In geometry, how many different types of platonic solids are there?
And the answer: five.
Platonic solids are three-dimensional geometric shapes whose faces are all identical, like a pyramid or cube. There are five types of platonic solids, including the tetrahedron or 4-sided pyramid, the 6-sided cube, the 8-sided octahedron, the 12-sided dodecahedron, and the icosahedron, which has 20 faces.
These oh-so-friendly-sounding solids have been known since antiquity. Plato himself wrote of the solids in his famous work Timaeus. In it, Plato associated four of the shapes with the elements: earth was associated with the cube, air with the octahedron, fire with the tetrahedron, and water with the icosahedron. The fifth platonic solid he associated with the cosmos. Interestingly, modern investigations into the shape of our universe suggest that it actually may indeed resemble a higher-dimensional version of the dodecahedron. Whether this relation is fate, coincidence or some higher power...we'll let you decide.
The platonic solids get their name from Plato's extensive writings on the subject, though he was not the first to discover them. Egyptian scholars had published work on the subject prior to Plato and his predecessor Pythagoras. However, Plato's work Theaetetus was the first to offer a mathematical description of the platonic solids and to make the first concrete argument that there could only ever be five.
Dice can only be one of the five platonic solids. Otherwise, the dice would never land on a distinct face. (No dice, if you will.)