Seven-Set Venn Diagrams with Rotational and Polar Symmetry
Combinatorics, Probability and Computing
Generating all simple convexly-drawable polar symmetric 6-venn diagrams
IWOCA'11 Proceedings of the 22nd international conference on Combinatorial Algorithms
Generating simple convex Venn diagrams
Journal of Discrete Algorithms
Diagrams'12 Proceedings of the 7th international conference on Diagrammatic Representation and Inference
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An n-Venn diagram consists of n curves drawn in the plane in such a way that each of the 2n 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.