Vector Walk
Suppose that I am given a large supply of basic vectors $b_1=(2,1)$
and $b_2=(0,1)$.
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?
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.