Discrete lines: the birth of a new geometry

You are familiar with the so-called Euclidean line, an ideal geometric entity containing infinitely many points.

It has a certain self-evident quality: two points determine it, it obeys the axioms of classical geometry, and no one doubts what it is—everyone knows what we mean, even without a definition.

Instead, consider our lines not in Euclidean geometry but in discrete geometry. Recall that discrete geometry studies the properties and structures of geometric objects formed from a finite or countable set of points. Here, then, the only points that exist are those with integer coordinates. They can also be represented by a grid of pixels on a computer screen—for example, by taking the centers of the pixels to have integer coordinates.