Not a proof, but a thought experiment. Imagine you have a bunch of equilateral triangles made of plastic that have mangetout sides so they stick to each other. How many can you have around a point?

Two triangles just stick to each other flat and you don’t get a 3D shape.

Three triangles makes the top of a triangular pyramid and you can add one more on the base to get a tetrahedron.

Four triangles makes the top of a square based pyramid and you can make another one like that and put it underneath, giving you an octahedron.

Five triangles makes the top of a pentagonal based pyramid. You could make another one and stick it underneath, but it wouldn’t be regular because the top and bottom points would be surrounded by five triangles whereas the ones on the sides would be surrounded by just four. The trick here is to make a band of ten additional triangles pointing up, down, up down etc to go between the two layers. Then two triangles from the top layer touch at a point where a down, up, down junction is and you get five triangles around the point there too. Same for the bottom layer. Now you have an icosahedron.

Six triangles makes a flat plane and you’ll just tile a surface rather than make a solid.

Seven or more triangles is too many to even lie flat and it’ll go all wrinkly on the large scale.

You may be thinking of a D60 dice, which is based on a stellated dodecahedron - stick three pentagons together at each point but turn each pentagon into the base of a small pentagonal based pyramid. It’s not regular because the points don’t all have the same number of sides touching them, but it is very pretty and rolls very nicely.

There are only 5 polyhedrons that can be made from regular polygons. They’re called the Platonic Solids.

Tetrahedron (triangle), cube (square), octahedron (triangle), dodecahedron (pentagon), icosahedron (triangle)