Line intersects points grid

"No matter how spaced apart a rectangular grid of points is, any line that isn't parallel or orthogonal to the grid will cross infinitely many of the points"

What you’ve written is a true and beautiful fact from geometry and number theory:

Intuition

If you place points on the integer lattice (a rectangular grid, say all points with integer coordinates), then any line that is neither parallel to the grid lines (horizontal or vertical) nor at a rational slope will pass through infinitely many lattice points.

Why this happens

• A line on the plane can be written as

  $$ y = mx + b $$

  where $m$ is the slope.

• The integer grid is made of all pairs $(x, y)$ with $x, y \in \mathbb{Z}$.

• The condition for the line to pass through infinitely many integer points depends on $m$:

  1. If $m$ is rational (say $m = p/q$ with integers $p, q$): Then as $x$ steps through multiples of $q$, the corresponding $y$ will also be integer (shifted by $b$, if $b$ is rational with denominator dividing $q$). This guarantees infinitely many integer points.

  2. If $m$ is irrational: The line still intersects infinitely many vertical grid lines $x = n$ at real points $(n, mn+b)$. While not every one is an integer $y$, by Dirichlet’s approximation theorem, you can always find infinitely many integers $n$ such that $mn+b$ is very close to an integer. That means the line passes arbitrarily close to infinitely many grid points, but not exactly through them.

The subtle correction

So strictly speaking:

• If the slope is rational (not vertical, not horizontal), the line will pass through infinitely many lattice points exactly.
• If the slope is irrational, the line will never pass through a second lattice point (other than possibly one special point), but it will pass arbitrarily close to infinitely many.

Rephrasing your claim

The exact, precise statement is:

Any line in the plane that is not horizontal or vertical passes through infinitely many points of the integer lattice if and only if its slope is rational. If the slope is irrational, it will instead come arbitrarily close to infinitely many lattice points, but not pass exactly through them.

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👉 Do you want me to illustrate this with a diagram showing examples of both rational and irrational slopes crossing a grid?