Crystallography Restriction theorem

Question

I'm trying to understand the Crystallography Restriction theorem, but most proofs I have found include some assumptions that are non-obvious to me. For example, http://mathworld.wolfram.com/CrystallographyRestriction.html has a very simple proof, but it assumes that ,,the sum of the interior angles divided by the number of sides is a divisor of 360 degrees'', that is, $(180^\circ (n-2))/n=(360^\circ)/m$, where $n$ is a number of sides and $m$ is some integer. I don't see why this is true.

Answer 1

That proof assumes the existence of some rotational axis plus the crystal itself being provided by some 3D lattice. Therefore some plane orthogonal to the axis, when containing at least a single lattice point, then shall contain a whole subdimensional 2D lattice.

Wrt. this sublattice the rotational symmetry around that former axis would be a rotation around the intersection point. This is where $360°/m$ stems from.

Now consider the according Delone complex given by that 2D lattice. Obviously it would be given by polygons. By full rotational symmetry (including their mirrors) the polygonal vertices incident to that intersection point all are the same, moreover the internal angle of some regular $n$-gon. This is where $180°(1-2/n)$ derives from.

The remainder then is equating those and solving for integral values of both $m$ and $n$.

--- rk