Difference between revisions of "Lattice point"
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| + | A '''lattice point''' is a [[point]] in a [[Cartesian coordinate system]] such that both its <math>x</math>- and <math>y</math>-coordinates are [[integer]]s. A lattice point is a point at the [[intersection]] of two or more grid lines in a regularly spaced array of points, which is a ''' point lattice'''. In a [[plane]], point lattices can be constructed having unit cells in the shape of a [[square]], [[rectangle]], [[hexagon]], and other shapes. If not specified, a point lattice is usually a point in a square array. Lattice points are complicated, so don't get stressed if you don't get it right away! Here's an example to help you to understand it better: | ||
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| + | ==Example== | ||
| + | A point lattice is constructed by plotting all of the points <math>(a,b)</math> such that <math>a</math> and <math>b</math> are positive integers. How many points in the point lattice lie on the line <math>y = -3x + 8</math>? | ||
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| + | ==Solution== | ||
| + | Notice that <math>y > 0 \implies -3x + 8 > 0 \implies x \leq 2</math>. So, <math>(1,5)</math> and <math>(2,2)</math> are the only such points, giving us <math>\boxed{2}</math> points. | ||
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| + | ~advanture | ||
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| + | ==See Also== | ||
| + | * [[Pick's Theorem]] | ||
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| + | [[Category:Geometry]] | ||
| + | [[Category:Definition]] | ||
Latest revision as of 10:45, 2 August 2024
This article is a stub. Help us out by expanding it.
A lattice point is a point in a Cartesian coordinate system such that both its 
- and 
-coordinates are integers. A lattice point is a point at the intersection of two or more grid lines in a regularly spaced array of points, which is a point lattice. In a plane, point lattices can be constructed having unit cells in the shape of a square, rectangle, hexagon, and other shapes. If not specified, a point lattice is usually a point in a square array. Lattice points are complicated, so don't get stressed if you don't get it right away! Here's an example to help you to understand it better:
Example
A point lattice is constructed by plotting all of the points 
such that 
and 
are positive integers. How many points in the point lattice lie on the line 
?
Solution
Notice that 
. So, 
and 
are the only such points, giving us 
points.
~advanture
