = 219–259
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| | year = 1981
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| | doi = 10.1002/prop.19810290503
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| | url = http://www.physik.fu-berlin.de/~kleinert/75/75.pdf| ref = harv| postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}
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| </ref> See the review article [http://chemgroups.northwestern.edu/seideman/Publications/The%20liquid-crystalline%20blue%20phases.pdf here].
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| In aluminum the icosahedral structure was discovered three years after this by [[Dan Shechtman]], which earned him the Nobel Prize in Chemistry in 2011.
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| ==Related polyhedra and polytopes==
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| ===Uniform polyhedra===
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| There exist four regular polyhedra which are not convex, called [[Kepler–Poinsot polyhedra]]. These all have [[icosahedral symmetry]] and may be obtained as [[stellation]]s of the dodecahedron and the icosahedron.
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| {| style="float: right; margin-left: 1em; text-align: center; border-collapse: collapse; border: 1pt solid #aaa;"
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| |-
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| |style="padding: 3pt;"|[[Image:Cuboctahedron.svg|80px]]<br />[[cuboctahedron]]
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| |-
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| |style="padding: 3pt;"|[[Image:Icosidodecahedron.svg|80px]]<br />[[icosidodecahedron]]
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| |}
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| The next most regular convex polyhedra after the Platonic solids are the [[cuboctahedron]], which is a [[rectification (geometry)|rectification]] of the cube and the octahedron, and the [[icosidodecahedron]], which is a rectification of the dodecahedron and the icosahedron (the rectification of the self-dual tetrahedron is a regular octahedron). These are both ''quasi-regular'', meaning that they are vertex- and edge-uniform and have regular faces, but the faces are not all congruent (coming in two different classes). They form two of the thirteen [[Archimedean solid]]s, which are the convex [[uniform polyhedron|uniform polyhedra]] with polyhedral symmetry.
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| The uniform polyhedra form a much broader class of polyhedra. These figures are vertex-uniform and have one or more types of [[regular polygon|regular]] or [[star polygon]]s for faces. These include all the polyhedra mentioned above together with an infinite set of [[prism (geometry)|prisms]], an infinite set of [[antiprism]]s, and 53 other non-convex forms.
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| The [[Johnson solid]]s are convex polyhedra which have regular faces but are not uniform.
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| ===Regular tessellations===
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| The three [[regular tessellation]]s of the plane are closely related to the Platonic solids. Indeed, one can view the Platonic solids as the five regular tessellations of the [[sphere]]. This is done by projecting each solid onto a concentric sphere. The faces project onto regular [[spherical polygon]]s which exactly cover the sphere. One can show that every regular tessellation of the sphere is characterized by a pair of integers {''p'', ''q''} with 1/''p'' + 1/''q'' > 1/2. Likewise, a regular tessellation of the plane is characterized by the condition 1/''p'' + 1/''q'' = 1/2. There are three possibilities:
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| * {4, 4} which is a [[square tiling]],
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| * {3, 6} which is a [[triangular tiling]], and
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| * {6, 3} which is a [[hexagonal tiling]] (dual to the triangular tiling).
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| In a similar manner one can consider regular tessellations of the [[hyperbolic geometry|hyperbolic plane]]. These are characterized by the condition 1/''p'' + 1/''q'' < 1/2. There is an infinite family of such tessellations.
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| ===Higher dimensions===
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| In more than three dimensions, polyhedra generalize to [[polytope]]s, with higher-dimensional convex [[regular polytope]]s being the equivalents of the three-dimensional Platonic solids.
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| In the mid-19th century the Swiss mathematician [[Ludwig Schläfli]] discovered the four-dimensional analogues of the Platonic solids, called [[convex regular 4-polytope]]s. There are exactly six of these figures; five are analogous to the Platonic solids, while the sixth one, the [[24-cell]], has one lower-dimension analogue (truncation of a simplex-faceted polyhedron that has simplices for ridges and is self-dual): the [[hexagon]].
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| In all dimensions higher than four, there are only three convex regular polytopes: the [[simplex]], the [[hypercube]], and the [[cross-polytope]].<ref>{{citation|title=Regular Polytopes|first=H. S. M.|last=Coxeter|authorlink=Harold Scott MacDonald Coxeter|publisher=Dover|year=1973|page=136}}.</ref> In three dimensions, these coincide with the tetrahedron, the cube, and the octahedron.
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| ==See also==
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| * [[Regular skew polyhedron]]
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| * [[Archimedean solid]]
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| * [[Catalan solid]]
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| * [[Johnson solid]]
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| * [[Kepler-Poinsot polyhedron|Kepler solids]]
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| * [[List of regular polytopes]]
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| * [[Metatron's Cube]]
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| * [[Project Euler]] uses platonic solids to denote scoring levels.
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| * [[Regular polytope]]s
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| * [[Toroidal polyhedron]]
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| ==Notes==
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| <references />
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| ==References==
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| * {{cite journal
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| | last = Atiyah
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| | first = Michael
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| | authorlink = Michael Atiyah
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| | coauthors = and Sutcliffe, Paul
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| | year = 2003
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| | title = Polyhedra in Physics, Chemistry and Geometry
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| | journal = Milan J. Math
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| | volume = 71
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| | pages = 33–58
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| | doi = 10.1007/s00032-003-0014-1
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| | ref = harv
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| }}
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| * {{cite book
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| | first = Boyer
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| | last = Carl
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| | coauthors = Merzbach, Uta
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| | year = 1989
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| | title = A History of Mathematics
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| | edition = 2nd
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| | publisher = Wiley
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| | isbn = 0-471-54397-7
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| }}
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| * {{cite book
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| | first = H. S. M.
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| | last = Coxeter
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| | authorlink = H. S. M. Coxeter
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| | year = 1973
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| | title = [[Regular Polytopes (book)|Regular Polytopes]]
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| | edition = 3rd
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| | publisher = Dover Publications
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| | location = New York
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| | isbn = 0-486-61480-8
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| }}
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| * {{cite book
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| | author = [[Euclid]]
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| | year = 1956
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| | title = The Thirteen Books of Euclid's Elements, Books 10–13
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| | editor-first = Thomas L. | editor-last = Heath | editor-link = Thomas Little Heath
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| | edition = 2nd unabr.
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| | publisher = Dover Publications
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| | location = New York
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| | isbn = 0-486-60090-4
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| }}
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| * Haeckel, E. (1904). ''Kunstformen der Natur''. Available as Haeckel, E. (1998); ''Art forms in nature'', Prestel USA. ISBN 3-7913-1990-6, or online at [http://caliban.mpiz-koeln.mpg.de/~stueber/haeckel/kunstformen/natur.html].
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| * {{cite book
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| | first = Hermann
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| | last = Weyl
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| | authorlink = Hermann Weyl
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| | year = 1952
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| | title = Symmetry
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| | publisher = Princeton University Press
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| | location = Princeton, NJ
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| | isbn = 0-691-02374-3
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| }}
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| * "Strena seu de nive sexangula" (On the Six-Cornered Snowflake), 1611 paper by Kepler which discussed the reason for the six-angled shape of the snow crystals and the forms and symmetries in nature. Talks about platonic solids.
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| * {{Cite web|title=New Explorations with The Moon Model|first=Laurence|last=Hecht|first2=Charles B.|last2=Stevens|date=Fall 2004|page=58|work=21st Century Science and Technology|url=http://www.21stcenturysciencetech.com/Articles%202005/MoonModel_F04.pdf|ref=harv|postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}
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| * {{cite book | author= Anthony Pugh | year= 1976 | title= Polyhedra: A visual approach | publisher= University of California Press Berkeley | location= California | isbn= 0-520-03056-7 }}
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| ==External links==
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| {{commons category|Platonic solids}}
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| * [http://www.encyclopediaofmath.org/index.php/Platonic_solids ''Platonic solids'' at Encyclopaedia of Mathematics]
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| * {{Mathworld | urlname=PlatonicSolid | title=Platonic solid }}
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| * [http://aleph0.clarku.edu/~djoyce/java/elements/bookXIII/propXIII13.html Book XIII] of Euclid's ''Elements''.
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| * [http://ibiblio.org/e-notes/3Dapp/Convex.htm Interactive 3D Polyhedra] in Java
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| * [http://kovacsv.github.com/JSModeler/documentation/examples/solids.html Solid Body Viewer] is an interactive 3D polyhedron viewer which allows you to save the model in svg, stl or obj format.
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| * [http://www.mat.puc-rio.br/~hjbortol/mathsolid/mathsolid_en.html Interactive Folding/Unfolding Platonic Solids] in Java
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| * [http://www.software3d.com/Platonic.php Paper models of the Platonic solids] created using nets generated by [[Stella (software)|Stella]] software
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| * [http://www.korthalsaltes.com/cuadros.php?type=p Platonic Solids] Free paper models(nets)
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| * {{cite web|title=Platonic Solids|url=http://www.numberphile.com/videos/platonic_solids.html|work=Numberphile|publisher=[[Brady Haran]]|author=Grime, James|coauthors=Steckles, Katie}}
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| * [http://www.ldlewis.com/Teaching-Mathematics-with-Art/Polyhedra.html Teaching Math with Art] student-created models
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| * [http://www.ldlewis.com/Teaching-Mathematics-with-Art/instructions-for-polyhedra-project.html Teaching Math with Art] teacher instructions for making models
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| * [http://www.bru.hlphys.jku.at/surf/Kepler_Model.html Frames of Platonic Solids] images of [[algebraic surface]]s
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| * [http://whistleralley.com/polyhedra/platonic.htm Platonic Solids] with some [http://whistleralley.com/polyhedra/derivations.htm formula derivations]
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| {{Polyhedron navigator}}
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| {{DEFAULTSORT:Platonic Solid}}
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| [[Category:Platonic solids| ]]
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