Hmm. Not my idea of perfection, but oh well.
Platonic solids, or regular solids, are perfect in form. Each face is a regular n-gon, and all faces look alike. There are infinitely many n-gons, but there are only five regular solids. If you want to hold them in your hand, buy the game Dungeons and Dragons. The five dice are shaped like the five platonic solids.
The analysis is similar to that presented earlier, for tiling the plane, but the polygons that meet at a corner don't lay flat, so their angles don't add up to 360°. In fact they are pulled together a bit, hence their angles sum to something less then 360°. That's the only difference, so on we go.
If the chosen n-gon is a hexagon or higher, three interior angles sum to 360° or more, hence they can't fit together to make a corner. Platonic solids are based on the triangle, square, or pentagon.
Let's start with the triangle. If three triangles meet at a corner then a fourth triangle completes the shape. This is called a tetrahedron, 4 faces, 6 edges, 4 vertices.
Let four triangles meet at each corner. This is called an octahedron, 8 faces, 12 edges, 6 vertices.
Let five triangles meet at each corner. This is called an icosahedron, 20 faces, 30 edges, 12 vertices.
If 6 or more triangles meet at a point, the angles sum to 360° or more, so lets move on to squares.
Let three squares meet at each corner. This is called a cube, 6 faces, 12 edges, 8 vertices.
Let three pentagons meet at each corner. This is called a dodecahedron, 12 faces, 30 edges, 20 vertices.