## Visual Differential Geometry and Forms – a mathematical drama in five acts, by Tristan Needham – a review

NB. I was sent this book as a review copy.

Studying physics, some two decades ago at The University of Bristol, I found the majority of what we covered relatively intuitive. Even the arcane world of quantum mechanics, while impossible to truly visualise, is, paradoxically, often relatively simple to calculate, and the objects that you use are directly from the world of complex numbers, differential equations and linear algebra. What stumped me however were tensors. I found it so hard to really picture what was going on with these objects. Vectors were ok, and the metric tensor I could handle, but as soon as you got onto differential forms, all my intuition went out the window. The world of differential geometry, while I could plug and chug, felt like putting together sentences in a foreign language where all I had were rules for using the syntax and grammar, without a deep understanding of what the objects were

This book would have answered all of my prayers back then.…

## Curves for the Mathematically Curious – an anthology of the unpredictable, historical, beautiful and romantic, by Julian Havil – a review

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:

1. The Euler Spiral – an elegant spiral which leads to many other interesting parametrically defined curves
2. The Weierstrass Curve – an everywhere continuous but nowhere differentiable function
3. Bezier Curves – which show up in computer graphics and beyond
4. The Rectangular Hyperbola – which leads to the investigation of logarithms and exponentials
5. The Quadratrix of Hippies – which are tightly linked to the impossible problems of antiquity
6. Peano’s Function and Hilbert’s Curve – space filling curves which lead to a completely flipped understanding of the possibilities of infinitely thin lines
7. Curves of Constant Width – curves which can perfectly fit down a hallway as they rotate.

## Prime Suspects – The anatomy of integers and permutations, by Andrew Granville and Jennifer Granville, illustrated by Robert Lewis – a review

NB I was sent this book as a review copy.

What a spectacular book! I am rather blown away by it. This is a graphic novel written about two bodies discovered by cops in an American city some time around the present day, and the forensic investigation which goes into solving the case, and somehow the authors have managed to make the whole book about number theory and combinatorics.

I have to admit that when I started reading the book I was worried that it was going to have the all-too-common flaw of starting off very simple and then suddenly getting way too complicated for the average reader, but they have managed to somehow avoid that remarkably well.

It is however a book that should be read with pen and paper, or preferably computer by one’s side. As I read through and mathematical claims were made, about prime factors of the integers and about cycle groups of permutations, I coded up each one to see if I was following along, and I would recommend this to be a good way to really follow the book.…

## How to Fall Slower Than Gravity And Other Everyday (and Not So Everyday) Uses of Mathematics and Physical Reasoning – by Paul J. Nahin, a review

NB. I was sent this book as a review copy.

This book is without a doubt the most enjoyable, stimulating book of mathematical physics (and occasionally more pure branches of maths) puzzles that I have ever read. It’s essentially a series of cleverly, and occasionally fiendishly put-together mathematics and physics challenge questions, each of which gets you thinking in a new and fascinating way.

The level of mathematics needed is generally only up to relatively basic calculus, though there is the occasional diversion into a slightly more complex area, though anyone with basic first year university mathematics, or even a keen high school student who has done a little reading ahead, would be able to get a lot from the questions.

I found that there were a number of ways of going through the questions. Some of them are enjoyable to read, and simply ponder. For me, occasionally figuring out what should be done, without writing anything down, was enough to be pretty confident that I saw the ingenuity in the puzzle and the solution and I was happy to leave it at that.…

## Millions, Billions, Zillions – Defending Yourself in a World of Too Many Numbers – by Brian W. Kernighan, a review

NB. I was sent this book as a review copy.

I have to admit that I was skeptical about this book when I first saw it, and even on browsing through it became more so (read on for the but…). I count myself as a highly numerate person who has a reasonable awareness of the world of numbers around me and I thought that the book probably wouldn’t help me to navigate through the world that I already feel comfortable in.

The book is essentially a series of short chapters which discuss some of the ways that numbers are used, misused and mistakenly used in the media, from errors in units, to orders of magnitude, to the ways that graphs can misrepresent data either intentionally or unintentionally to the improbable precision so often used online and in print. Each chapter uses news headlines and quotes to highlight how such mistakes come about and the examples are extremely clear.…

## The Mathematics of Secrets – by Joshua Holden, a review

NB. I was sent this book as a review copy.

This is an extremely clearly, well-written book covering a lot of ground in the mathematics of cyphers. It starts from the very basics with simple transposition cyphers and goes all the way through to elliptic cyphers, public key cryptography and quantum cryptography. Each section gives detailed examples where you can follow precisely the mathematics of what underlies the encryption. Indeed the mathematics is non-trivial in a fair number of places, but it is always explained well, and I think that anyone with a first year university level of mathematics should be able to understand the bulk of it. I think that if you were to come at this book with a high-school level of mathematics, there would be some aspects which would be pretty hard work, but with some persistence, even those would be understandable, and perhaps the breakthroughs in understanding would feel like a great (though doable) achievement for the maths enthusiast.…

## Music by the Numbers, From Pythagoras to Schoenberg – By Eli Maor, a review

NB. I was sent this book as a review copy.

Music by the numbers leads us on a journey, as stated in the title, from Pythagoras to Schoenberg. In many ways the endpoint is stated early on, giving us clues that a revolution in mathematical thinking about musical scales will be encountered in the early twentieth century. Indeed the journey through musical practice, mathematics, physics and the biology of hearing is woven rather beautifully together, giving the account of our step by step explorations of tonal systems and their links to the physics of vibration. The development of calculus and the triumph of Fourier take as from the somewhat numerological and empiric realms of musical experimentation to the age of a true understanding of timbre – the way different instruments express harmonics and their overtones in different admixtures. A lot of emphasis is placed on the development of scales based on subtly different frequency ratios, which were developed over the years (particularly within European music, non-European music being given only very brief comment) to balance the physical, mathematical and aesthetic qualities of the various possible tunings of instruments.…

## What can be computed? A practical guide to the theory of computation – by John MacCormick, a review

NB. I was sent this book as a review copy.

It’s not often that a textbook comes along that is compelling enough that you want to read it from cover to cover. It’s also not often that the seed of inspiration of a textbook is quoted as being Douglas Hofstadter’s Pulitzer prize-winning book Godel, Escher Bach. However, in the case of “What can be computed”, both of these things are true.

I am not a computer scientist, but I have spent some time thinking about computability, Turing machines, automata, regular expressions and the like, but to read this book you don’t even need to have dipped your toes into such waters. This is a textbook of truly outstanding clarity, which feels much more like a popular science book in terms of the journey that it takes you on. If it weren’t for the fact that it is a rigorous guide to the theory of computability and computational complexity, complete with a lot of well thought through exercises, formal definitions and huge numbers of examples, you might be fooled by the easy-reading nature of it into thinking that this book couldn’t take you that far.…

## On Gravity, a brief tour of a weighty subject – By Tony Zee, a review

NB. I was sent this book as a review copy.

In the era when our eyes are being opened to the Universe in the gravitational spectrum via the recent gravitational wave observations, this book is exactly what is needed to communicate to the general public the beauty and depth of Einstein’s theory of gravity, as well as the interplay between gravity and quantum mechanics which takes place at the event horizon of a black hole.

Starting with the observations of merging black holes black in 2016, Tony Zee takes the reader on a clear and swift journey through the ideas of the incredible weakness of gravity, the basics of field theory, relativity, curved space-times, quantum weirdness, black holes and Hawking radiation, back, full circle to the consequences of General Relativity including the existence of gravitational waves and the detectors now observing them and those which will give us a far clearer picture of the gravitational universe in the near future.…

## Hypatia, The Life and Legend of an Ancient Philosopher – by Edward J. Watts, a review by Henri Laurie

Review written by Henri Laurie.

This is an important book for anybody interested in the history of mathematics and in the history of women intellectuals.

To recap very briefly: Hypatia is well-known as the mathematician/philosopher who was murdered by a Christian mob in 415 CE in Alexandria. She is one of the best-attested woman philosophers in the Greek tradition.

Watts turns this on its head: he tells the story of a life, one of singular achievement, and one in which the manner of death is not the most important part. The picture he paints is of a very remarkable woman, who became the head of her father’s school at a relatively young age and came to dominate the scholarly activity of her city, at the time one of the three most important centres of learning in the Mediterranean.

It is important to realise that although women did study philosophy at the time, and therefore also mathematics, which was seen as preparation for philosophy, very few of them were able to continue well into adulthood.…