I ran into this interesting problem while thinking about some stats models the other day. The context was efficient estimation of anisotropic covariance structures in geostatistical models, but it appears to be more of a geometry or number theory problem. This is really not my area, so please forgive any abuse of notation:
Let $L_{(0,1)}$ be a square lattice in the Euclidean plane with unit spacing and let $L_{(\theta,d)}$ be the lattice obtained by rotating $L_{(0,1)}$ by angle $\theta$ and scaling (increasing the spacing) by factor $d\geq1$. Now suppose we partition $L_{(0,1)}$ into subsets that, up to translation, are perfectly aligned with $L_{(\theta,d)}$. To make this precise, let $S(\theta,d)$ be the set of partitions $P = \left\lbrace L_1, L_2, ... \right\rbrace$ of $L_{(0,1)}$ that satisfy: $$ \forall L_i \in P, \quad \exists \quad (x_i, y_i) \in R^2 \quad \text{such that} \quad L_i + (x_i, y_i) \subseteq L_{(\theta,d)} $$
The set $S(\theta,d)$ is nonempty, since it always contains the partition of singletons (which can be shuffled around however we like). Sometimes it also contains partitions of finite cardinality. Symmetry in the square lattice provides some easy examples: with a right angle and no scaling we can just use the trivial partition, $$ \left\lbrace L_{(0,1)} \right\rbrace \in S \left( \frac{\pi}{2},1 \right), $$
and when $\theta$ is a multiple of $\pi/4$ we can scale by $\sqrt{2}$ and get a partition of cardinality 2, $$ \left\lbrace \left\lbrace (i,j) \in L_{(0,1)} \mid |i-j| \text{ odd} \right\rbrace, \left\lbrace (i,j) \in L_{(0,1)} \mid |i-j| \text{ even} \right\rbrace \right\rbrace \in S \left( \frac{\pi}{4},\sqrt{2} \right). $$
I would like to know in general when $S(\theta,d)$ will contain a partition of finite cardinality (and for my application, I'm really interested in special angles for which we get partitions of low cardinality, say less than 10). Any insights or references to help me better understand the problem are appreciated, but I do have these specific questions:
Is there any theory that concisely characterizes the relationship between $(\theta$, $d)$ and $S(\theta,d)$?
When does a partition of finite cardinality exist? When does a partition of minimal cardinality exist? Is it unique? Is it easy to find?
My intuition is that if $\theta = \arctan(p/q)$ for integers $p$ and $q$ then the set $$ S \left( \theta,\sqrt{p^2 + q^2} \right) $$ should contain a partition of cardinality $p^2 + q^2$, and this would be minimal. But I'm really not sure if this is true, nor how to approach the problem rigorously. Thanks!
Here is an example diagram for the case $p=2$, $q=1$. The dotted blue lines are $L_{(0,1)}$ and the dashed green is $L_{(\theta,\sqrt{5})}$. The green box is the rotated and scaled version of the blue box. I count $5 = 1^2 + 2^2$ points (one blue and four green) representing subsets of $L_{(0,1)}$ (together, a partition) that can be translated to lie on the lattice $L_{(\theta,\sqrt{5})}$.
