Two Lattice Points and are mutually visible if the line segment joining them contains no further Lattice Points. This corresponds to the requirement that , where 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 . This is also the probability that two Integers picked at random are Relatively Prime. If a Lattice Point is picked at random in -D, the probability that it is visible from the Origin is , where 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 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
© 1996-9 Eric W. Weisstein