genus one, a fundamental convex polygon is sought for the action by translation of Λ = Z a ⊕ Z b on R2 = C where a and b are linearly independent over R. (After performing a real linear transformation on R2, it can be assumed if necessary that Λ = Z2 = Z + Z i; for a genus one Riemann surface it can be taken to have the form Λ = Z2 = Z + Z ω, with Im ω > 0.) A fundamental domain is given by the parallelogram for where and are generators of Λ. If C is the interior of a fundamental convex polygon, then the translates + x cover R2 as x runs over Λ. It follows that the boundary points of C are formed of intersections ∩ ( + x). These are compact convex sets in ∂C and thus either vertices of C or sides of C. It follows that every closed side of C can be written this way. Translating by −x it follows that ∩ ( − x) is also a side of C. Thus sides of C occur in parallel pairs of equal length. The end points of two such parallel segments of equal length can be joined so that they intersect and the intersection occurs at the midpoints of the line segments joining the endpoints. It follows that the intersections of al such segments occur at the same point. Translating that point to the origin, it follows that the polygon is centrally symmetric; that is, if a point z is in the polygon, so too is −z. It is easy to see translates of a centrally symmetric convex hexagon tessellate the plane. If A is a point of the hexagon, then the lattice is generated by the displacement vectors AB and AC where B and C are the two vertices which are not neighbours of A and not opposite A. Indeed, the second picture shows how the hexagon is equivalent to the parallelogram obtained by displacing the two triangles chopped off by the segments AB and AC. Equally well the first picture shows another way of matching a tiling by parallelograms with the hexagonal tiling. If the centre of the hexagon is 0 and the vertices in order are a, b, c, −a, −b and −c, then Λ is the Abelian group with generators and . Examples of Fundamental