Challenge Level

Suppose that I am given a large supply of basic vectors $b_1=\pmatrix{2\cr 1}$ and $b_2=\pmatrix{0\cr 1}$.

Starting at the origin, I take a 2-dimensional 'vector walk' where each step is either a $b_1$ vector or a $b_2$ vector, either forwards or backwards.

Investigate the possible coordinates for the end destinations of my walk.

Can you find any other pairs of basic vectors which yield exactly the same set of destinations?

Can you find any pairs of basic vectors which yield none of these destinations?

Can you find any pairs of basic vectors which allow you to visit all integer coordinates?

NOTES AND BACKGROUND

In more formal mathematics, the points visited by such a vector walk would be called a lattice and the two basic vectors would be called the generators . Lattices which repeat themselves are structurally interesting; the symmetry properties of such lattices are important in both pure mathematics and its applications to, for example, the properties of crystals.

In more formal mathematics, the points visited by such a vector walk would be called a lattice and the two basic vectors would be called the generators . Lattices which repeat themselves are structurally interesting; the symmetry properties of such lattices are important in both pure mathematics and its applications to, for example, the properties of crystals.