Lattice — Definition, Formula & Examples

A lattice is a regular, repeating arrangement of points in space, like a grid. In two dimensions, the most common lattice consists of all points whose coordinates are both integers.

A lattice in $\mathbb{R}^n$ is the set of all integer linear combinations of $n$ linearly independent vectors $\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n$ . That is, $L = \{ a_1 \mathbf{v}_1 + a_2 \mathbf{v}_2 + \cdots + a_n \mathbf{v}_n \mid a_i \in \mathbb{Z} \}$ . The standard integer lattice $\mathbb{Z}^2$ uses the standard basis vectors $\mathbf{e}_1 = (1,0)$ and $\mathbf{e}_2 = (0,1)$ .

Key Formula

L = \{ a_1 \mathbf{v}_1 + a_2 \mathbf{v}_2 \mid a_1, a_2 \in \mathbb{Z} \}

Where:

$\mathbf{v}_1, \mathbf{v}_2$ = Linearly independent basis vectors that generate the lattice
$a_1, a_2$ = Integer coefficients
$L$ = The resulting lattice (set of all grid points)

How It Works

Think of graph paper: every intersection where a horizontal and vertical line cross is a lattice point. A lattice point in

\mathbb{Z}^2

has the form

(m, n)

where both

m

and

n

are integers. You can count lattice points inside geometric shapes to solve area and counting problems. Pick's theorem, for example, computes the area of a polygon whose vertices are lattice points by counting interior and boundary lattice points.

Worked Example

Problem: How many lattice points lie inside or on the boundary of a circle of radius 2 centered at the origin?

Set up the condition: A lattice point

(m, n)

lies inside or on the circle when

m^2 + n^2 \leq 4

List the points: Check all integers

m, n

from

-2

2

. The qualifying points are:

(0,0)

(\pm1, 0)

(0, \pm1)

(\pm1, \pm1)

(\pm2, 0)

, and

(0, \pm2)

Count them: Tally: 1 (origin) + 4 (axis, distance 1) + 4 (diagonals, distance

\sqrt{2}

) + 4 (axis, distance 2) = 13.

Answer: There are 13 lattice points inside or on the circle of radius 2.

Visualization

Why It Matters

Lattice points appear throughout competition math, especially in problems involving Pick's theorem and counting arguments. In number theory, questions about which integers can be written as a sum of two squares reduce to counting lattice points on circles. Cryptography systems like NTRU rely on the difficulty of finding short vectors in high-dimensional lattices.

Common Mistakes

Mistake: Confusing lattice points with all points on a grid line.

Correction: Lattice points are only the intersection points where both coordinates are integers, not every point along a grid line.