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An n-Venn diagram consists of n curves drawn in the plane in such a way that each of the 2 n 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… (More)

Every permutation of {1, 2,. .. , N } can be written as the product of two involutions. As a consequence, any permutation of the elements of an array can be performed in-place using simultaneous swaps in two rounds of swaps. In the case where the permutation is the k-way perfect shuffle we develop two methods for efficiently computing the pair of… (More)

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… (More)

An n-Venn diagram consists of n curves that divide the plane into 2 n connected regions, one region for each possible intersection of the interiors of the curves. We show there are exactly 406 6-Venn diagrams that (a) have 6 curves, (b) are simple (at most two curves intersect at any point), (c) can be drawn with all curves convex, and (d) are invariant… (More)

Every permutation of {1, 2,. .. , N } can be written as the product of two involutions. As a consequence, any permutation of the elements of an array can be performed in-place using simultaneous swaps in two rounds of swaps. In the case where the permutation is the k-way perfect shuffle we develop two methods for efficiently computing the pair of… (More)

A symmetric Venn diagram is one that is invariant under rotation, up to a relabel-ing of curves. A simple Venn diagram is one in which at most two curves intersect at any point. In this paper we introduce a new property of Venn diagrams called crosscut symmetry, which is related to dihedral symmetry. Utilizing a computer search restricted to crosscut… (More)

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… (More)

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