Platonic solids

The Platonic solids (or: ideal solids, regular polyhedra“polyhedra”) are convex solids with the greatest possible regularity, named after the Greek philosopher Plato (427–347 BC). (Thereby a body is called convex, if with each two of its points P and Q also all points on the connecting line \overline{PQ} belong to it).

This (“greatest possible”) regularity consists in the fact that for each of these solids all side faces are congruent (“congruent”) to each other and that they meet in every corner in the same way.

There are exactly five Platonic solids:

Figure 1: The Five Platonic Solids

Specifically, these five solids have the following properties:

 Side facesNumber of surfacesNumber of cornersNumber of edgesNumber of faces on a corner
Tetrahedronequilateral triangles4463
Cube (Hexameter)Squares68123
Octahedronequilateral triangles86124
Dodecahedronregular pentagons1220303
Icosahedronequilateral triangles2012305
Figure 1: The properties of the five Platonic solids

The Platonic solids play an important role in the history of ideas from Greek antiquity through the Middle Ages to our own time. Tetrahedron, hexahedron (cube) and dodecahedron were well known to the students of Pythagoras in the 6th century BC. Theaitetos (4th century BC) was also familiar with octahedron and icosahedron.

Plato described the solids later named after him in detail in his work Timaisos and assigned them to the four elements, which were the “world building blocks” according to the view of that time, in the following way:

  • Tetrahedron — Fire;
  • Hexahedron (cube) — Earth;
  • Octahedron — Air;
  • Icosahedron — Water.

The later added fifth element “ether” (which was interpreted as “upper heaven” in the antiquity and whose existence played a special role in physics until the 19th century) was assigned to the dodecahedron.

Famous is also the attempt of the astronomer Johannes Kepler (1571–1630), in 1596 in his work Mysterium Cosmographicum, to describe the (mean) orbital radii of the six planets known at that time (Mercury, Venus, Earth, Mars, Jupiter, Saturn) by a certain sequence of the five Platonic solids and their inner and outer spheres:

Figure 2: Model of the solar system in Kepler’s Mysterium Cosmographicum (1596).

And now … the mathematics of it:

Already Euclid (about 300 BC) proved in his famous work The Elements that there are exactly five of these Platonic solids.

The following considerations lead to this:

The sum of the interior angles in an n-corner is (n-1)\cdot180^\circ. So each interior angle in a regular n-corner has the value

    \[\frac{n-2}{n}\cdot 180^\circ.\]

(e.g., for an equilateral triangle 60°, for a square 90°, for a regular pentagon 108°, etc.).

If m denotes the number of faces meeting in a corner of the Platonic solid, the sum of their angles must be less than 360°, i.e.

    \[m\cdot\frac{n-2}{n}\cdot 180^\circ<360^\circ.\]

It now follows m(n-2)<2n, which we further convert to

    \[(m-2)\cdot(n-2)<4\quad (\ast)\]

rearrange.

Now, since n>2 (each of the bounding faces has at least three corners) and m>2 (at least three faces meet in each corner of the body), only the following five pairs (m, n) of natural numbers (each greater than 2) satisfy the inequality (\ast):

  • (3,3) — Tetrahedron;
  • (4,3) — Octahedron;
  • (5,3) — Icosahedron;
  • (3,4) — Cube;
  • (3,5) — Dodecahedron.

This ends the proof.


Literature

[1] Adam, P. und Wyss, A.: Platonische und Archimedische Körper, ihre Sternformen und polaren Gebilde, Stuttgart, 1994.

[2] Beutelspacher, A. u.a: Mathematik zum Anfassen, Mathematikum, Gießen, 2005.

[3] Euklid: Die Elemente, Buch XIII, Hrsg. u. übs. v. Clemens Thaer, 4. Auflage, Frankfurt am Main, 2003.

[4] Kepler, J.: Mysterium cosmographicum. De stella nova, Hrsg. Max Caspar, München, 1938.

[5] Tiberiu, R.: Reguläre und halbreguläre Polyeder, Berlin, 1987.