Visions of Numberland, by Alex Bellos

One of the great perks of my job - writing about mathematics – is that I am always learning new mathematics. And I learned more new maths in my latest book than probably in any other project I have been involved in.

You might find this surprising since Visions of Numberland is a colouring book. That is, it contains black-and-white line drawings and the reader is invited to colour them in. It is accessible to anyone who can hold a crayon or pencil.

Yet for me and my collaborator Edmund Harriss, our aim for the book was more than just to provide a pretty canvas. It was to curate a gallery of beautiful images that would introduce readers to deep mathematical ideas. Our hope was that simply by contemplating the patterns, you gain some mathematical insight. If colouring is a meditative activity, why not meditate with deep ideas that over the last few centuries have fascinated the greatest mathematical minds!

We looked all across the discipline for images that were both aesthetically appealing and conceptually interesting. If there another popular mathematics book that contains as wide a spread of fields, we haven’t seen it. We have included images from number theory, topology, projective geometry, higher dimensional geometry, statistical physics, combinatorics, fractals, computer science, calculus, group theory, modular forms, complex arithmetic, Lie groups, tessellations, dynamical systems and many more.

Already your heart may be sinking – these include areas you would only ever discover at degree level or beyond. Yet for each image we believe it is possible to engage with the ideas behind it. It’s not just because together with each image we have an explanation that is understandable to the lay reader. It is that merely by looking at these mathematical images you are thinking about them.

Sometimes an image explains a thousand equations.

Take the Collatz conjecture, one of the most famous unsolved problems in mathematics. You can explain the conjecture in two sentences: take any number, if it is even half it, and if it is odd triple it and add one. If you repeat the procedure as many times as you need to, every initial number will lead to 1. Mathematicians are not close to proving the Collatz conjecture, although they believe it to be true. Why is it so difficult? Well, look at the picture of the Collatz conjecture in the book. All we have done is to follow the paths of every number under 10,000 under the simple ‘half it or triple and add one’ rule. And it looks like a mesh of seaweed. The image looks organic, irregular, complicated – there is nothing simple about it at all! The picture beautifully illustrates how such a basic piece of arithmetic leads to a stunning complexity.

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