Rotationally Symmetric Venn Diagrams

Everyone knows the classic Venn diagram for 3 sets consisting of 3 circles.

This is rotationally symmetric and it is not hard to see that such can exist only when n is prime. The cases of 5 and 7 had been settled some time ago (when n = 5 one can do it with 5 ellipses), but the cases beyond that were challenging. Peter Hamburger obtained a rotationally symmetric Venn diagram when n = 11 in 2002, and more recently Griggs, Killian, and Savage showed that such exist for every prime. Much more information can be found in the excellent survey at <<http://www.combinatorics.org/Surveys/ds5/VennEJC.html>>

Using a Mathematica package I had written some years ago for four-coloring planar maps, I was able to manipulate the various maps and graphs used in these constructions, and so can present here the Venn diagram based on the Boolean structure of the dual found by Killian, Ruskey, Savage, and Weston, which is an extension of the GKS work.

Here is the map one first needs. This is a planar map with 2048 vertices (one is a point at infinity) and 1397 faces, and was found by KRSW. It is 4-colored using my 4-coloring algorithm. The dual of this map gives the Venn diagram.

Here is the dual to the Venn model, which is an exact Venn diagram. The map has 2048 regions, including the exterior, which corresponds to the empty set; there are 1397 intersections (out of a maximum possible 2046). They are colored according to Venn rank. This color scheme was worked out by Ruskey, Savage, and Wagon.

Here is one generator. Rotating this 10 times gives the 2048 regions.

And here is an image that combines everything. We see the Venn regions inside one generator only, and it is all superimposed on the 4-colored Venn model.

Created by Wolfram Mathematica 6.0 (28 October 2007) |