Given an integer k greater than 1, let S(k) be the space of complete embedded singly periodic minimal surfaces in R^3, which in the quotient have genus zero and 2k Scherk-type ends. It is well known that S(2) consists of the 1-parameter family of singly periodic Scherk minimal surfaces. We prove that for each k greater than 2, there exists a natural one-to-one correspondence between S(k) and the space of convex unitary nonspecial polygons through the map which assigns to each M in S(k) the polygon whose edges are the flux vectors at the ends of M. (Here a special polygon is a parallelogram with two sides of length 1 and two sides of length k-1). As consequence, S(k) reduces to the Saddle Towers constructed by Karcher.