Venn diagram for 11 sets of objects (Image: Khalegh Mamakani and Frank Ruskey)

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A new rose has blossomed in the garden of mathematics: a flowery Venn diagram for 11 sets of objects.

Venn diagrams use overlapping circles to show all possible relationships between sets. But diagrams for more than two or three sets often require circles to be stretched, squeezed and turned in on themselves to cover the increased number of set relationships.


Such geometrical gymnastics were distasteful to British logician John Venn, who created the diagrams in 1880. What’s more, the results of these mathematical acrobatics tend to be too elaborate to be useful.

So, instead, mathematicians hunt for symmetrical diagrams, which are easier to understand and are proven to exist only for Venn diagrams with a prime number of sets. For purity’s sake, these diagrams must also be “simple”, meaning no more than two curves cross at any point.

Lucky strike

Previously, examples for simple, symmetric Venn diagrams with five and seven sets had been found – but no higher. Now Khalegh Mamakani and Frank Ruskey at the University of Victoria in British Columbia, Canada, have hit on the first simple, symmetric 11-set Venn diagram (pictured).

One of the sets is outlined in white, and the colours correspond to the number of overlapping sets. The team called their creation Newroz, Kurdish for “the new day”. The name also sounds like “new rose” in English, reflecting the diagram’s flowery appearance.

To find the rose-like diagram, the pair had to comb through myriad potential diagrams, represented as lists of numbers corresponding to the way the curves cross. Sifting through all of the possibilities for an 11-set diagram would be an impossible task even for the combined might of Earth’s computers, so the researchers narrowed the options by restricting the search to diagrams with a property called crosscut symmetry, meaning that a segment of each set crosses all the other sets exactly once.

Hardcore geometer

The same method has been used to find simple, symmetric seven-set diagrams. Still, the researchers knew there were no guarantees of success. “After searching for them for so long, the big surprise was to find one at all,” says Ruskey.

“I love the picture,” says Peter Cameron , a mathematician at Queen Mary, University of London. He says the computational techniques used to find Newroz might prove useful in representing other complex geometric objects.

However, the diagram itself is unlikely to have direct practical applications. “We use two and three-set Venn diagrams for working out simple logical puzzles,” Cameron says. “Beyond that, I don’t think anyone but the most hardcore geometer would use a Venn diagram.”

Reference: arxiv.org/abs/1207.6452