Why are there only 5 regular polyhedra?

Answer 1

understand you need at leas 3 faces per corner to make a 3d object and all shapes on a regular polyhedron must be regular.

triangles:

tetrahedron,(3 per corner) octohedron,(4 per corner) icosahedron,(5 per corner) there is none with 6per corner because that would be 2d as all shapes must be regular

squares:

cube(3 per corner) is the only one because 2 or 4 would both be 2d.

pentagons:

dodecaahedron(3 per corner) is the only one because the pentagon fits together in no other way.

hexagons(non-existant)

there are none because the simplest way of maching them(3 per corner) is 2d.