Why is there a limited number of platonic solids?

Answer 1

Let's try to make our own platonic solid.

First we need to choose a regular polygon for our faces. Let's pick the n-gon.

Now we need to decide how many n-gons will meet in each vertex of our platonic solid. Let's call this number m.

Notice that not all combinations of n and m are good choices. If we pick m too large our solid will never close! For instance for n = m = 4, we would have to glue four squares together in every vertex, but this just gives a plane, not a solid.

The right criterion for our solid to become 3D is that the sum of the angles in each vertex should be LESS than 360 degrees, because in this case gluing the edges together forces the shape to 'curl up'. Now, it's not so hard to calculate the angle of a corner in a regular n-gon: it's just 180 degrees times (n-2)/n.

So we get the following angles:

Triangle: 60 degrees

Square: 90 degrees

pentagon: 108 degrees

hexagon: 120 degrees

etc.

Now, since in each vertex at least 3 faces must meet (if two faces would meet it would just be an edge) we can already see that for hexagons and beyond we can never get less than 360 degrees in a vertex, so platonic solids can only be of the following form:

Three triangles meeting in every vertex. I.E. the tetrahedron

Four triangles meeting in every vertex. I.E. the octagon

Five triangle meeting in every vertex. I.E. the icosahedron

three squares meeting in every vertex. I.E. the cube

three pentagons meeting in every vertex. I.E. the dodecahedron

These are indeed exactly the platonic solids in 3 dimensions.
Why are there a limited number of platonic solids?

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