What’s the easiest way to explain why only 5 polyhedrons exist?

Question

What’s the easiest way to explain why only 5 polyhedrons exist?

In: Mathematics

Answer 1

You can basically go through every regular polygon and show that everything above 5 sides has too wide of an angle to connect with itself, the same way that a sphere doesn’t take up all space.

Then, you go through triangles, squares, and pentagons, and you can find that triangles have lots of wiggle room, they can connect with three different dihedral angles before it starts hitting itself. These three angles form the tetrahedron, octahedron, and icosahedron. The cube has less freedom, and can only connect to form the cube before it hits itself. Similarly, the pentagon is only able to get the dodecahedron.