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 two dimensions, 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 dimensions, 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 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 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 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 which is completely invisible since its interior
point is invisible in addition to its edge midpoints and vertices.
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. General27, 2669-2674, 1994.Baake, M.;
Moody, R. V.; and Pleasants, P. A. B. "Diffraction from Visible
Lattice Points and th
Power Free Integers." 19 Jun 1999. http://arxiv.org/abs/math.MG/9906132.Gardner,
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.
http://www.inwap.com/pdp10/hbaker/hakmem/number.html#item48.Herzog,
F. and Stewart, B. M. "Patterns of Visible and Nonvisible Lattice Points."
Amer. Math. Monthly78, 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.
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.
. This is also the probability that two integers
picked at random are relatively prime. If a lattice point is picked at random in 
dimensions, the probability that it is visible from the origin is 
, where 
is the Riemann zeta
function.
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).
invisible squares occur at 
, 
, 
, 
, 
, ... (OEIS A157426
and A157427). The The first few 
invisible squares occur at 
, 
, 
, 
, 
, ... (OEIS A157428
and A157429). Both of these sets are plotted
above for the first 1000 such squares.
square with 
which is completely invisible since its interior
point is invisible in addition to its edge midpoints and vertices.
th
Power Free Integers." 19 Jun 1999.