Visible Point

A visible point relative to a specified lattice point is another lattice point for which the open line segment from to contains no lattice point. Thus distinct points and of the integer lattice are mutually visible iff , where denotes the nonnegative greatest common divisor, with . The plots above show the first few points visible from the origin.

In an orchard interpretation, trees are idealized as fixed lattice points, and intervening lattice points block visibility. The orchard-planting problem uses the same imagery for a different task, arranging a fixed number of trees to maximize the number of lines containing exactly a prescribed number of them.

Here the probability of visibility is a limiting proportion obtained by choosing a lattice point uniformly from and letting . In the integer lattice with , the limiting probability that the point is visible from the origin is , where is the Riemann zeta function (Baake et al. 1994, pp. 2669-2670). In two dimensions this is , also the limiting probability that two independently and uniformly chosen integers from are relatively prime. In one dimension, only and 1 are visible from the origin, so the limiting probability is 0.

An invisible figure is a polygon all of whose vertices are lattice points invisible from the origin. For the invisible squares below, every lattice point on the boundary is included as a polygon vertex, allowing consecutive vertices on the same straight edge. Every finite set of lattice points can be translated so that all its points are invisible from the origin, by the Chinese remainder theorem. The lower left-hand corner of the invisible squares on a square grid with having smallest -coordinate and side lengths 1 and 2 are (20, 14) and (54, 20), respectively. The first invisible square has lower left-hand corner at (42273, 35397) (E. Weisstein, Mar. 1, 2009).

The first few invisible squares occur at , , , , , ... (OEIS A157426 and A157427). The first few invisible squares occur at , , , , , ... (OEIS A157428 and A157429). Both of these sets are plotted above for the first 1000 such squares.

The filled square with lower left-hand corner at (1308, 1274) is the first square with for which all nine of its lattice points, including the point in its interior, are invisible from the origin.