Everything in its place

Do you own a dog? Do you ride a bike to school? Do you have black hair? Each of these questions divides people into two groups – you either have a dog, or you don’t. And if you had a large circle drawn on the ground, all the dog owners in your class could stand in that circle, so you could tell them apart from the people who don’t own a dog.

What if you wanted to know who owned a dog, and who rode a bike to school? You’d need a second circle for bike riders. But you have to be careful where you draw the second circle. Bike-riding dog-owners also need a place to stand. You’ll need to make the circles overlap so they can be in both circles at the same time.

A diagram with overlapping areas like this is called a Venn diagram. The picture below is a Venn diagram with three regions. You can check for yourself that everyone can find a space, whether they have a dog, ride a bike, have dark hair, or any combination of those three things.

Venn diagrams become much more complicated as you add more regions. Larger Venn diagrams can’t just use circles – they often have confusing shapes and are hard to understand. There are two things important rules that mathematicians like to follow to make large Venn diagrams clearer. First, they have to be symmetrical. This means a diagram can be rotated by a certain angle and it looks exactly the same. Secondly, they need to be simple. This means a diagram should not have any places where three or more lines cross.

Mathematicians from the University of Victoria in Canada discovered new techniques for generating large Venn diagrams that are symmetrical and simple. They found some important rules that the shapes had to follow, and used them to generate Venn diagrams.

Using this method, the mathematicians created Newroz, the first simple symmetric Venn diagram with 11 shapes. This means you could ask 11 yes or no questions, and everyone would have a place to stand, no matter how they answered!