 TOPICS # Integer Lattice A regularly spaced array of points in a square array, i.e., points with coordinates , where , , ... are integers. Such an array is often called a grid or mesh, and is a special case of a point lattice.

The fraction of lattice points visible from the origin, as derived in Castellanos (1988, pp. 155-156), is   (1)   (2)   (3)

Therefore, this is also the probability that two randomly picked integers will be relatively prime to one another.

The number of the lattice points which can be picked with no four concyclic is (Guy 1994, p. 241). Any parallelogram on the lattice in which two opposite sides each have length 1 has unit area (Hilbert and Cohn-Vossen 1999, pp. 33-34).

A special set of polygons defined on the regular lattice are the golygons. A necessary and sufficient condition that a linear transformation transforms a lattice to itself is that it be unimodular. M. Ajtai has shown that there is no efficient algorithm for finding any fraction of a set of spanning vectors in a lattice having the shortest lengths unless there is an efficient algorithm for all of them (of which none is known). This result has potential applications to cryptography and authentication (Cipra 1996).

Lattice, No-Three-in-a-Line-Problem, Point Lattice

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## References

Apostol, T. Introduction to Analytic Number Theory. New York: Springer-Verlag, 1995.Castellanos, D. "The Ubiquitous Pi." Math. Mag. 61, 67-98, 1988.Cipra, B. "Lattices May Put Security Codes on a Firmer Footing." Science 273, 1047-1048, 1996.Eppstein, D. "Lattice Theory and Geometry of Numbers." http://www.ics.uci.edu/~eppstein/junkyard/lattice.html.Gardner, M. "The Lattice of Integers." Ch. 21 in The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 208-219, 1984.Guy, R. K. "Gauss's Lattice Point Problem," "Lattice Points with Distinct Distances," "Lattice Points, No Four on a Circle," and "The No-Three-in-a-Line Problem." §F1, F2, F3, and F4 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 240-244, 1994.Hammer, J. Unsolved Problems Concerning Lattice Points. London: Pitman, 1977.Hilbert, D. and Cohn-Vossen, S. "Regular Systems of Points." Ch. 2 in Geometry and the Imagination. New York: Chelsea, pp. 32-93, 1999.Knupp, P. and Steinberg, S. Fundamentals of Grid Generation. Boca Raton, FL: CRC Press, 1994.Nagell, T. "Lattice Points and Point Lattices." §11 in Introduction to Number Theory. New York: Wiley, pp. 32-34, 1951.Thompson, J. F.; Soni, B.; and Weatherill, N. Handbook of Grid Generation. Boca Raton, FL: CRC Press, 1998.

Integer Lattice

## Cite this as:

Weisstein, Eric W. "Integer Lattice." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/IntegerLattice.html