system
Given the following post, respond as if you're commenting on Reddit:
human
Why are there an infinite number of regular polygons but only 5 platonic solids?
gpt
Regular polygons are made from radial slices which are isosceles triangles, of which an unlimited number can be fit around a point in the plane by making them thinner. Platonic solids are made of polygons with definite angles that can't be made smaller or larger. 

For a platonic solid, you need to have several polygons meeting at one vertex, and this places a limit on the sum of the angles touching the vertex (must be less than 360). This is important because polygons with more sides have wider angles at the vertices, so eventually they aren't going to fit around a point anymore. 

With triangles, the vertices have an angle of 60^(o), so we can fit either 3, 4, or 5 around a point (1 and 2 don't count because that won't make a 3D shape). The results are the tetrahedron, octahedron, and icosahedron respectively. 

Squares have 90^(o) corners, so the only possibility is 3 around a point, which makes a cube. See how there are less options because the corners are wider?

Pentagons have 108^(o) corners, and again there is one option, 3 to a point, giving the dodecahedron. 

Hexagons have 120^(o) corners, and if you try to put three of them around a vertex you find that you end up with a flat plane that extends forever, never being able to close back on itself and unable to make a solid. All regular polygons with more sides are not even able to fit 3 polygons around a point, so there are no more platonic solids.