How many regular polyhedra? (Investigation)

Earlier we were introduced to regular solids.

So, we can use squares as faces to come up with the cube, or regular pentagons as faces to come up with the dodecahedron.

Now, what if we try to make a regular solid with regular hexagons for faces?

As it turns out, this is impossible! In fact, most regular polygons can't be used to make a regular solid. Only five convex (no dents) regular solids exist! These are the five Platonic solids.

But why is this? Why aren't there any more?

Notice that in 3D solids, at least 3 faces always meet at every vertex. Here are 3 hexagons meeting at a vertex.

It's impossible to fold this. It's stuck in a flat position, and we can't make 3D shapes from that!

Once we start getting even more sides, like heptagons, we can't even put the faces together at all!

This is why there are only 5 Platonic solids.

Follow these tutorials on Mathigon to build your own Platonic solids out of origami!