Fundamental Pair of Periods

The fundamental pair of periods is a pair of complex numbers such that their ratio ω₂/ω₁ is not real. In other words, considered as vectors in, the two are not collinear. The lattice generated by ω₁ and ω₂ is

This lattice is also sometimes denoted as Λ(ω₁, ω₂) to make clear that it depends on ω₁ and ω₂. It is also sometimes denoted by Ω or Ω(ω₁, ω₂), or simply by 〈ω₁, ω₂〉. The two generators ω₁ and ω₂ are called the lattice basis.

The parallelogram defined by the vertices 0, and is called the fundamental parallelogram.

It is important to note that, while a fundamental pair generates a lattice, a lattice does not have any unique fundamental pair, that is, many (in fact, an infinite number) fundamental pairs correspond to the same lattice.