FREE ARRANGEMENTS AND RHOMBIC TILINGS

Let Z be a centrally symmetric polygon with integer side lengths. We a nswer the following two questions: (1) When is the associated discrimi nantal hyperplane arrangement free in the sense of Saito and Terao? (2 ) When are all of the tilings of Z by unit rhombi coherent in the sens e of Billera and Sturmfels? Surprisingly, the answers to these two que stions are very similar. Furthermore, by means of an old result of Mac Mahon on plane partitions and some new results of Elnitsky on rhombic tilings, the answer to the first question helps to answer the second. These results then also give rise to some interesting geometric coroll aries. Consideration of the discriminantal arrangements for some parti cular octagons leads to a previously announced counterexample to the c onjecture by Saito [ER2] that the complexified complement of a real fr ee arrangement is a K(pi, 1) space.