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Point Lattice — Definition, Formula & Examples

A point lattice is a regularly spaced grid of points formed by taking all possible integer combinations of a set of basis vectors. In two dimensions, the simplest example is the set of all points with integer coordinates, like (1,3)(1, 3) or (−2,5)(-2, 5).

A point lattice Λ\Lambda in Rn\mathbb{R}^n is the set of all points p=a1v1+a2v2+⋯+anvn\mathbf{p} = a_1 \mathbf{v}_1 + a_2 \mathbf{v}_2 + \cdots + a_n \mathbf{v}_n, where v1,v2,…,vn\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n are linearly independent basis vectors and a1,a2,…,ana_1, a_2, \ldots, a_n range over all integers. The basis vectors determine the geometry of the lattice, including the spacing and angles between neighboring points.

Key Formula

p=a1v1+a2v2+⋯+anvn,ai∈Z\mathbf{p} = a_1 \mathbf{v}_1 + a_2 \mathbf{v}_2 + \cdots + a_n \mathbf{v}_n, \quad a_i \in \mathbb{Z}
Where:
  • p\mathbf{p} = A point in the lattice
  • vi\mathbf{v}_i = The $i$-th basis vector of the lattice
  • aia_i = An integer coefficient
  • nn = The dimension of the lattice

How It Works

You build a point lattice by choosing basis vectors and then forming every integer combination of them. In 2D with the standard basis v1=(1,0)\mathbf{v}_1 = (1, 0) and v2=(0,1)\mathbf{v}_2 = (0, 1), the lattice is simply Z2\mathbb{Z}^2, the set of all integer-coordinate points. Changing the basis vectors skews or stretches the lattice. For instance, using v1=(1,0)\mathbf{v}_1 = (1, 0) and v2=(1/2, 3/2)\mathbf{v}_2 = (1/2,\, \sqrt{3}/2) produces a hexagonal lattice. The fundamental domain (or unit cell) is the parallelogram spanned by the two basis vectors, and tiling it fills the plane with lattice points at every vertex.

Worked Example

Problem: Given basis vectors v1=(2,0)\mathbf{v}_1 = (2, 0) and v2=(1,3)\mathbf{v}_2 = (1, 3), find the lattice point corresponding to a1=3a_1 = 3 and a2=−1a_2 = -1.
Step 1: Multiply each basis vector by its integer coefficient.
3v1=3(2,0)=(6,0),−1⋅v2=−1(1,3)=(−1,−3)3\mathbf{v}_1 = 3(2,0) = (6,0), \quad -1\cdot\mathbf{v}_2 = -1(1,3) = (-1,-3)
Step 2: Add the results to get the lattice point.
p=(6,0)+(−1,−3)=(5,−3)\mathbf{p} = (6,0) + (-1,-3) = (5, -3)
Answer: The lattice point is (5,−3)(5, -3).

Visualization

Why It Matters

Point lattices are central to crystallography, where atoms arrange in lattice patterns, and to number theory, where lattice-based arguments prove results like Minkowski's theorem. They also appear in modern cryptography — lattice-based encryption schemes are leading candidates for post-quantum security.

Common Mistakes

Mistake: Assuming every evenly spaced grid of points is a lattice.
Correction: A point lattice must include the origin and be closed under addition and subtraction of its points. A shifted grid (e.g., all points (x+0.5, y+0.5)(x + 0.5,\, y + 0.5) with integer x,yx, y) is not a lattice because it does not contain the origin.