Fundamental Region
Define τ = ω2/ω1 to be the half-period ratio. Then the lattice basis can always be chosen so that τ lies in a special region, called the fundamental domain. Alternately, there always exists an element of PSL(2,Z) that maps a lattice basis to another basis so that τ lies in the fundamental domain.
The fundamental domain is given by the set D, which is composed of a set U plus a part of the boundary of U:
where H is the upper half-plane.
The fundamental domain D is then built by adding the boundary on the left plus half the arc on the bottom:
If τ is not i and is not t=exp(1/3*pi*i), then there are exactly two lattice bases with the same τ in the fundamental region: namely, and . If then four lattice bases have the same τ: the above two and . If t=exp(1/3*pi*i) then there are six lattice bases with the same τ:, and their negatives. Note that and t=exp(1/3*pi*i) in the closure of the fundamental domain.
Read more about this topic: Fundamental Pair Of Periods
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