Editing Platonic solid (section)

=== 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.
}}