invisible point | invisible square | visibility | visible square

Two lattice points (x, y) and (x', y') are mutually visible if the line segment joining them contains no further lattice points. This corresponds to the requirement that (x' - x, y' - y) = 1, where (m, n) denotes the greatest common divisor. The plots above show the first few points visible from the origin. If a lattice point is selected at random in two dimensions, the probability that it is visible from the origin is 6/π^2. This is also the probability that two integers picked at random are relatively prime. If a lattice point is picked at random in n dimensions, the probability that it is visible from the origin is 1/ζ(n), where ζ(n) is the Riemann zeta function.

Euclid's orchard | lattice point | orchard-planting problem | orchard visibility problem | relatively prime | Riemann zeta function