1. The (regular, 3D) Platonic Solids
All faces, all edges, all corners, are the same.
They are composed of regular 2D polygons:
There were infinitely many 2D n-gons!
How many of these regular 3D solids are there?
OK – Let’s step back and look at some beautiful geometry that we _can_ see: the Platonic Solids! They have been known to mankind for 2 millennia.
“Regular” means that all faces, edges, and corners are indistinguishable; they are all exactly the same.
The surfaces of these objects are composed of regular 2D polygons. There are infinitely many of those 2D n-gons.
But how many are there of these regular 3D solids?
>>> There are indeed just the 5 shown here. But could you prove this to your friends -- or to your grandmother? -- Soon you will be able to do this!

2. Making a Corner for a Platonic Solid
Put at least 3 polygons around a shared vertex to form a real physical 3D corner!
Putting 3 squares around a vertex leaves a large (90º) gap;
Forcefully closing this gap makes the structure pop out into 3D space, forming the corner of a cube.
We can also do this with 3 pentagons:  dodecahedron.
In order to construct the surface of some Platonic solid, we have to assemble some 2D polygons so that they form real physical 3D corners; we thus need at least 3 faces coming together at each corner.
When we put 3 squares around a shared vertex, then there is still a large gap left. When we forcefully close this gap, the structure pops out into the 3rd dimension, and a real cube corner is formed.
We can also use 3 pentagons, and the result will be the dodecahedron.

3. Why Only 5 Platonic Solids?
Lets try to build all possible ones:
from triangles: 3, 4, or 5 around a corner:
from squares: only 3 around a corner:
from pentagons: only 3 around a corner:
from hexagons:  “floor tiling”, does not bend!
higher n-gons:  do not fit around a vertex without undulations (forming saddles);  Now the edges are no longer all alike!
Now we can see why there are only 5 Platonic solids: When we use triangles for the surface facets, we have a choice, we can use 3 or 4 or 5 around a point to form a regular corner. This leads to the Tetrahedron, the Octahedron, and the Icosahedron which uses 20 triangles. …
But if we use squares there is only one option: 3 squares around a corner – forming a cube.
And we have already seen that with 3 pentagons we can make a dodecahedron.
All larger n-gons are too “round” and are not able to make true 3D corners.