Visible Point

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 2-D, the probability that it is visible from the origin is $6/\pi^2$. This is also the probability that two Integers picked at random are Relatively Prime. If a Lattice Point is picked at random in $n$-D, the probability that it is visible from the Origin is $1/\zeta(n)$, where $\zeta(n)$ is the Riemann Zeta Function.

An invisible figure is a Polygon all of whose corners are invisible. There are invisible sets of every finite shape. The lower left-hand corner of the invisible squares with smallest $x$ coordinate of Areas 2 and 3 are (14, 20) and (104, 6200).

See also Lattice Point, Orchard Visibility Problem, Riemann Zeta Function

References

Apostol, T. §3.8 in Introduction to Analytic Number Theory. New York: Springer-Verlag, 1976.

Baake, M.; Grimm, U.; and Warrington, D. H. ``Some Remarks on the Visible Points of a Lattice.'' J. Phys. A: Math. General 27, 2669-2674, 1994.

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972.

Herzog, F. and Stewart, B. M. ``Patterns of Visible and Nonvisible Lattice Points.'' Amer. Math. Monthly 78, 487-496, 1971.

Mosseri, R. ``Visible Points in a Lattice.'' J. Phys. A: Math. Gen. 25, L25-L29, 1992.

Schroeder, M. R. ``A Simple Function and Its Fourier Transform.'' Math. Intell. 4, 158-161, 1982.

Schroeder, M. R. Number Theory in Science and Communication, 2nd ed. New York: Springer-Verlag, 1990