Classification of the Platonic solids

Builds on Platonic solid Euler's polyhedron formula

Statement

Let $P$ be a Platonic solid whose faces have $p$ sides each, with $q$ faces at each vertex. Then $\{p, q\}$ is one of $\{3, 3\}, \quad \{4, 3\}, \quad \{3, 4\}, \quad \{5, 3\}, \quad \{3, 5\}.$ Each of these five cases occurs, by the regular tetrahedron, the cube, the regular octahedron, the regular dodecahedron and the regular icosahedron. Up to similarity, there is exactly one Platonic solid for each case.

Remarks

A proof: let $V$ , $E$ and $F$ be the numbers of vertices, edges and faces. Then $pF = 2E = qV$ , and Euler's formula $V - E + F = 2$ gives $\frac{1}{p} + \frac{1}{q} = \frac{1}{2} + \frac{1}{E} > \frac{1}{2}$ . With $p, q \geq 3$ , this inequality has exactly the five solutions above. A geometric proof uses the angles: at each vertex, the $q$ face angles $\frac{(p - 2)\pi}{p}$ must add up to less than $2\pi$ .

History

Theaetetus, a contemporary of Plato, described all five solids. He may have given the first proof that no other convex regular polyhedra exist. Euclid's Elements contains a geometric proof.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Platonic solid concept
Polyhedron concept
Euler's polyhedron formula statement
Classification of the Platonic solids statement

Tag	`0M4TCS0RRG`
URI	`m2://0M4TCS0RRG`
Kind	statement, theorem
Writing	W0 stub
MSC	51M20
Status	This copy of the site has no build data. Run `m2 status 0M4TCS0RRG`.
Uses	Platonic solid, Euler's polyhedron formula

Classification of the Platonic solids

Statement

Remarks

History

Reading path

Further reading

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