Pretty drawings. More doodles and doilies, symmetric venn diagrams

Abstract

In this paper we generalize the concept of a doodle, and the method that was developed in [11] for constructing doilies. Using this generalized method we systematically study doilies, that is, symmetric Venn diagrams, for small numbers of curves. We show that a p-doily exists for any possible size of vertex set, if p = 1, 2, 3, 5, 7. This is the ground-work for showing that 11-doilies exist for any possible size of vertex set, answering a question of Grünbaum. This will appear in a series of forthcoming papers, [12]. In this paper we show only that there are at least 50 non-isomorphic 11-doilies with vertex sets, with sizes 462 through 1001, increasing in every step by 11. In fact without checking all the details we show that with these properties there are at least 2 ⁴⁹ such (non-isomorphic) diagrams. This result is an immediate consequence of the results of the paper [11].