Pythagoras (c. 570 - c. 495 BCE) knew about at least three (tetrahedron, cube, dodecahedron) of the Platonic Solids
Possible that Theaetetus (c. 417 - c. 369 BCE) discovered the other two (octahedron and icosahedron)
First proof that there are exactly five regular polyhedra
Plato (c. 426 - c. 347 BCE) wrote about these regular polyhedra in his Timaeus
Cube was associated with Earth
Tetrahedron was associated with fire
Octahedron was associated with air
Icosahedron was associated with water
Dodecahedron was a model for the whole universe
Euclid (c. 325 BCE - 265 BCE) proved that there are exactly five Platonic Solids in his The Elements
Archimedes (c. 287 - c. 212 BCE) first described the 13 Archimedean solids, but his work has since then been lost
Kepler (1571 - 1630) listed all 13 Archimedean solids in his second book of Harmonice mundi.
There are exactly five Platonic Solids and exactly 13 Archimedean Solids.