Short answers, pulled from the story.
The five regular polyhedra are the tetrahedron, cube, octahedron, icosahedron, and dodecahedron. Euclid described all five forms in Book XIII of his Elements during antiquity. Proposition eighteen argues no further convex regular polyhedra exist within Euclidean space.
Plato wrote about these geometric forms in his dialogue Timaeus around 360 B.C. He associated earth with the cube and air with the octahedron while water connected to the icosahedron. Fire linked to the tetrahedron and the god used the dodecahedron for arranging constellations across the whole heaven.
Theaetetus, a contemporary of Plato, likely proved the existence of all five convex regular polyhedra. He provided the first mathematical description of each shape and established that no other such solids exist within Euclidean space. Pythagoras may have known only three of these forms before Theaetetus completed the proof.
Johannes Kepler published Mysterium Cosmographicum in 1596 with an attempt to model planetary orbits using nested Platonic solids. He proposed that five extraterrestrial planets known at that time fit inside a series of inscribed and circumscribed spheres. His research produced three laws of orbital dynamics despite the original idea requiring abandonment as astronomers gathered better data.
Ernst Haeckel described species of Radiolaria in the early twentieth century whose skeletons resemble various regular polyhedra. Dan Shechtman discovered an icosahedral structure in aluminum which earned him the Nobel Prize in Chemistry in 2011. Many viruses such as herpes adopt the shape of a regular icosahedron for structural efficiency while global atmospheric models sometimes employ geodesic grids based on an icosahedron refined by triangulation.