Lattice Point
A Point at the intersection of two or more grid lines in a ruled array. (The array of grid lines could be orientedto form unit cells in the shape of a square, rectangle, hexagon, etc.) However, unless otherwise specified, lattice pointsare generally taken to refer to points in a square array, i.e., points with coordinates 
, where 
, 
,... are Integers.
An -D 
-lattice 
lattice can be formally defined as a free -Module in complex-D space 
.
The Fraction of lattice points Visible from theOrigin, as derived in Castellanos (1988, pp. 155-156), is
Therefore, this is also the probability that two randomly picked integers will be Relatively Prime to oneanother.
For 
, it is possible to select 
lattice points with 
such that no three are in a straightLine. The number of distinct solutions (not counting reflections and rotations) for 
, 2, ..., are 1, 1, 4, 5,11, 22, 57, 51, 156 ... (Sloane's A000769). For large , it is conjectured that it is only possible to select at most
lattice points with no three Collinear, where
(Guy and Kelly 1968; Guy 1994, p. 242). The number of the
lattice points which can bepicked with no four Concyclic is 
(Guy 1994, p. 241).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 itbe Unimodular. M. Ajtai has shown that there is no efficient Algorithm forfinding any fraction of a set of spanning vectors in a lattice having the shortest lengths unless there is an efficientalgorithm for all of them (of which none is known). This result has potential applications to cryptography andauthentication (Cipra 1996).
See also Barnes-Wall Lattice, Blichfeldt's Theorem, Browkin's Theorem, Circle Lattice Points,Coxeter-Todd Lattice, Ehrhart Polynomial, Gauss's Circle Problem, Golygon, IntegrationLattice, Jarnick's Inequality, Lattice Path, Lattice Sum, Leech Lattice, MinkowskiConvex Body Theorem, Modular Lattice, N-Cluster, Nosarzewska's Inequality, Pick's Theorem,Poset, Random Walk, Schinzel's Theorem, Schröder Number, VisiblePoint, Voronoi PolygonReferences
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. 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. Guy, R. K. and Kelly, P. A. ``The No-Three-in-Line-Problem.'' Canad. Math. Bull. 11, 527-531, 1968. Hammer, J. Unsolved Problems Concerning Lattice Points. London: Pitman, 1977. Sloane, N. J. A. SequenceA000769/M3252in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.
- Geometric Progression
- Geometric Sequence
- Geometric Series
- Geometrization Conjecture
- Geometrography
- Geometry
- Gergonne Line
- Gergonne Point
- Germain Primes
- Gerono Lemniscate
- Gersgorin Circle Theorem
- G-Function
- g-Function
- Ghost
- Gibbs Constant
- Gibbs Effect
- Gibbs Phenomenon
- Gigantic Prime
- Gilbrat's Distribution
- Gilbreath's Conjecture
- Gill's Method
- Gingerbreadman Map
- Girard's Spherical Excess Formula
- Girko's Circular Law
- Girth