Generating Simple Convex Venn Diagrams
Khalegh Mamakani,
Department of Computer Science,
University of Victoria, Canada.
Wendy Myrvold,
Department of Computer Science,
University of Victoria, Canada.
Frank Ruskey,
Department of Computer Science,
University of Victoria, Canada.
Abstract:
In this paper we are concerned with producing exhaustive lists of
simple monotone Venn diagrams that have some symmetry (nontrivial
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 7Venn diagrams with a 7fold rotational
symmetry about the polar axis, and that 6 of these have an additional
2fold rotational symmetry about an equatorial axis.
In the case of simple monotone 6Venn diagrams, we show that there are
39020 nonisomorphic planar diagrams in total, and that 375 of them have
a 2fold symmetry by rotation about an equatorial axis, and amongst
these we determine all those that have a richer isometry group on the sphere.
Additionally, 270 of the 6Venn diagrams also have the 2fold symmetry
induced by reflection about the center of the sphere.
Since such exhaustive searches are prone to error, we have implemented
the search in a couple of ways, and with independent programs.
These distinct algorithms are described.
We also prove that the Grübaum encoding can be used to efficiently
identify any monotone Venn diagram.

This paper is the journal version of the merging and improvement of the two
conference papers listed below. You should definitely read this journal paper
rather than either of these two.

Files: pdf.

A full list of all 375 6Venn diagrams with a nontrivial isometry:
PolarSixVenn.html.

Submitted to Journal of Discrete Algorithms, DATE?, 2012.

Final acceptance, April 25, 2012.

Some of the code in this paper is available:
Code for 6Venn and 7Venn Diagrams.

Please send me a note if
you download one of these files.
It's always nice to know who's reading your papers.

Selected citations:
Back to list of publications.