Spherical Venn Diagrams with Involutory Isometries

Abstract

In this paper we give a construction, for any $n$, of an $n$-Venn diagram on the sphere that has antipodal symmetry; that is, the diagram is fixed by the map that takes a point on the sphere to the corresponding antipodal point. Thus, along with certain diagrams due to Anthony Edwards which can be drawn with rotational and reflective symmetry, for any isometry of the sphere that is an involution, there exists an $n$-Venn diagram on the sphere invariant under that involution. Our construction uses a recursively defined chain decomposition of the Boolean lattice.