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.…

By | December 11th, 2021|Book reviews, Reviews|0 Comments

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.
By | March 15th, 2020|Book reviews, Reviews, Uncategorized|0 Comments

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.…

By | July 9th, 2019|Book reviews, Reviews|1 Comment

Relativity, The Special and General Theory, 100th anniversary edition – by Albert Einstein

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

In 1917, two years after publishing his work on The General Theory of Relativity, Einstein published a popular science account of both The Special, and General Theories of relativity. It is with some embarrassment that I have to admit that I’d never read this before, despite taking a number of undergraduate and postgraduate courses in relativity. Einstein understood the importance that his results had on our understanding of the universe, but also that the profundity of them could not truly be grasped by the general public, despite the headlines which covered many newspapers around the world on his results, without a popular exposition. 1917 was the publication of the first edition of this explication, but he continued to update them up until 1954. This allowed him to extend the theoretical discussion with the experimental verifications and discoveries which occurred over the next decades, including that of the expanding cosmology, spearheaded by Hubble’s observations.…

By | April 19th, 2019|Book reviews, Reviews|0 Comments

Data Visualization, a practical introduction – by Kieran Healy, a review

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

I’m not an expert on the R programming language, but I have dabbled, which meant that while this book is perhaps aimed at slightly more advanced users (I’ve used it a half a dozen times for Coursera courses), I had enough to appreciate the value of this really lovely resource.

The book can be seen, I think, in two ways. One of the ways, which is the one which most interests me, is in explaining what it is that makes good data visualization captivating, clear and unambiguous. Interleaved in these ideas of aesthetics are the precisel methods to go about making such visualizations using the ggplot package in R.

The other way to look at the book is as a way to really get to grips with the advanced features of the ggplot package, which is taught via interesting examples of data visualization.…

By | March 27th, 2019|Book reviews, Reviews|0 Comments

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.…

By | January 10th, 2019|Book reviews, Reviews, Uncategorized|1 Comment

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.…

By | December 13th, 2018|Book reviews, Reviews|2 Comments

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.…

By | November 20th, 2018|Book reviews, Reviews|0 Comments

How Behavior Spreads: The Science of Complex Contagions, by Damon Centola, a review

NB. This book was sent to me as a review copy.


The idea of this book is relatively simple, but the consequences are huge, and in fact some of the ideas are far more subtle and complex than they may first appear.

Essentially this book is based on a series of experiments which Damon Centola has run, which are all related to changes in behaviour which can be tracked, and made to occur, through a social network (in the broadest sense of the word). This is the study diffusion in a network.

The fundamentals of the research lie on two distinctions: One in the complexity of a contagion/behaviour, meaning how many connections with others who have the contagion/behaviour do you need until you adopt it, and the other in the topology of the social network, meaning loosely, how much like a street where each person only talks to their neighbours, versus a small world-network where there are a lot of disparate connections does the network look like.…

By | August 21st, 2018|Book reviews, Reviews|2 Comments

Calculus for the ambitious, by Tom Korner, a review, by Henri Laurie

Amazon link

This is a lovely book: strong emphasis on ideas; a lively sense of humour; a sure logical touch; historical detail that is accurate, relevant, yet quirky (takes some doing!). What’s not to like?

Well, there’s this: it is not easy to decide whether to recommend the book to anybody who doesn’t already know calculus. I’ll return to that. Let me start by describing why this is such a good book.

Firstly, the light touch and the clarity, which together make it wonderfully accessible. Fans of Tom Korner, including yours truly, will be happy to hear that it it as good as his “The Pleasures of Counting” and “Fourier Analysis”, two of the best books on maths ever. Like them, it discusses applications, social context and history but always in a way that supports the maths, which remains the main focus.

Secondly, the balance between rigour and intuition is superbly judged and maintained.…

By | June 20th, 2018|Book reviews, Reviews|0 Comments