Symmetries of Venn diagrams on the sphere




Weston, Mark Richard Nicholas

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A diagram on a surface is a collection of coloured simple closed curves which generally intersect only at points, and a Venn diagram of n curves has the additional property that there are exactly 2^n faces in the diagram, each corresponding to a unique intersection of the interiors of a subset of the curves. A diagram has rotational symmetry if it can be constructed by rotating a single closed curve in the plane n times, each time by 2 \pi /n, and changing the colour of the curve for each rotation; equivalently, the diagram can be constructed from a region forming a "pie-slice" of the diagram and containing a section of each curve, and then copying and rotating this region n times, recolouring the sections of curves in the region appropriately. This and reflective symmetries are the only non-trivial ways a finite plane diagram can have some kind of symmetry. In this thesis, we extend the notion of planar symmetries for diagrams onto the sphere by constructing and projecting diagrams onto the sphere and examining the much richer symmetry groups that result. Restricting our attention to Venn diagrams gives a rich combinatorial structure to the diagrams that we examine and exploit. We derive several constructions of Venn diagrams with interesting symmetries on the sphere by modifying the landmark work of Griggs, Killian and Savage from 2004 which provided some important answers to questions about planar symmetric diagrams. We examine a class of diagrams that exhibit a rotary reflection symmetry (a rotation of the sphere followed by a reflection), in which we make some initial steps towards a general construction for n-Venn diagrams realizing a very rich symmetry group of order 2n, for n prime or a power of two. We also provide a many-dimensional construction of very simple Venn diagrams which realize any subgroup of an important type of symmetry group that use only reflection symmetries. In summary, we exhibit and examine at least one Venn diagram realizing each of the 14 possible different classes of finite symmetry groups on the sphere, many of these diagrams with different types of colour symmetry. All of these investigations are coupled with a theoretical and practical framework for further investigation of symmetries of diagrams and discrete combinatorial objects on spheres and higher-dimensional surfaces.



symmetry, groups, Venn diagrams, sphere