Symmetric Venn Diagrams on the Sphere I: Chain Decompositions and Isometry Groups of Order Two
نویسندگان
چکیده
Let f be an isometry of the sphere which is an involution (that is, f−1 = f). For any n ≥ 1 we prove that there is an n-curve Venn diagram on the sphere whose isometry group is generated by f and which has the property that f(Θ) = Θ, where Θ is a curve of the diagram. One of the constructions uses a new chain decomposition of the Boolean lattice, a decomposition which is based on the well-known decomposition of N. G. de Bruijn, C. A. van Ebbenhorst Tengbergen, and D. R. Kruyswijk.
منابع مشابه
Venn Diagrams and Symmetric Chain Decompositions in the Boolean Lattice Preliminary Draft
In this paper we show that symmetric Venn diagrams for n sets exist for every prime n, settling an open question. Until this time, n = 11 was the largest prime for which the existence of such diagrams had been proven. We show that the problem can be reduced to finding a symmetric chain decomposition, satisfying a certain cover property, in a subposet of the Boolean lattice Bn, and prove that su...
متن کاملSimple Spherical Venn diagrams with Isometry Groups of Order Eight
For each n ≥ 4 we show how to construct simple Venn diagrams of n curves embedded on the sphere with the following sets of isometries: (a) a 4-fold rotational symmetry about the polar axis, together with an additional involutional symmetry about an axis through the equator, and (b) an involutional symmetry about the polar axis together with two reflectional symmetries about orthogonal planes th...
متن کاملGenerating simple convex Venn diagrams
In this paper we are concerned with producing exhaustive lists of simple monotone Venn diagrams that have some symmetry (non-trivial isometry) when drawn on the sphere. A 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. We show that there are 23 such 7-Venn diagrams with a 7-fold rotational s...
متن کاملVenn Diagrams and Symmetric Chain Decompositions in the Boolean Lattice
We show that symmetric Venn diagrams for n sets exist for every prime n, settling an open question. Until this time, n = 11 was the largest prime for which the existence of such diagrams had been proven, a result of Peter Hamburger. We show that the problem can be reduced to finding a symmetric chain decomposition, satisfying a certain cover property, in a subposet of the Boolean lattice Bn, an...
متن کاملSpherical Venn Diagrams with Involutory Isometries
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, t...
متن کامل