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 two dimensions, 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 dimensions, 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 vertices (with possibly degenerate edges when restricted on a grid) are invisible from the origin. There are invisible sets of every finite shape. The lower left-hand corner of the invisible squares on a square grid with 0<x<y having smallest x-coordinate and side lengths 1 and 2 are (20, 14) and (54, 20), respectively. The first 3×3 invisible square has lower left-hand corner at (42273, 35397) (E. Weisstein, Mar. 1, 2009).


The first few 1×1 invisible squares occur at (20,14), (35,14), (35,20), (54,44), (65,39), ... (OEIS A157426 and A157427). The The first few 2×2 invisible squares occur at (54,20), (174,98), (550,114), (574,368), (588,494), ... (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 2×2 square with 0<x<y which is completely invisible since its interior point is invisible in addition to its edge midpoints and vertices.

See also

Euclid's Orchard, Lattice Point, Orchard-Planting Problem, Orchard Visibility Problem, Relatively Prime, Riemann Zeta Function

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Apostol, T. §3.8 in Introduction to Analytic Number Theory. New York: Springer-Verlag, 1976.Asano, T.; Ghosh, S. K.; and Shermer, T. C. "Visibility in the Plane." Ch. 19 in Handbook of Computational Geometry (Ed. J.-R. Sack and J. Urrutia). Amsterdam, Netherlands: North-Holland, pp. 829-876, 2000.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.Baake, M.; Moody, R. V.; and Pleasants, P. A. B. "Diffraction from Visible Lattice Points and kth Power Free Integers." 19 Jun 1999., M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 208-210, 1984.Gosper, R. W. and Schroeppel, R. Item 48 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 17, Feb. 1972., 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.Sloane, N. J. A. Sequences A157426, A157427, A157428, and A157429 in "The On-Line Encyclopedia of Integer Sequences."Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 100-101, 1999.

Referenced on Wolfram|Alpha

Visible Point

Cite this as:

Weisstein, Eric W. "Visible Point." From MathWorld--A Wolfram Web Resource.

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