Searching for Simple Symmetric Venn Diagrams

Date

2013-07-24

Authors

Ahmadi Mamakani, Abdolkhalegh

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Abstract

An n-Venn diagram is defined as a collection of n finitely intersecting closed curves dividing the plane into 2^n distinct regions, where each region is in the interior of a unique subset of the curves. A Venn diagram is simple if at most two curves intersect at any point, and it is monotone if it has some embedding on the plane in which all curves are convex. An n-Venn diagram has n-fold rotational symmetry if a rotation of 180 degrees about a centre point in the plane leaves the diagram unchanged, up to a relabeling of the curves. It has been known that rotationally symmetric Venn diagrams could exist only if the number of curves is prime. Moreover, non-simple Venn diagrams with rotational symmetry have been proven to exist for any prime number of curves. However, the largest prime for which a simple rotationally symmetric Venn diagram was known prior to this, was 7. In this thesis, we are concerned with generating simple monotone Venn diagrams, especially those that have some type(s) of symmetry. Several representations of these diagrams are introduced and different backtracking search algorithms are provided based on these representations. Using these algorithms we show that there are 39,020 non-isomorphic simple monotone 6-Venn diagrams in total. In the case of drawing Venn diagrams on a sphere, we prove that there exists a simple symmetric n-Venn diagram, for any n >= 6, with the following set(s) of isometries : (a) a 4-fold rotational symmetry about the polar axis, together with an additional involutional symmetry about an axis through the equator, or (b) an involutional symmetry about the polar axis together with two reflectional symmetries about orthogonal planes that intersect at the polar axis. Finally, we introduce a new type of symmetry of Venn diagrams which leads us to the discovery of the first simple rotationally symmetric Venn diagrams of 11 and 13 curves.

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Keywords

Simple Venn diagrams, Bounded Venn diagrams, Crosscut symmetry, Newroz, Polar symmetry, Rotational symmetry

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