*This post was motivated by the book “Euler’s Gem: The Polyhedra Formula and the Birth of Topology” by David R. Richeson.*

Plato was a greek philosopher and founder of the Academy in Athens, the first institution of higher learning in the western world. A lecturer of the Academia, and also a friend of Plato, Theaetetus is credited to be the first to give a complete proof of the existence of five (and no more than five) regular convex polyhedra. Plato believed that it must be some cosmological reason for this fact and he proposed an atomic model in which all the matter was composed by four elements and that

these four elements must be shaped as regular convex polyhedra. The fifth element, the dodecahedron, would be the material from what the gods created the universe itself.

In the following, we are going to present three different arguments proving that only five convex regular polyhedra can exist. We start saying few words on polyhedra’s cousin in a lower dimension, the polygons.