Symmetric Monotone Venn Diagrams with Seven Curves

An n-Venn diagram consists of n curves drawn in the plane in such a way that each of the 2 possible intersections of the interiors and exteriors of the curves forms a connected non-empty region. A k-region in a diagram is a region that is in the interior of precisely k curves. A n-Venn diagram is symmetric if it has a point of rotation about which rotations of the plane by 2π/n radians leaves the diagram fixed; it is polar symmetric if it is symmetric and its stereographic projection about the infinite outer face is isomorphic to the projection about the innermost face. A Venn diagram is monotone if every k-region is adjacent to both some (k − 1)region (if k > 0) and also to some k+1 region (if k < n). A Venn diagram is simple if at most two curves intersect at any point. We prove that the “Grünbaum ” encoding uniquely identifies monotone simple symmetric n-Venn diagrams and describe an algorithm that produces an exhaustive list of all of the monotone simple symmetric n-Venn diagrams. There are exactly 23 simple monotone symmetric 7-Venn diagrams, of which 6 are polar symmetric.