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Academic Acknowledgements
Professor Martin Liebeck of Imperial College London, who taught me most of what I have since forgotten about Group Theory (no fault implied on his side). As well as lecturing me as an undergraduate, [then] Dr Liebeck was my MSc. supervisor and I rather wish that I had spent more time listening to him than in other pursuits.

Professor Robert Vaughan previously of Imperial College London, now at Penn State, who fulfilled a similar role with respect to Number Theory. Professor Vaughan lectured me as both an undergraduate and post graduate. He was guilty of awarding me a mark of 120% for my MSc. Number Theory exam, something which had both compelling logic and a sense of the surreal. It’s probably fair to say that I have never worked my brain harder than in preparation for my MSc. Number Theory exam.
Literary Acknowledgements

As explicitly acknowledged in the Foreword, reading Marcus Du Sautoy’s latest book, What we Cannot Know could be described as the distal cause of this book. I have read a few of Marcus’s tomes and – as I mention – he is maybe a shadowy reflection of what I might have been had I continued to pursue a career in academic Mathematics [1]. While I enjoyed reading Marcus’s work immensely (and would not want this comment to detract from my assessment of its value), a final unexplained reference to the mysterious SU(3) × SU(2) × U(1) was enough to push me into writing this book.

I have been an avid reader of popular science and Mathematics books since learning to read. Going back in time, two titles that had a profound impact on me were Riddles in Mathematics (which was awarded to my father as a school prize and opened a new world of Mathematical possibilities to me) and
Mathematics and the Imagination
(which broadened this view to include a larger universe).

About half way through writing this book, I decided to re-read one of Professor Ian Stewart‘s books on Group Theory, Why Beauty Is Truth: The History of Symmetry. The clue is in the title and this work is more a mathematical history than a maths book (not that I am questioning Professor Stewart’s ability, he is an esteemed Mathematician). While – at the author’s express intent – the Mathematics in Why Beauty is Truth is very much behind the scenes, revisiting some of the key events and players in the development of what is now called Group Theory helped me to structure elements of this book and to recall some interesting areas and results, which I might otherwise have overlooked.
At the beginning of this book, I stated that I am a Mathematician, not a Physicist. I have a background that enables me to remember or rederive most of the Mathematics that appears in these pages; this is not the case where it comes to Physics. Given this, I wanted some reliable guide to the terra incognita I was planning to visit. My able companion in this respect has been Group Theory in a Nutshell for Physicists by Theoretical Physicist Anthony Zee from the University of California at Santa Barbara. While I turned to Anthony’s book to educate me on the application of Group Theory to Physics, he ended up teaching me a few things about Group Theory itself that I either never knew or had forgotten. At the very least he provided a more tangible perspective on a number of topics, particularly Lie Theory. I would recommend both this book and the ones he has written for a more general audience.
Personal Acknowledgments

To be completed
Image Credits

Except where listed below, all images that appear in this book are drawn by the author and therefore © Peter James Thomas 2016-2017. Published under a Creative Commons Attribution 4.0 International License.


Contents Background Adapted by the author from RobertLovesPi’s Blog


Hexagon Drawn by the author, but based on numerous similar diagrams from multiple sources

Chapter 1 – Introduction

Standard Model Sourced from: Wikimedia Commons – Fermilab, Office of Science, United States Department of Energy, Particle Data Group, Creative Commons BY 3.0

Chapter 2 – What is a Group?

Group Theory Sourced from:
Black Box Drawn by the author, including gears photograph by Jay Divinagracia. Sourced from:
Plus Sign Sourced from:
Kalidoscope II Sourced from:

Chapter 3 – Shifting Shapes

Matryoshka Doll Sourced from:

Chapter 4 – Rationality & Reality

Sense and Sensibility Photo: REX. Sourced from:
Divide et impera Sourced from:
Escher mirror By M. C. Escher. Sourced from:
New Horizons Sourced from:

Chapter 5 – Tabular Amasser

The Matrix © 1999 Warner Bros Entertainment & Village Roadshow Films Limited. Sourced from:
Cards on the Table Ownership unknown. Sourced from:

Chapter 6 – Matrix Revolutions Matrix Transform © Randall Munroe 2006. Sourced from #184
Direction of Travel Ownership unknown. Sourced from
Geeneric Gyrations Ownership unknown. Sourced from Pinterest

Chapter 7 – Imaginary Battleships

Battleships © Juego. Sourced from
Fleet manoeuvres Photo released by the United States Navy. Sourced from Wikimedia Commons
A mad hat Ownership unknown. Sourced from myaliceinwonderlandblog.-
Mirror mirror on the wall Ownership unknown. Sourced from

Chapter 8 – Simplicity

Exception to the Rule © Lord Bullgod. Sourced from: Flicka/td>
Zeta function reciprocal © Eric W. Weisstein 1996-9. Sourced from:

Chapter 9 – Normality

Normal Illinois Ownership unknown. Sourced from: Chicago Tribune
The deciding factor Ownership unknown. Sourced from:

Chapter 10 – Profundity

De profundis Ownership unknown. Sourced from:
A simple algorithm © Sourced from:
Multiplication Redux © Sourced from:

Chapter 11 – Root of the Problem

Roots © AB Paisatgistes. Sourced from: Pinterest

Chapter 12 – Mont Évariste

Évariste Galois (1811-1832) Ownership unknown. Sourced from: Wikimedia Commons.

Chapter 13 – First Contact – U(1)

Radio telescope Milky Way © Wayne England. Sourced from:
Grandes Complications © Audemars Piguet 2016. Sourced from:
Buzz Lightyear © 1995 Pixar Animation Studios. Sourced from: cantorontheshore.-

Chapter 14 – Determination – U(2) & SU(2)

Determinant definition Ownership unknown. Sourced from:
It's complex © Mike Massee. Sourced from:
The Shape of Things to Come © Penguin Books (Random House), excerpted from the cover of H. G. Wells The Shape of Things to Come. Sourced from:
The other U2 Ownership unknown. Sourced from:

Chapter 15 – It’s Space Jim…

Space - The Nearby Frontier Ownership [one would assume] NASA. Sourced from:
The Red Arrows Ownership unknown. Sourced from:
An Unkindness of Vectors? Ownership unknown. Sourced from:
Scales and Scalars Ownership unknown. Sourced from:

Chapter 16 – …But not as we know it

Warped space © Warner Brothers Entertainment (2014). Sourced from:
John Constable - West End fields © The Alfred Felton Bequest. Sourced from:
Hubble image of a star field © NASA. Sourced from:
I was 21 years when I found this set, I’m 22 now, but I won’t feel regret © 2010 Billy Bragg’s Emporium. Sourced from:
Exempli Locis © 2017 Oxford University Press. Sourced from:

Chapter 17 – MatricesRedux

Symmetrical eigenfunction © Lance J. Putnam, JoAnn Kuchera-Morin and Luca Peliti. Sourced from
Rabbits Ownership unknown. Sourced from:
Establishing Ownership Ownership unknown. Sourced from: Twitter Pics.
Sturm-Liouville eigenvalues Ownership unknown. Sourced from: YouTube.
Hydrogen atom wave functions Ownership unknown. Sourced from: Wikimedia Commons.

Chapter 18 – The Lie of the Land

Sophus Lie (1842-1899) © The Municipal Archives of Trondheim. Sourced from: Wikimedia Commons.
Space Odyssey © Warner Brothers Entertainment. Sourced from:
Escher Ball By M. C. Escher. Sourced from:

Chapter 19 – Making Connections

The Creation of Adam By Michelangelo di Lodovico Buonarroti Simoni. Sourced from: Wikimedia Commons.
A torus By Oleg Alexandrov. Adapted from: Wikimedia Commons.

Chapter 20 – Power to Truth

Exponential Chessboard Ownership unknown. Sourced from:
There and Back Again By J.R.R. Tolkien. Sourced from:

Chapter 21 – SU(3) and the Meaning of Lie


Chapter 22 – The Final Frontier

David Hilbert Göttingen Faculty photograph, ownership unknown. Sourced from: Wikimedia Commons.


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Acknowledgements – Notes

With apologies to both Lawrence Kasdan and Marcus himself.

Text: © Peter James Thomas 2016-17.
Images: © Peter James Thomas 2016-17, unless stated above.
Published under a Creative Commons Attribution 4.0 International License.