Editing Platonic solid

{{Short description|Any of the five regular polyhedra}}

In [[geometry]], a '''Platonic solid''' is a [[Convex polytope|convex]], [[regular polyhedron]] in [[three-dimensional space|three-dimensional Euclidean space]]. Being a regular polyhedron means that the [[face (geometry)|faces]] are [[congruence (geometry)|congruent]] (identical in shape and size) [[regular polygon]]s (all [[angle]]s congruent and all [[edge (geometry)|edge]]s congruent), and the same number of faces meet at each [[Vertex (geometry)|vertex.]] There are only five such polyhedra:

{| class="wikitable center" style="text-align: center"
|- 
! scope="col" style="font-weight:normal" | [[Regular tetrahedron|Tetrahedron]]
! scope="col" style="font-weight:normal" | [[Cube]]
! scope="col" style="font-weight:normal" | [[Regular octahedron|Octahedron]]
! scope="col" style="font-weight:normal" | [[Regular dodecahedron|Dodecahedron]]
! scope="col" style="font-weight:normal" | [[Regular icosahedron|Icosahedron]]<!--PLEASE DO NOT SWAP THE DODECAHEDRON AND ICOSAHEDRON, THE NAMES CURRENTLY GIVEN ARE CORRECT-->
|-
| Four faces
| Six faces
| Eight faces
| Twelve faces
| Twenty faces
|-
| [[File:Tetrahedron.svg|80px]]
| [[File:Hexahedron.svg|80px]]
| [[File:Octahedron.svg|80px]]
| [[File:Dodecahedron.svg|80px]]
| [[File:Icosahedron.svg|80px]]
|- style="font-size: 0.85rem"
| {{cslist | [[:File:tetrahedron.gif|Animation]] | [[:File:tetrahedron.stl|3D model]] }}
| {{cslist | [[:File:hexahedron.gif|Animation]] | [[:File:hexahedron.stl|3D model]] }}
| {{cslist | [[:File:octahedron.gif|Animation]] | [[:File:octahedron.stl|3D model]] }}
| {{cslist | [[:File:dodecahedron.gif|Animation]] | [[:File:dodecahedron.stl|3D model]] }}
| {{cslist | [[:File:icosahedron.gif|Animation]] | [[:File:Icosahedron.stl|3D model]] }}
|}

[[Geometer]]s have studied the Platonic solids for thousands of years.<ref>Gardner (1987): [[Martin Gardner]] wrote a popular account of the five solids in his December 1958 [[Mathematical Games column]] in Scientific American.</ref> They are named for the ancient Greek philosopher [[Plato]], who hypothesized in one of his dialogues, the ''[[Timaeus (dialogue)|Timaeus]]'', that the [[classical element]]s were made of these regular solids.<ref name="The Stanford Encyclopedia of Philosophy">{{cite encyclopedia |last=Zeyl|first=Donald|encyclopedia=The Stanford Encyclopedia of Philosophy|title=Plato's Timaeus|year=2019|url=http://plato.stanford.edu/entries/plato-timaeus/}}</ref>

== 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.

==Cartesian coordinates==
For Platonic solids centered at the origin, simple [[Cartesian coordinate system|Cartesian coordinates]] of the vertices are given below. The Greek letter <math>\varphi</math> is used to represent the [[golden ratio]] <math>\frac{1+\sqrt{5}}{2}\approx 1.6180</math>.

{| class=wikitable style="text-align:center;"
|+ Parameters
! scope="row" | Figure
! colspan=2 | Tetrahedron !! Octahedron !! Cube !! colspan=2 | Icosahedron !! colspan=2 | Dodecahedron
|-
! scope="row" | Faces
|colspan=2|4||8||6||colspan=2|20||colspan=2|12
|-
! scope="row" | Vertices
|colspan=2|4||6 (2&nbsp;×&nbsp;3)||8||colspan=2|12 (4&nbsp;×&nbsp;3)||colspan=2|20 (8&nbsp;+&nbsp;4&nbsp;×&nbsp;3)
|-
! Position|| 1||2 || |||| 1||2|| 1||2
|- style="vertical-align:top;"
! scope="row" style="vertical-align:middle;" | Vertex <br/>coordinates
| {{nowrap|(1, 1, 1)}}<BR/>{{nowrap|(1, −1, −1)}}<BR/>{{nowrap|(−1, 1, −1)}}<BR/>{{nowrap|(−1, −1, {{fsp}}1)}}
| {{nowrap|(−1, −1, −1)}}<BR/>{{nowrap|(−1, 1, 1)}}<BR/>{{nowrap|({{fsp}}1, −1, {{fsp}}1)}}<BR/>{{nowrap|({{fsp}}1, {{fsp}}1, −1)}}
| {{nowrap|(±1, {{fsp}}0, {{fsp}}0)}}<BR/>{{nowrap|({{fsp}}0, ±1, {{fsp}}0)}}<BR/>{{nowrap|({{fsp}}0, {{fsp}}0, ±1)}}
| {{nowrap|(±1, ±1, ±1)}}
| {{nowrap|({{fsp}}0, ±1, ±''φ'')}}<BR/>{{nowrap|(±1, ±''φ'', {{fsp}}0)}}<BR/>{{nowrap|(±''φ'', {{fsp}}0, ±1)}}||{{nowrap|({{fsp}}0, ±''φ'', ±1)}}<BR/>{{nowrap|(±''φ'', ±1, {{fsp}}0)}}<BR/>{{nowrap|(±1, {{fsp}}0, ±''φ'')}}
| {{nowrap|(±1, ±1, ±1)}}<BR/>{{nowrap|({{fsp}}0, ±{{sfrac|1|''φ''}}, ±''φ'')}}<BR/>{{nowrap|(±{{sfrac|1|''φ''}}, ±''φ'', {{fsp}}0)}}<BR/>{{nowrap|(±''φ'', {{fsp}}0, ±{{sfrac|1|''φ''}})}}
| {{nowrap|(±1, ±1, ±1)}}<BR/>{{nowrap|({{fsp}}0, ±''φ'', ±{{sfrac|1|''φ''}})}}<BR/>{{nowrap|(±''φ'', ±{{sfrac|1|''φ''}}, {{fsp}}0)}}<BR/>{{nowrap|(±{{sfrac|1|''φ''}},{{fsp}}}} 0, ±''φ'')
|}

The coordinates for the tetrahedron, dodecahedron, and icosahedron are given in two positions such that each can be deduced from the other: in the case of the tetrahedron, by changing all coordinates of sign ([[central symmetry]]), or, in the other cases, by exchanging two coordinates ([[reflection (geometry)|reflection]] with respect to any of the three diagonal planes).

These coordinates reveal certain relationships between the Platonic solids: the vertices of the tetrahedron represent half of those of the cube, as {4,3} or {{CDD|node_1|4|node|3|node}}, one of two sets of 4 vertices in dual positions, as h{4,3} or {{CDD|node_h|4|node|3|node}}. Both tetrahedral positions make the compound [[stellated octahedron]].

The coordinates of the icosahedron are related to two alternated sets of coordinates of a nonuniform [[truncated octahedron]], t{3,4} or {{CDD|node_1|3|node_1|4|node}}, also called a ''[[Icosahedron#Pyritohedral symmetry|snub octahedron]]'', as s{3,4} or {{CDD|node_h|3|node_h|4|node}}, and seen in the [[compound of two icosahedra]].

Eight of the vertices of the dodecahedron are shared with the cube. Completing all orientations leads to the [[compound of five cubes]].

== Combinatorial properties ==
A convex polyhedron is a Platonic solid if and only if all three of the following requirements are met.
* All of its faces are [[Congruence (geometry)|congruent]] convex [[regular polygon]]s.
* None of its faces intersect except at their edges.
* The same number of faces meet at each of its [[vertex (geometry)|vertices]].

Each Platonic solid can therefore be assigned a pair {''p'',&nbsp;''q''} of integers, where ''p'' is the number of edges (or, equivalently, vertices) of each face, and ''q'' is the number of faces (or, equivalently, edges) that meet at each vertex. This pair {''p'',&nbsp;''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.

{| class="wikitable sortable"
|+Properties of Platonic solids
|-
!scope="col" colspan=2 | Polyhedron
!scope="col" |[[Vertex (geometry)|Vertices]]
!scope="col" |[[Edge (geometry)|Edges]]
!scope="col" |[[Face (geometry)|Faces]]
!scope="col" |[[Schläfli symbol]]
!scope="col" |[[Vertex configuration]]
|- align=center
|scope="row"| [[Regular tetrahedron]]
| [[Image:tetrahedron.svg|50px|Tetrahedron]]
| 4 || 6 || 4 || {3, 3} || 3.3.3
|- align=center
|scope="row"| [[cube]]
| [[Image:hexahedron.svg|50px|Hexahedron (cube)]]
| 8 || 12 || 6 || {4, 3} || 4.4.4
|- align=center
|scope="row"| [[Regular octahedron]]
| [[Image:octahedron.svg|50px|Octahedron]]
| 6 || 12 || 8 || {3, 4} || 3.3.3.3
|- align=center
|scope="row"| [[Regular dodecahedron|dodecahedron]]<!--PLEASE DO NOT SWAP THE DODECAHEDRON AND ICOSAHEDRON, IT IS CORRECT-->
| [[Image:Dodecahedron.svg|50px|Dodecahedron]]
| 20 || 30 || 12 || {5, 3} || 5.5.5
|- align=center
|scope="row"| [[Regular icosahedron|icosahedron]]
| [[Image:icosahedron.svg|50px|Icosahedron]]
| 12 || 30 || 20 || {3, 5} || 3.3.3.3.3
|}

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:

<math display="block">pF = 2E = qV.\,</math>

The other relationship between these values is given by [[Euler characteristic|Euler's formula]]:

<math display="block">V - E + F = 2.\,</math>

This can be proved in many ways. Together these three relationships completely determine ''V'', ''E'', and ''F'':

<math display="block">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>

Swapping ''p'' and ''q'' interchanges ''F'' and ''V'' while leaving ''E'' unchanged. For a geometric interpretation of this property, see {{section link||Dual polyhedra}}.

=== As a configuration===
The elements of a polyhedron can be expressed in a [[Configuration (polytope)#Higher dimensions|configuration matrix]]. The rows and columns correspond to vertices, edges, and faces. The diagonal numbers say how many of each element occur in the whole polyhedron. The nondiagonal numbers say how many of the column's element occur in or at the row's element. Dual pairs of polyhedra have their configuration matrices rotated 180 degrees from each other.<ref>Coxeter, Regular Polytopes, sec 1.8 Configurations</ref>

{| class=wikitable
! {p,q}
! colspan=5 | Platonic configurations
|- style="vertical-align:top;"
! [[Group order]]: <br/>''g'' = 8''pq''/(4 − (''p'' − 2)(''q'' − 2))
! ''g'' = 24
! colspan=2 | ''g'' = 48
! colspan=2 | ''g'' = 120
|-
|
{| class=wikitable style="margin: auto;"
! !! v !! e !! f
|- align=center
!v
| ''g''/2''q'' || ''q'' || ''q''
|- align=center
! e
| 2 || ''g''/4 || 2
|- align=center
! f
| ''p'' || ''p'' || ''g''/2''p''
|}

| style="background-color:#e0f0e0;" |
{| class=wikitable
|+ {3,3}
|- align=center
| 4 || 3 || 3
|- align=center
| 2 || 6 || 2
|- align=center
| 3 || 3 || 4
|}
| style="background-color:#f0e0e0;" |
{| class=wikitable
|+ {3,4}
|- align=center
| 6 ||  4 || 4
|- align=center
| 2 || 12 || 2
|- align=center
| 3 ||  3 || 8
|}
| style="background-color:#e0e0f0;" |
{| class=wikitable
|+ {4,3}
|- align=center
| 8 ||  3 || 3
|- align=center
| 2 || 12 || 2
|- align=center
| 4 ||  4 || 6
|}
| style="background-color:#f0e0e0" |
{| class=wikitable
|+ {3,5}
|- align=center
| 12 ||  5 ||  5
|- align=center
|  2 || 30 ||  2
|- align=center
|  3 ||  3 || 20
|}
| style="background-color:#e0e0f0;" |
{| class=wikitable
|+ {5,3}
|- align=center
| 20||  3 ||  3
|- align=center
|  2|| 30 ||  2
|- align=center
|  5||  5 || 12
|}
|}

== Classification ==
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 requires an explicit construction.

=== Geometric proof ===
{| class="wikitable floatright" style="text-align:center"
|+ Polygon nets around a vertex
|- style="vertical-align:bottom;"
| [[File:Polyiamond-3-1.svg|80px]]<BR/>{3,3}<BR/>Defect 180°
| [[File:Polyiamond-4-1.svg|80px]]<BR/>{3,4}<BR/>Defect 120°
| [[File:Polyiamond-5-4.svg|80px]]<BR/>{3,5}<BR/>Defect 60°
| style="background-color:#e0e0ff;" | [[File:Polyiamond-6-11.svg|80px]]<BR/>{3,6}<BR/>Defect 0°
|- style="vertical-align:bottom;"
| [[File:TrominoV.svg|80px]]<BR/>{4,3}<BR/>Defect 90°
| style="background-color:#e0e0ff;" | [[File:Square tiling vertfig.svg|80px]]<BR/>{4,4}<BR/>Defect 0°
| [[File:Pentagon_net.svg|80px]]<BR/>{5,3}<BR/>Defect 36°
| style="background-color:#e0e0ff;" | [[File:Hexagonal tiling vertfig.svg|80px]]<BR/>{6,3}<BR/>Defect 0°
|-
| colspan=4 | A vertex needs at least 3 faces, and an [[angle defect]]. <BR/>A 0° angle defect will fill the Euclidean plane with a regular tiling. <BR/>By [[angular defect#Descartes' theorem|Descartes' theorem]], the number of vertices is 720°/''defect''.
|}

The following geometric argument is very similar to the one given by [[Euclid]] in the [[Euclid's Elements|''Elements'']]:

{{ordered list
| Each vertex of the solid must be a vertex for at least three faces.
| At each vertex of the solid, the total, among the adjacent faces, of the angles between their respective adjacent sides must be strictly less than 360°. The amount less than 360° is called an [[angle defect]].
| The angles at all vertices of all faces of a Platonic solid are identical: each vertex of each face must contribute less than {{sfrac|360°|3}}&nbsp;=&nbsp;120°.
| 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. For these different shapes of faces the following holds:
; [[Triangle|Triangular]] faces: Each vertex of a regular triangle is 60°, so a shape may have three, four, or five triangles meeting at a vertex; these are the tetrahedron, octahedron, and icosahedron respectively.
; [[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.
; [[Pentagon]]al faces: Each vertex is 108°; again, only one arrangement of three faces at a vertex is possible, the dodecahedron.

Altogether this makes five possible Platonic solids.
}}

=== Topological proof ===
A purely [[topology|topological]] proof can be made using only combinatorial information about the solids. The key is [[Euler characteristic|Euler's observation]] that ''V''&nbsp;−&nbsp;''E''&nbsp;+&nbsp;''F''&nbsp;=&nbsp;2, and the fact that ''pF''&nbsp;=&nbsp;2''E''&nbsp;=&nbsp;''qV'', 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

{{Hamiltonian_platonic_graphs.svg}}

<math display="block">\frac{2E}{q} - E + \frac{2E}{p} = 2.</math>

Simple algebraic manipulation then gives

<math display="block">{1 \over q} + {1 \over p}= {1 \over 2} + {1 \over E}.</math>

Since ''E'' is strictly positive we must have

<math display="block">\frac{1}{q} + \frac{1}{p} > \frac{1}{2}.</math>

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'',&nbsp;''q''}:
{{block indent|{3,&nbsp;3}, {4,&nbsp;3}, {3,&nbsp;4}, {5,&nbsp;3}, {3,&nbsp;5}.}}

== Geometric properties ==
=== Angles ===
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

<math display="block">\sin(\theta/2) = \frac{\cos(\pi/q)}{\sin(\pi/p)}.</math>

This is sometimes more conveniently expressed in terms of the [[tangent (trigonometric function)|tangent]] by

<math display="block">\tan(\theta/2) = \frac{\cos(\pi/q)}{\sin(\pi/h)}.</math>

The quantity ''h'' (called the [[Coxeter number]]) is 4, 6, 6, 10, and 10 for the tetrahedron, cube, octahedron, dodecahedron, and icosahedron respectively.

The [[angular deficiency]] at the vertex of a polyhedron is the difference between the sum of the face-angles at that vertex and 2{{pi}}. The defect, ''δ'', at any vertex of the Platonic solids {''p'',''q''} is

<math display="block">\delta = 2\pi - q\pi\left(1 - {2 \over p}\right).</math>

By a theorem of Descartes, this is equal to 4{{pi}} divided by the number of vertices (i.e. the total defect at all vertices is 4{{pi}}).

The three-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

<math display="block">\Omega = q\theta - (q - 2)\pi.\,</math>

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.

The solid angle of a face subtended from the center of a platonic solid is equal to the solid angle of a full sphere (4{{pi}} steradians) divided by the number of faces. This is equal to the angular deficiency of its dual.

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'' = {{sfrac|1 + {{sqrt|5}}|2}} is the [[golden ratio]].

{| class="wikitable" style="text-align:center"
! Polyhedron
! [[Dihedral angle|Dihedral <br/>angle]] <br/>(''θ'')
! tan&nbsp;{{sfrac|''θ''|2}}
! [[Defect (geometry)|Defect]] <br/>(''δ'')
! Vertex [[solid angle]] (''Ω'')
! Face <br/>solid <br/>angle
|-
| [[tetrahedron]] || 70.53° || <math>1 \over {\sqrt 2}</math> || <math>\pi</math>
| <math>\arccos\left(\frac{23}{27}\right) \quad \approx 0.551286</math>
| <math>\pi</math>
|-
| [[cube]] || 90° || <math>1</math> || <math>\pi \over 2</math>
| <math>\frac{\pi}{2} \quad \approx 1.57080</math>
| <math>2\pi \over 3</math>
|-
| [[octahedron]] || 109.47° || <math>\sqrt 2</math> || <math>{2\pi} \over 3</math>
| <math>4\arcsin\left({1 \over 3}\right) \quad \approx 1.35935</math>
| <math>\pi \over 2</math>
|-
| [[Regular dodecahedron|dodecahedron]] || 116.57° || <math>\varphi</math> || <math>\pi \over 5</math>
| <math>\pi - \arctan\left(\frac{2}{11}\right) \quad \approx 2.96174</math>
| <math>\pi \over 3</math>
|-
| [[Regular icosahedron|icosahedron]] || 138.19° || <math>\varphi^2</math> || <math>\pi \over 3</math>
| <math>2\pi - 5\arcsin\left({2\over 3}\right) \quad \approx 2.63455</math>
| <math>\pi \over 5</math>
|}

=== Radii, area, and volume ===
Another virtue of regularity is that the Platonic solids all possess three concentric spheres:

* the [[circumscribed sphere]] that passes through all the vertices,
* the [[midsphere]] that is tangent to each edge at the midpoint of the edge, and
* the [[inscribed sphere]] that is tangent to each face at the center of the face.

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'',&nbsp;''q''} with edge length ''a'' are given by

<math display="block">\begin{align}
  R &= \frac{a}{2} \tan\left(\frac{\pi}{q}\right)\tan\left(\frac{\theta}{2}\right) \\[3pt]
  r &= \frac{a}{2} \cot\left(\frac{\pi}{p}\right)\tan\left(\frac{\theta}{2}\right)
\end{align}</math>

where ''θ'' is the dihedral angle. The midradius ''ρ'' is given by

<math display="block">\rho = \frac{a}{2} \cos\left(\frac{\pi}{p}\right)\,{\csc}\biggl(\frac{\pi}{h}\biggr)</math>

where ''h'' is the quantity used above in the definition of the dihedral angle (''h'' = 4, 6, 6, 10, or 10). The ratio of the circumradius to the inradius is symmetric in ''p'' and ''q'':

<math display="block">\frac{R}{r} =
  \tan\left(\frac{\pi}{p}\right) \tan\left(\frac{\pi}{q}\right) =
  \frac{{\sqrt{{\csc^{2}}\Bigl(\frac\theta2\Bigr) - {\cos^{2}}\Bigl(\frac\alpha2\Bigr)}}}{\sin\Bigl(\frac{\alpha}{2}\Bigr)}.
</math>

The [[surface area]], ''A'', of a Platonic solid {''p'',&nbsp;''q''} is easily computed as area of a regular ''p''-gon times the number of faces ''F''. This is:

<math display="block">A = \biggl(\frac{a}{2}\biggr)^2 Fp\cot\left(\frac{\pi}{p}\right).</math>

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,

<math display="block">V = \frac{1}{3} rA.</math>

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.

{| class="wikitable" style="text-align:center"
|-
! rowspan=2 | Polyhedron, <br/>''a''&nbsp;=&nbsp;2
! colspan=3 | Radius
! rowspan=2 | Surface area, <br/>''A''
! colspan=2 | Volume
|-
! In-, ''r''
! Mid-, ''ρ''
! Circum-, ''R''
! ''V''
! Unit edges
|-
| [[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}\approx 0.942809</math> || <math>\approx0.117851</math>
|- align=center
| [[cube]] || <math>1\,</math> || <math>\sqrt 2</math> || <math>\sqrt 3</math> || <math>24\,</math> || <math>8\,</math> || <math>1\,</math>
|-
| [[octahedron]] || <math>\sqrt{2\over 3}</math> || <math>1\,</math> || <math>\sqrt 2</math> || <math>8\sqrt 3</math> || <math>\frac{\sqrt {128}}{3}\approx 3.771236</math> || <math>\approx 0.471404</math>
|-
| [[regular dodecahedron|dodecahedron]] || <math>\frac{\varphi^2}{\xi}</math> || <math>\varphi^2</math> || <math>\sqrt 3\,\varphi</math> || <math>12 {\sqrt {25+10\sqrt5}}</math> || <math>\frac{20\varphi^3}{\xi^2}\approx 61.304952</math> || <math>\approx 7.663119</math>
|-
| [[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}\approx 17.453560</math> || <math>\approx 2.181695</math>
|}

The constants ''φ'' and ''ξ'' in the above are given by

<math display="block">
  \varphi = 2\cos{\pi\over 5} = \frac{1+\sqrt 5}{2},\qquad
  \xi = 2\sin{\pi\over 5} = \sqrt{\frac{5-\sqrt 5}{2}} = \sqrt{3 - \varphi}.
</math>

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.

===Point in space===
For an arbitrary point in the space of a Platonic solid with circumradius ''R'', whose distances to the centroid of the Platonic solid and its ''n'' vertices are ''L'' and ''d<sub>i</sub>'' respectively, and

<math display="block">S^{(2m)}_{[n]}= \frac 1n\sum_{i=1}^n d_i^{2m}</math>,

we have<ref name=Mamuka >{{cite journal| last1= Meskhishvili |first1= Mamuka| date=2020|title=Cyclic Averages of Regular Polygons and Platonic Solids |journal= Communications in Mathematics and Applications|volume=11|pages=335–355|doi= 10.26713/cma.v11i3.1420|doi-broken-date= 1 November 2024|arxiv= 2010.12340|url= https://www.rgnpublications.com/journals/index.php/cma/article/view/1420/1065}}</ref>

<math display="block">\begin{align}
S^{(2)}_{[4]} = S^{(2)}_{[6]} = S^{(2)}_{[8]}= S^{(2)}_{[12]}= S^{(2)}_{[20]} &= R^2+L^2, \\[4px]
S^{(4)}_{[4]} = S^{(4)}_{[6]} = S^{(4)}_{[8]}= S^{(4)}_{[12]}= S^{(4)}_{[20]} &= \left(R^2+L^2\right)^2 + \frac 43 R^2L^2, \\[4px]
S^{(6)}_{[6]} = S^{(6)}_{[8]} = S^{(6)}_{[12]}= S^{(6)}_{[20]}&= \left(R^2+L^2\right)^3 + 4R^2L^2 \left(R^2+L^2\right), \\[4px]
S^{(8)}_{[12]} = S^{(8)}_{[20]} &= \left(R^2+L^2\right)^4 + 8R^2L^2 \left(R^2+L^2\right)^2+\frac {16}{5} R^4L^4, \\[4px]
S^{(10)}_{[12]} = S^{(10)}_{[20]} &= \left(R^2+L^2\right)^5 +\frac {40}{3}R^2L^2\left(R^2+L^2\right)^3+16R^4L^4\left(R^2+L^2\right).
\end{align}</math>
For all five Platonic solids, we have<ref name= Mamuka />

<math display="block">S^{(4)}_{[n]}+\frac {16}{9}R^4= \left(S^{(2)}_{[n]}+ \frac 23R^2\right)^2.</math>

If ''d<sub>i</sub>'' are the distances from the ''n'' vertices of the Platonic solid to any point on its circumscribed sphere, then<ref name= Mamuka />

<math display="block">4\left(\sum_{i=1}^n d_i^2\right)^2=3n \sum_{i=1}^n d_i^4.</math>

===Rupert property===
A polyhedron ''P'' is said to have the [[Rupert property]] if a polyhedron of the same or larger size and the same shape as ''P'' can pass through a hole in ''P''.<ref name="AllFive">{{cite journal | first1=Richard P. | last1=Jerrard  |first2=John E. | last2=Wetzel | first3=Liping | last3 = Yuan | title = Platonic Passages | journal = Mathematics Magazine | date = April 2017 | volume = 90 | issue = 2 | pages = 87–98 | publisher = [[Mathematical Association of America]] | location = Washington, DC | doi = 10.4169/math.mag.90.2.87| s2cid=218542147 }}</ref>
All five Platonic solids have this property.<ref name="AllFive" /><ref>{{citation|last=Schrek|first= D. J. E.|title=Prince Rupert's problem and its extension by Pieter Nieuwland|journal=[[Scripta Mathematica]]|volume=16|year=1950|pages=73–80 and 261–267}}</ref><ref>{{citation | last = Scriba | first = Christoph J. | issue = 9 | journal = Praxis der Mathematik | language = de | mr = 0497615 | pages = 241–246 | title = Das Problem des Prinzen Ruprecht von der Pfalz | volume = 10 | year = 1968}}</ref>

== Symmetry ==
=== Dual polyhedra ===
{{multiple image
 | align = right  | direction=vertical | width=150
 | image1 = Dual compound 4 max.png
 | image2 = Dual compound 8 max.png
 | image3 = Dual compound 20 max.png
 | footer = [[Dual compound]]s
}}

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.

* The tetrahedron is [[self-dual polyhedron|self-dual]] (i.e. its dual is another tetrahedron).
* The cube and the octahedron form a dual pair.
* The dodecahedron and the icosahedron form a dual pair.

If a polyhedron has Schläfli symbol {''p'',&nbsp;''q''}, then its dual has the symbol {''q'',&nbsp;''p''}. Indeed, every combinatorial property of one Platonic solid can be interpreted as another combinatorial property of the dual.

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.

More generally, one can dualize a Platonic solid with respect to a sphere of radius ''d'' concentric with the solid. The radii (''R'',&nbsp;''ρ'',&nbsp;''r'') of a solid and those of its dual (''R''*,&nbsp;''ρ''*,&nbsp;''r''*) are related by

<math display="block">d^2 = R^\ast r = r^\ast R = \rho^\ast\rho.</math>

Dualizing with respect to the midsphere (''d''&nbsp;=&nbsp;''ρ'') is often convenient because the midsphere has the same relationship to both polyhedra. Taking ''d''<sup>2</sup>&nbsp;=&nbsp;''Rr'' yields a dual solid with the same circumradius and inradius (i.e. ''R''*&nbsp;=&nbsp;''R'' and ''r''*&nbsp;=&nbsp;''r'').

=== Symmetry groups ===
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]].

The symmetry groups of the Platonic solids are a special class of [[point groups in three dimensions|three-dimensional point groups]] known as [[polyhedral group]]s. 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 (mathematics)|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]].

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:

* the [[tetrahedral group]] ''T'',
* the [[octahedral group]] ''O'' (which is also the symmetry group of the cube), and
* the [[icosahedral group]] ''I'' (which is also the symmetry group of the dodecahedron).

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]].

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 parentheses (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.

{| class="wikitable" style="text-align:center"
|-
!rowspan=2|Polyhedron
!rowspan=2|[[Schläfli symbol|Schläfli<br/>symbol]]
!rowspan=2|[[Wythoff symbol|Wythoff<br/>symbol]]
!rowspan=2|[[Dual polyhedron|Dual<br/>polyhedron]]
!colspan=5|[[Symmetry group]] (reflection, rotation)
|-
![[Polyhedral group|Polyhedral]]
![[Schönflies notation|Schön.]]
![[Coxeter notation|Cox.]]
![[Orbifold notation|Orb.]]
![[group order|Order]]
|-
| [[tetrahedron]]
| {3, 3} || 3 {{pipe}} 2 3 || tetrahedron
| style="text-align:right;" | [[tetrahedral symmetry|Tetrahedral]] [[File:Disdyakis 6 spherical.png|50px]]
| ''T''<sub>d</sub><BR/>''T''
| [3,3]<BR/>[3,3]<sup>+</sup>
| *332<BR/>332
| 24<BR/>12
|-
| [[cube]]
| {4, 3} || 3 {{pipe}} 2 4 || octahedron
| rowspan=2 style="text-align:right;" | [[octahedral symmetry|Octahedral]] [[File:Disdyakis 12 spherical.png|50px]]
| rowspan=2 | ''O''<sub>h</sub><BR/>''O''
| rowspan=2 | [4,3]<BR/>[4,3]<sup>+</sup>
| rowspan=2 | *432<BR/>432
| rowspan=2 | 48<BR/>24
|-
| [[octahedron]]
| {3, 4} || 4 {{pipe}} 2 3 || cube
|-
| [[dodecahedron]]
| {5, 3} || 3 {{pipe}} 2 5 || icosahedron
| rowspan=2 style="text-align:right;" | [[icosahedral symmetry|Icosahedral]] [[File:Disdyakis 30 spherical.png|50px]]
| rowspan=2 | ''I''<sub>h</sub><BR/>''I''
| rowspan=2 | [5,3]<BR/>[5,3]<sup>+</sup>
| rowspan=2 | *532<BR/>532
| rowspan=2 | 120<BR/>60
|-
| [[icosahedron]]
| {3, 5} || 5 {{pipe}} 2 3 || dodecahedron
|}

== In nature and technology ==
{{more citations needed|section|date=April 2025}}
[[Image:Circogoniaicosahedra ekw.jpg|frame|''Circogonia icosahedra'', a species of [[radiolaria]], shaped like a [[regular icosahedron]].]]

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. [[Allotropes of boron]] and many [[crystal structure of boron-rich metal borides|boron compounds]], such as [[boron carbide]], include discrete B<sub>12</sub> icosahedra within their crystal structures. [[Carborane acid]]s also have molecular structures approximating regular icosahedra.

In the early 20th century, [[Ernst Haeckel]] described 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.<ref>[[Ernst Haeckel|Haeckel, Ernst]], E. (1904). ''Kunstformen der Natur''. Available as Haeckel, E. (1998); ''[https://web.archive.org/web/20090627082453/http://caliban.mpiz-koeln.mpg.de/~stueber/haeckel/kunstformen/natur.html Art forms in nature]'', Prestel USA. {{isbn|3-7913-1990-6}}.</ref>

Many [[virus]]es, such as the [[herpes]]<ref>{{cite journal |title= Why large icosahedral viruses need scaffolding proteins |author=Siyu Li, [[Polly Roy]], Alex Travesset, and [[Roya Zandi]] |date= October 2018 |journal=Proceedings of the National Academy of Sciences |volume= 115 |issue= 43 |pages= 10971–10976 |doi= 10.1073/pnas.1807706115 |pmid= 30301797 |pmc= 6205497 |bibcode= 2018PNAS..11510971L |quote=|doi-access= free }}</ref> 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]].

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.

Geometry of [[space frame]]s is often based on platonic solids. In the MERO system, Platonic solids are used for naming convention of various space frame configurations. For example, {{sfrac|1|2}}O+T refers to a configuration made of one half of octahedron and a tetrahedron.

Several [[Platonic hydrocarbons]] have been synthesised, including [[cubane]] and [[dodecahedrane]] and not [[tetrahedrane]].

<gallery>
Image:Tetrahedrane-3D-balls.png |[[Tetrahedrane]]
Image:Cubane-3D-balls.png |[[Cubane]]
Image:Dodecahedrane-3D-balls.png|[[Dodecahedrane]]
</gallery>

=== Liquid crystals with symmetries of Platonic solids ===
For the intermediate material phase called [[liquid crystal]]s, the existence of such symmetries was first proposed in 1981 by [[Hagen Kleinert|H. Kleinert]] and K. Maki.<ref>Kleinert and Maki (1981)</ref><ref>{{cite web |title=''The liquid-crystalline blue phases'' (1989). by Tamar Seideman, Reports on Progress in Physics, Volume 53, Number 6 |url=http://chemgroups.northwestern.edu/seideman/Publications/The%20liquid-crystalline%20blue%20phases.pdf}}</ref>
In aluminum the icosahedral structure was discovered three years after this by [[Dan Shechtman]], which earned him the [[Nobel Prize in Chemistry]] in 2011.

== In culture ==
[[Image:BluePlatonicDice2.jpg|thumb|A set of polyhedral dice.]]

Platonic solids are often used to make [[dice]], because dice of these shapes can be made [[fair dice|fair]]. 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.

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]].

=== Architecture ===
Architects liked the idea of Plato's timeless [[Form (architecture)|forms]] that can be seen by the soul in the objects of the material world, but turned these shapes into more suitable for construction [[sphere]], [[cylinder]], [[cone]], and [[square pyramid]].{{sfn | Gelernter | 1995 | pp=50-51}} In particular, one of the leaders of [[neoclassicism]], [[Étienne-Louis Boullée]], was preoccupied with the architects' version of "Platonic solids".{{sfn | Gelernter | 1995 | pp=172-173}}
<gallery mode="nolines" widths="250" heights="250">
File:Icosahedron-spinoza.jpg|Icosahedron as a part of [[Baruch Spinoza|Spinoza]] monument in [[Amsterdam]]
</gallery>

== Related polyhedra and polytopes ==
=== Uniform polyhedra ===
There exist four regular polyhedra that are not convex, called [[Kepler–Poinsot polyhedron|Kepler–Poinsot polyhedra]]. These all have [[icosahedral symmetry]] and may be obtained as [[stellation]]s of the dodecahedron and the icosahedron.

{| class="wikitable floatright" style="text-align:center"
|- style="vertical-align:bottom;"
| [[Image:Cuboctahedron.svg|120px]]<br />[[cuboctahedron]]
| [[Image:Icosidodecahedron.svg|120px]]<br />[[icosidodecahedron]]
|}

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. Their duals, the [[rhombic dodecahedron]] and [[rhombic triacontahedron]], are edge- and face-transitive, but their faces are not regular and their vertices come in two types each; they are two of the thirteen [[Catalan solid]]s.

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.

The [[Johnson solid]]s are convex polyhedra which have regular faces but are not uniform. Among them are five of the eight convex [[deltahedron|deltahedra]], which have identical, regular faces (all equilateral triangles) but are not uniform. (The other three convex deltahedra are the Platonic tetrahedron, octahedron, and icosahedron.)

=== Regular tessellations ===
{| class="wikitable floatright"
|+ Regular spherical tilings
! colspan=5 | Platonic
|-
|[[File:Uniform tiling 332-t0-1-.svg|60px]]
|[[File:Uniform tiling 432-t0.png|60px]]
|[[File:Uniform tiling 432-t2.png|60px]]
|[[File:Uniform tiling 532-t0.png|60px]]
|[[File:Uniform tiling 532-t2.png|60px]]
|-
!{3,3}
!{4,3}
!{3,4}
!{5,3}
!{3,5}
|-
! colspan=5 | Regular dihedral
|-
|[[Image:Digonal dihedron.png|60px]]
|[[Image:Trigonal dihedron.png|60px]]
|[[Image:Tetragonal dihedron.png|60px]]
|[[Image:Pentagonal dihedron.png|60px]]
|[[Image:Hexagonal dihedron.png|60px]]
|-
!{2,2}
!{3,2}
!{4,2}
!{5,2}
!{6,2}...
|-
! colspan=5 | Regular hosohedral
|-
|[[Image:Spherical digonal hosohedron.svg|60px]]
|[[Image:Spherical trigonal hosohedron.svg|60px]]
|[[Image:Spherical square hosohedron.svg|60px]]
|[[Image:Spherical pentagonal hosohedron.svg|60px]]
|[[Image:Spherical hexagonal hosohedron.svg|60px]]
|-
!{2,2}
!{2,3}
!{2,4}
!{2,5}
!{2,6}...
|}

The three [[Euclidean tilings by convex regular polygons#Regular tilings|regular tessellation]]s of the plane are closely related to the Platonic solids. Indeed, one can view the Platonic solids as 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. Spherical tilings provide two infinite additional sets of regular tilings, the [[hosohedra]], {2,''n''} with 2 vertices at the poles, and [[Lune (mathematics)|lune]] faces, and the dual [[dihedra]], {''n'',2} with 2 hemispherical faces and regularly spaced vertices on the equator. Such tesselations would be degenerate in true 3D space as polyhedra.

Every regular tessellation of the sphere is characterized by a pair of integers {''p'',&nbsp;''q''} with {{sfrac|1|''p''}}&nbsp;+&nbsp;{{sfrac|1|''q''}}&nbsp;>&nbsp;{{sfrac|1|2}}. Likewise, a regular tessellation of the plane is characterized by the condition {{sfrac|1|''p''}}&nbsp;+&nbsp;{{sfrac|1|''q''}}&nbsp;=&nbsp;{{sfrac|1|2}}. There are three possibilities:

{| class=wikitable
|+ The three regular tilings of the Euclidean plane
|[[File:Uniform tiling 44-t0.svg|100px]]
|[[File:Uniform tiling 63-t2-red.svg|100px]]
|[[File:Uniform tiling 63-t0.svg|100px]]
|-
! [[square tiling|{4, 4}]]
! [[triangular tiling|{3, 6}]]
! [[hexagonal tiling|{6, 3}]]
|}
In a similar manner, one can consider regular tessellations of the [[hyperbolic geometry|hyperbolic plane]]. These are characterized by the condition {{sfrac|1|''p''}}&nbsp;+&nbsp;{{sfrac|1|''q''}}&nbsp;<&nbsp;{{sfrac|1|2}}. There is an infinite family of such tessellations.
{| class=wikitable
|+ Example regular tilings of the hyperbolic plane
|[[File:H2-5-4-dual.svg|100px]]
|[[File:H2-5-4-primal.svg|100px]]
|[[File:Heptagonal tiling.svg|100px]]
|[[File:Order-7 triangular tiling.svg|100px]]
|-
! [[Order-4 pentagonal tiling|{5, 4}]]
! [[Order-5 square tiling|{4, 5}]]
! [[Heptagonal tiling|{7, 3}]]
! [[Order-7 triangular tiling|{3, 7}]]
|}

=== Higher dimensions ===
{{Further|List of regular polytopes}}
{| class="wikitable floatright" style="text-align:center; max-width: 22em"
|-
! {{nowrap|Number of}} dimensions
! {{nowrap|Number of convex}} regular polytopes
|-
| 0 || 1
|-
| 1 || 1
|-
| 2 || &infin;
|-
| '''3''' || '''5'''
|-
| 4 || 6
|-
| &gt; 4 || 3
|}

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.

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 : [[5-cell]] as {3,3,3}, [[16-cell]] as {3,3,4}, [[600-cell]] as {3,3,5}, [[tesseract]] as {4,3,3}, and [[120-cell]] as {5,3,3}, and a sixth one, the self-dual [[24-cell]], {3,4,3}.

In all dimensions higher than four, there are only three convex regular polytopes: the [[simplex]] as {3,3,...,3}, the [[hypercube]] as {4,3,...,3}, and the [[cross-polytope]] as {3,3,...,4}.{{sfn|Coxeter|1973|p=136}} In three dimensions, these coincide with the tetrahedron as {3,3}, the cube as {4,3}, and the octahedron as {3,4}.

== See also ==
{{Columns-list|colwidth=22em|
* [[Archimedean solid]]
* [[Catalan solid]]
* [[Deltahedron]]
* [[Johnson solid]]
* [[Goldberg polyhedron]]
* [[Kepler-Poinsot polyhedron]]
* [[List of regular polytopes]]
* [[Prince Rupert's cube]]
* [[Regular polytope]]
* [[Regular skew polyhedron]]
* [[Toroidal polyhedron]]
}}

== Citations ==
{{Reflist|30em}}

== General and cited sources ==
* {{cite journal
| last1 = Atiyah
| first1 = Michael
| author2-link = Paul Sutcliffe
| author-link = Michael Atiyah
| last2=Sutcliffe
| first2=Paul
| year = 2003
| title = Polyhedra in Physics, Chemistry and Geometry
| journal = Milan J. Math.
| volume = 71
| pages = 33–58
| doi = 10.1007/s00032-003-0014-1
| arxiv = math-ph/0303071
| bibcode = 2003math.ph...3071A
| s2cid = 119725110
}}
* {{cite book |first1       = Carl
 |last1        = Boyer
 |author1-link = Carl Benjamin Boyer
 |first2       = Uta
 |last2        = Merzbach
 |author2-link = Uta Merzbach
 |year         = 1989
 |title        = A History of Mathematics
 |edition      = 2nd
 |publisher    = Wiley
 |isbn         = 0-471-54397-7
 |url-access   = registration
 |url          = https://archive.org/details/historyofmathema00boye
}}
* {{cite book
| first      = H. S. M.
| last       = Coxeter
| author-link = H. S. M. Coxeter
| year       = 1973
| title      = Regular Polytopes
| title-link = Regular Polytopes (book)
| edition    = 3rd
| publisher  = Dover Publications
| location   = New York
| isbn        = 0-486-61480-8
}}
* {{cite book | author = Euclid | author-link = Euclid | year = 1956 | title = The Thirteen Books of Euclid's Elements, Books 10–13 | editor-first = Thomas L. | editor-last = Heath | editor-link = Thomas Little Heath | edition = 2nd unabr. | publisher = Dover Publications | location = New York | isbn = 0-486-60090-4 | url-access = registration | url = https://archive.org/details/thirteenbooksofe00eucl }}
*  [[Martin Gardner|Gardner, Martin]] (1987). ''The 2nd Scientific American Book of Mathematical Puzzles & Diversions'', University of Chicago Press, Chapter 1: The Five Platonic Solids, {{isbn|0226282538}}
* {{cite book | last=Gelernter | first=Mark | title=Sources of Architectural Form: A Critical History of Western Design Theory | publisher=[[Manchester University Press]] | year=1995 | isbn=978-0-7190-4129-7 | url=https://books.google.com/books?id=Ri6qER8Ej6kC | access-date=2024-02-12}}
* [[Johannes Kepler|Kepler. Johannes]] ''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.
* {{Cite journal| title = Lattice Textures in Cholesteric Liquid Crystals
| author =  Kleinert, Hagen
| author-link =  Hagen Kleinert
| author2 =  Maki, K.
| name-list-style =  amp
| journal = Fortschritte der Physik
| volume = 29
| issue = 5
| pages = 219–259
| year = 1981
| doi = 10.1002/prop.19810290503
| url = http://users.physik.fu-berlin.de/~kleinert/75/75.pdf| bibcode = 1981ForPh..29..219K}}
* {{cite journal
| last1 = Lloyd
| first1 = David Robert
| year = 2012
| title = How old are the Platonic Solids?
| journal = BSHM Bulletin: Journal of the British Society for the History of Mathematics
| volume = 27
| issue = 3
| pages = 131–140
| doi = 10.1080/17498430.2012.670845
| s2cid = 119544202
}}
* {{cite book | first=Anthony|last= Pugh | year= 1976 | title= Polyhedra: A visual approach | publisher= University of California Press Berkeley | location= California | isbn= 0-520-03056-7  }}
* {{cite book
| first      = Hermann
| last       = Weyl
| author-link = Hermann Weyl
| year       = 1952
| title      = Symmetry
| url      = https://archive.org/details/symmetry0000weyl
| url-access = registration
| publisher  = Princeton University Press
| location   = Princeton, NJ
| isbn        = 0-691-02374-3
}}
* Wildberg, Christian (1988). [https://books.google.com/books?id=af3XzdAvB_cC&pg=PA11 ''John Philoponus' Criticism of Aristotle's Theory of Aether''].  Walter de Gruyter. pp.&nbsp;11–12. {{isbn|9783110104462}}.

== External links ==
{{external links|section|date=December 2019}}
{{Commons category|Platonic solids}}
* [http://www.encyclopediaofmath.org/index.php/Platonic_solids ''Platonic solids'' at Encyclopaedia of Mathematics]
* {{MathWorld | urlname=PlatonicSolid | title=Platonic solid }}
* {{MathWorld | urlname = Isohedron | title = Isohedron}}
* [http://aleph0.clarku.edu/~djoyce/java/elements/bookXIII/propXIII13.html Book XIII] of Euclid's ''Elements''.
* [https://web.archive.org/web/20050403235101/http://ibiblio.org/e-notes/3Dapp/Convex.htm Interactive 3D Polyhedra] in Java
* [http://dmccooey.com/polyhedra/Platonic.html Platonic Solids] in Visual Polyhedra
* [https://archive.today/20130411004747/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.
* [http://www.mat.puc-rio.br/~hjbortol/mathsolid/mathsolid_en.html Interactive Folding/Unfolding Platonic Solids] {{Webarchive|url=https://web.archive.org/web/20070209043012/http://www.mat.puc-rio.br/~hjbortol/mathsolid/mathsolid_en.html |date=2007-02-09 }} in Java
* [http://www.software3d.com/Platonic.php Paper models of the Platonic solids] created using nets generated by [[Stella (software)|Stella]] software
* [http://www.korthalsaltes.com/cuadros.php?type=p Platonic Solids] Free paper models (nets)
* {{cite web|title=Platonic Solids|url=http://www.numberphile.com/videos/platonic_solids.html|work=Numberphile|publisher=[[Brady Haran]]|author=Grime, James|author2=Steckles, Katie|access-date=2013-04-13|archive-date=2018-10-23|archive-url=https://web.archive.org/web/20181023183946/http://www.numberphile.com/videos/platonic_solids.html|url-status=dead}}
* [http://www.ldlewis.com/Teaching-Mathematics-with-Art/Polyhedra.html Teaching Math with Art] student-created models
* [http://www.ldlewis.com/Teaching-Mathematics-with-Art/instructions-for-polyhedra-project.html Teaching Math with Art] teacher instructions for making models
* [http://www.bru.hlphys.jku.at/surf/Kepler_Model.html Frames of Platonic Solids] images of [[algebraic surface]]s
* [http://whistleralley.com/polyhedra/platonic.htm Platonic Solids] with some [http://whistleralley.com/polyhedra/derivations.htm formula derivations]
* [http://woodenpolyhedra.web.fc2.com/making.pdf How to make four platonic solids from a cube]

{{Ancient Greek mathematics}}
{{Polyhedron navigator}}
{{Plato navbox}}
{{Authority control}}

{{DEFAULTSORT:Platonic Solid}}
[[Category:Platonic solids| ]]
[[Category:Multi-dimensional geometry]]
[[Category:Timaeus (Plato)]]