I have this:
k=3
180*(n-2)/n < 120
so
-360/n < -60
360 > 60*n

Since n has to be at least 3, then n can be 3, 4 or 5.

k=4

180*(n-2)/n < 90
-360/n < -90
360 > 90*n

n can be 3 only.

k=5
180*(n-2)/n < 72
-360/n < -108
360 > 108*n

n can be only 3.

We have showed that there can be at most 5 regular polyhedron, and since we know of 5 then we are done, we've totally classified all regular polyhedron!

What about analogs in yet higher dimensions?

to find out.

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