Moving an object in a lattice

Question

There's this problem I've been thinking about. Suppose we have a geometric shape (a ball, or a rectangle) of specific size, and we would like to know if it's possible to move it (by translating or rotating) from a point of $\mathbb{R^2}$ to another, with the condition that the object shouldn't cross $\mathbb{Z^2}$ (Assuming it doesn't intersect it in the initial and final positions). For now I just managed to show that it holds for a ball of radius smaller than $\mathbb{1/2}$, and a segment of length less than $\mathbb{\sqrt2}$. I've been wondering if this was a known problem that I could read on, and if there was a nice way to study it in terms of linear algebra perhaps.

Answer 1

I don't think this is much related to linear algebra.

We can view $\Bbb R^2/\Bbb Z^2$ as torus $T$, of which a single point $p$ represents the whole lattice. Then each possible position of our given shape is given by a point in $T$ and a direction, i.e., a unit tangential vector. The set of possible placements of the shape is thus a subset of the tangent bundle of $T$. Now the question whether the shape

• can move freely across the plane
• can move essentially only along one direction (which need not be horizontally or vertically, but could also be some (rational) diagonal direction)
• is locked into a single grid cell

can be answered by investigating topological proerties of said subset.