Crystallographic restriction

Statement
Suppose $$\Lambda$$ is a lattice in the Euclidean plane, and suppose there exists an angle $$\alpha$$ that is not a multiple of $$\pi$$ and such that rotation by $$\alpha$$ sends $$\Lambda$$ to itself. Then, there exist two vectors $$v,w$$ in $$\Lambda$$ of shortest length, such that $$v,w$$ span $$\Lambda$$, and the angle between $$v$$ and $$w$$ is either $$\pi/2$$ or $$2\pi/3$$.

In the statement, the center of rotation need not be a lattice point.

Proof when the center of rotation is a lattice point
We first give the proof assuming that the center of rotation is a lattice point. In this case, we can assume that the center of rotation is the origin.

Let $$v$$ be a nonzero vector of shortest length in $$\Lambda$$, i.e., a point closest to the origin. Let $$\theta$$ be the smallest positive angle of rotation sending $$\Lambda$$ to itself. Let $$w$$ be the image of $$v$$ under $$\theta$$. Then, $$w$$ has the same length as $$v$$.


 * 1) $$v$$ and $$w$$ span $$\Lambda$$: Given any point in the Euclidean plane, we can subtract an element in the sublattice spanned by $$v$$ and $$w$$ to arrive in the fundamental parallelogram bounded by the origin, $$v,w,v+w$$. It is easy to see that the maximum of the distances between this new point and the corners of the parallelogram is less than the length of $$v$$ or $$w$$. Thus, if there exists a lattice point not in the span of $$v$$ and $$w$$, we can obtain a lattice vector of shorter length. Thus, $$v$$ and $$w$$ span $$\Lambda$$.
 * 2) The angle between $$v$$ and $$w$$ divides $$2\pi$$: Suppose the angle is $$\theta$$. If $$\theta$$ does not divide $$2\pi$$, the smallest multiple of $$\theta$$ that is bigger than $$2\pi$$ gives an angle smaller than $$\theta$$ that can be realized as an angle between $$v$$ and one of its images under a rotation of the lattice -- in particular, this contradicts minimality of $$\theta$$.
 * 3) $$v$$ and $$w$$ have an angle of $$\pi/2$$ or $$\pi/3$$. Since $$v-w$$ is in $$\lambda$$, it has to be at least as large as $$v$$. If the angle between $$v$$ and $$w$$ is $$\theta$$, $$\| v - w \| = 2 \sin (\theta/2)$$, and thus, $$\theta \ge \pi/3$$. Thus, $$\theta \in \{ \pi/3, 2\pi/5, \pi/2, 2\pi/3 \}$$. $$\theta = 2\pi/3$$ is not possible because if $$v$$ and $$w$$ make an angle of $$2\pi/3$$, we can choose the spanning set $$\{ v, v+w \}$$, which make an angle of $$\pi/3$$, with a rotational symmetry of $$\pi/3$$. The angle $$2\pi/5$$ is not possible, because rotating by twice that angle gives a vector that makes an angle of $$\pi/5$$ with $$v$$. Thus, the angle is $$\pi/2$$ or $$\pi/3$$.

Proof when the center of rotation is not a lattice point
Suppose the center of rotation is $$u$$ and $$v$$ is any lattice point. Let $$\theta$$ be the smallest permissible positive angle of rotation about $$u$$. As in the previous case, $$\theta$$ divides $$2\pi$$, and the orbit of $$v$$ under rotation by multiples of $$\theta$$ forms the vertices of a regular polygon centered at $$u$$.

Let $$v_1, v_2, \dots, v_r$$ be the other points in this orbit. Then, $$(v_i - v) \in \Lambda$$ for each $$i$$, and thus, $$\sum (v_i - v) \in \Lambda$$. But an elementary trigonometric computation, or a use of complex roots of unity, shows that this sum is an integer multiple of $$u - v$$. In particular, some nonzero integer multiple of $$u - v$$ is in $$\Lambda$$. So, a nonzero integer multiple of $$u$$ is in $$\Lambda$$.

From this, we obtain that the span of $$\Lambda$$ and $$u$$ is also a lattice containing $$u$$. Moreover, rotation by $$\theta$$ sends this new lattice to itself, so we are reduced to the previous case of the center of rotation being a lattice point. Thus, the new lattice is generated by two vectors at an angle of either $$\pi/2$$ or $$2\pi/3$$.