Affine geometry has only one tool: parallelism. No ruler, no protractor. From this single primitive, an entire grid system can be built — and the grid that results is necessarily made of parallelograms, not squares.
1. Two families of parallel lines
Pick any direction and draw two parallel lines in it. Pick a different direction and draw two parallel lines in it. They cross to form one parallelogram ABCD — the seed of the whole grid. Notice: there is no choice of angle here, and no choice of length. The two directions are arbitrary.
2. Extending the grid with diagonals
Draw the diagonal AC of the parallelogram (green dashed). Now draw lines parallel to AC through the other two corners B and D. Where these parallel diagonals cross the original four lines, four new equally-spaced grid points appear (red). Repeat the trick on those new points and the grid extends indefinitely — all using nothing but parallelism.
3. The completed grid
The completed grid: a tessellation of identical parallelograms. Equal spacing within each family is guaranteed by the diagonal construction. Equal spacing across families — that the horizontal step matches the vertical step — is not guaranteed and not even meaningful. There is nothing to compare them with.
4. A vector on the grid
A vector v on the grid is described by a pair of numbers: how many steps along the first family of lines, and how many along the second. Here v is three steps in the e₁ direction and two steps in e₂, written v = (3, 2). The dashed path shows the construction; the red arrow is the vector itself. The same algebraic pair (3, 2) describes any vector with this same relative position on the grid.
5. Why parallelograms, not squares?
A natural question: if we want a grid, why not draw squares? The answer is that a square requires more structure than affine geometry has — it smuggles in a 90° angle and a length comparison across directions, neither of which exists yet.
On the left, the affine grid: parallelograms whose two directions are arbitrary. The "?" marks the angle that the geometry simply does not specify. On the right, a square grid: each cell carries a 90° claim (the small bracket) and an implicit equality between horizontal and vertical step length. Neither claim is available in affine geometry — there is no perpendicular, and no way to compare a step in one direction to a step in another. So the parallelogram grid is not a weaker choice. It is the most general thing affine geometry permits — and the only honest one.