Polygons Generated by Parallelograms There are exactly four topologies that can be created by identifying the sides of a rhombus in different ways. They are given below as directional edges A and B on a square, either as AABB or ABAB sequences. Fedorov's theorem Fedorov's theorem, established by the Russian crystallographer Evgraf Fedorov in 1891, asserts that parallelograms and centrally symmetric hexagons are the only convex polygons that are fundamental domains. There are several proofs of this, some of the more recent ones related to results in convexity theory, the geometry of numbers and circle packing, such as the Brunn–Minkowski inequality. Two elementary proofs due to H. S. M. Coxeter and Voronoi will be presented here. Coxeter's proof proceeds by assuming that there is a centrally symmetric convex polygon C with 2m sides. Then a large closed parallelogram formed from N2 fundamental parallelograms is tiled by translations of C which go beyond the edges of the large parallelogram. This induces a tiling on the torus C/NΛ. Let v, e and f be the number of vertices, edges and faces in this tiling (taking into account identifications in the quotient space). Then, because the Euler–Poincaré characteristic of a torus is zero, On the other hand, since each vertex is on at least 3 different edges and every edge is between two vertices, Moreover, since every edge is on exactly two faces, Hence so that as required. Voronoi's proof starts with the observation that every edge of C corresponds to an element x of Λ. In fact the edge is the orthogonal bisector of the radius from 0 to x. Hence the foot of the perpendicular from 0 to each edge lies in the interior of each edge. If y is any lattice point, then 1/2 y cannot lie in C; for if so, –1/2 y would also lie in C, contradicting C being a fundamental domain for Λ. Let ±x1, ..., ±xm be the 2m distinct points of Λ corresponding to sides of C. Fix generators a and b of Λ. Thus xi = αi a + βi b, where αi and βi are integers. It is not possible for both αi and βi to be even, since otherwise ± 1/2 xi would be a point of Λ on a side, which contradicts C being a fundamental domain. So there are three possibilities for the pair of integers (αi, βi) modulo