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).

**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

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1999-05-26