2: (0,1), (1,0) and (1,1). Consequently, if m > 3, there would be xi and xj with i ≠ j with both coordinates of xi − xj even, i.e. 1/2 (xi + xj) lies in Λ. But this is the midpoint of the line segment joining two interior points of edges and hence lies in C, the interior of the polygon. This again contradicts the fact that C is a fundamental domain. So reductio ad absurdum m ≤ 3, as claimed. Dirichlet–Voronoi domains For a lattice Λ in C = R2, a fundamental domain can be defined canonically using the conformal structure of C. Note that the group of conformal transformations of C is given by complex affine transformations with . These transformations preserve Euclidean metric up to a factor, as well as preserving the orientation. It is the subgroup of the Möbius group fixing the point at ∞. The metric structure can be used to define a canonical fundamental domain by (It is obvious from the definition that it is a fundamental domain.) This is an example of a Dirichlet domain or Voronoi diagram: since complex translations form an Abelian group, so commute with the action of Λ, these concepts coincide. The canonical fundamental domain for with is either a symmetric convex parallelogram or hexagon with centre 0. By conformal equivalence, the period ω can be further restricted to satisfy and . As Dirichlet showed ("Dirichlet's hexagon theorem", 1850), for almost all ω the fundamental domain is a hexagon. For , the midpoints of sides are given by ±1/2, ±ω/2 and ; the sides bisect the corresponding radii from 0 orthogonally, which determines the vertices completely. In fact the first vertex must have the form and with x and y real; so if , then and . Hence and . The six vertices are therefore and . Fundamental polygons in higher genus Overview Every compact Riemann surface X has a universal covering surface which is a simply connected Riemann surface . The fundamental group of X acts as deck transformations of and can be identified with a subgroup Γ of the group of biholomorphisms of . The group Γ thus acts freely on with compact quotient space /Γ, which can be identified with X. Thus the classification of compact Riemann surfaces can be reduced to the study of possible groups Γ. By the uniformization theorem is either the Riemann