Fearless Symmetry: Exposing the Hidden Patterns of Numbers (New Edition) [Paperback]

Whether you like this book or not, you cannot deny that the authors have been very brave to write a popular mathematics book on one of the most abstract parts of mathematics. The central topic of this book is Galois Theory. This theory was developed by Evariste Galois to show that it was possible to solve equations of one variable using addition, subtraction, multiplication, division, and the extraction of roots only for equations of order less than five. This theory has led to what is now called modern or abstract algebra, which probably ties together more areas of mathematics than any other. The problem is that it deals with many highly abstract mathematical objects. Equations, complex numbers, modular arithmetic, and permutations on the one hand are not so difficult to explain and understand. It is however very different for symmetry, groups and their representations, and elliptic curves. It is amazing how far the authors can reduce the complexity of the mathematics, but the book is not an easy one to read. However, if you are looking for an introductory book on abstract algebra, then this is certainly one to consider.

Let me say at the outset that I have not read all of this book. That's because I keep going back a chapter to make sure I have understood what has gone before.
I can only say that this book has taken me further than any other into the world of advanced pure mathematics. The text is pervaded with a sense that the authors really want the reader to understand this difficult and esoteric subject. What more can I say? Buy a copy. Some day it will be a classic.

Firstly, this book is not for the layman. This is despite the opening line on the back cover of the book. It may be the intention of the authors that this book is for everyone, but the level is far too high (even graduates and quite possibly maths professors outside of pure mathematics). The actual subject matter is an attempt to explain the proof of Andrew Wiles's proof of Fermat's Last Theorem. This greatly relies upon groups, and hence symmetry which is how the name of the book came about.

It could be that the publisher has prevented the book from being much longer but I could imagine a book twice the size being more fitting. The content of the book is incredibly interesting and is something I want to pursue. There is no shying away from algebra at all and there are no stories about the life of mathematicians. That, for me, is a big selling point. There are many more books out there that cover the lives of mathematicians past and present. This book just sticks to the mathematics, which is something I'd like to see more of going forward. I don't think that just sticking to mathematics means that a book has to be presented at academic research level. If it is possible for this book to be re-published with fuller explanations then it would be one of the greatest books ever: it explains a fascinating and difficult topic of mathematics in logically inductive way using clear examples that are simple to follow. That is the intention of this book, but it doesn't quite get there.

This book is getting a lot of high ratings on Amazon but I'm going to be bold and suggest the reason for the high ratings is due to the authors' *attempt* to delve into some incredibly interesting mathematics, but not based upon the actual delivery of the content. That said, the first few chapters were mostly fine. This seems to be a common theme of Amazon reviews for this book. The second half of this book quickly descends into some very opaque explanations of how everything fits together. Unfortunately, I felt slightly short changed.

The book starts with explaining what a function is. This is basic high school mathematics that even 13 or 14 year olds could grasp. The first few chapters are quite short and fairly easily to grasp if you have an interest in mathematics and at least a good grasp of high school mathematics. Group theory is presented in the 2nd chapter which is again quite clear, no prior knowledge is necessary. The next few chapters cover permutations, complex numbers, modular arithmetic, finding roots of polynomials, all of these together comprise the first ''part'' of the book and all of these topics are ones that I met at the latter end of high school. The language is slightly more abstract and advanced but not much more than what was covered in high school. Even matrices which are covered in part 2 of the book were covered at high school and again all the topics mentioned were repeated in my undergraduate course in physics.

What I lacked in my education was Group theory and all that goes with it. I've managed to pick up the basics of group theory since graduating which was enough to see me through the first half of this book. I struggled in parts, such as with Frobenius (Chapter 16, p 177) which is just over half way. From here onward the explanations are less clear and feel much quicker. Furthermore, most of the explanations rely upon the Frobenius methods which makes the latter half even more opaque if one's understanding of chapter 16 is not complete. I struggled onward and got the general flavour of what was being done and how everything fits together, and I dare say I could wing it during a 'party' if I had to re-explain it, but I would definitely say that I did not understand it fully.

There is at least some reward for trying this book which is a deeper look into mathematics. It should encourage the reader to find more supplementary material in order to gain a better understanding of the topic, however this book fails to provide the complete understanding that is implied from the cover of the book. This subject is definitely more accessible than it was before which is where the authors should be commended. I'd like to give this book a higher rating but I feel that the intentions and the content don't closely match each other enough to warrant it.