Let's start with a single floor tile with a side of 1 in the coordinate plane. To make it easy we will place one if its vertices at (0,0).
To cover a larger area we have to translate this tile. For example, we could translate it up and to the right like below.
The first translation is 1 step to the right and 1 step up. The second translation is 2 steps to the right and 2 steps up. We can describe these with the following translations:
&(x,y) → (x+1,y+1)
&(x,y) → (x+2,y+2)
Let's do two more translations of the original tile, one to the right and another directly above.
Lovely! The translation along the x-axis is a horizontal translation of 1 unit and a vertical translation of 0 units. The translation along the y-axis is a vertical translation of 1 unit and a horizontal translation of 0 units. We can describe these as
&(x,y) → (x+1,y+0)
&(x,y) → (x+0,y+1)
