NB I was sent this book as a review copy.

What a beautiful idea. What a beautiful book! In studying mathematics, one comes across various different curves while studying calculus, or number theory, or geometry in various forms and they are asides of the particular subject. The idea however of flipping the script and looking at curves themselves and from them gaining insight into: statistics, combinatorics, number theory, analysis, cryptography, fractals, Fourier series, axiomatic set theory and so much more is just wonderful.

This book looks at ten carefully chosen curves and from them shows how much insight one can get into vast swathes of mathematics and mathematical history. The curves chosen are:

- The Euler Spiral – an elegant spiralÂ which leads to many other interesting parametrically defined curves
- The Weierstrass Curve – an everywhere continuous but nowhere differentiable function
- Bezier Curves – which show up in computer graphics and beyond
- The Rectangular Hyperbola – which leads to the investigation of logarithms and exponentials
- The Quadratrix of Hippies – which are tightly linked to the impossible problems of antiquity
- Peano’s Function and Hilbert’s Curve – space filling curves which lead to a completely flipped understanding of the possibilities of infinitely thin lines
- Curves of Constant Width – curves which can perfectly fit down a hallway as they rotate. Naively the circle being the only one, but in fact there are infinitely many, and they tie in to Feynman’s investigation of the Challenger disaster.
- The Normal Curve – of course giving the seeds of so many statistical investigations
- The Catenary – something which deceived many and was important in the development of calculus
- Elliptic Curves – and their relationship to number theory and cryptography

If you can find another book which deals with so many disparate but linked areas of mathematics in 200 or so pages in such a beautiful way, I would love to see it.

The text is compelling, the mathematical proof are thorough, the historical connections are well-researched.

All in all it feels that this book should be bedside reading for all budding mathematicians.

## Leave a Reply