Editing Platonic solid (section)

== 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
|}
|}