Symmetry and the Monster: The Story of One of the Greatest Quests of Mathematics | |||

Ronan (Mark) | |||

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__Amazon Book Description__

- Mathematics is driven forward by the quest to solve a small number of major problems – the four most famous challenges being:-
__Fermat's Last Theorem___{1},__The Riemann Hypothesis___{2},__Poincaré's Conjecture___{3}, and- The quest for
__the "Monster" of Symmetry___{4}.

- Now, in an exciting, fast-paced historical narrative ranging across two centuries, Mark Ronan takes us on an exhilarating tour of this final mathematical quest.
- Ronan describes how the quest to understand symmetry really began with the tragic young genius Evariste Galois, who died at the age of 20 in a duel. Galois, who spent the night before he died frantically scribbling his unpublished discoveries, used symmetry to understand algebraic equations, and he discovered that there were building blocks or "atoms of symmetry." Most of these building blocks fit into a table, rather like the periodic table of elements, but mathematicians have found 26 exceptions.
- The biggest of these was dubbed "the Monster" – a giant snowflake in 196,884 dimensions. Ronan, who personally knows the individuals now working on this problem, reveals how the Monster was only dimly seen at first. As more and more mathematicians became involved, the Monster became clearer, and it was found to be not monstrous but a beautiful form that pointed out deep connections between symmetry, string theory, and the very fabric and form of the universe.
- This story of discovery involves extraordinary characters, and Mark Ronan brings these people to life, vividly recreating the growing excitement of what became the biggest joint project ever in the field of mathematics. Vibrantly written,
*Symmetry and the Monster*is a must-read for all fans of popular science-and especially readers of such books as "Singh (Simon) - Fermat's Last Theorem". is a Professor at the University of Illinois at Chicago, and a Visiting Professor of Mathematics at University College London.__Mark Ronan___{5}- This book tells for the first time the fascinating story of the biggest theorem ever to have been proved. Mark Ronan graphically describes not only the last few decades of the chase and the intriguing characters who led it, but also some of the more interesting byways, including my personal favourite, the one I called 'Monstrous Moonshine'.

→ John H. Conway, F.R.S.

- This is a history book about the history of research into group theory and the discovery of the "Monster", not a book about that Monster. The math has been simplified beyond recognition, and even after reading up on the subject in the Wikipedia and with a PhD in computer science, I could not make head or tail of it.
- The first problem is that the author does not make clear what he means by "a symmetry".
- We learn that the "zillions of symmetries" of the
__Rubik cube___{6}are "generated by 90 degree turns", which in the lines above are compared to "symmetry operators". This suggests that the 24 turns (4 on each of the 6 sides) are the operators and that the positions that can be achieved are the symmetries. But operators in a (mathematical) group have the property that the combination of two operators is again an operator in that group, so any configuration can be achieved with a single (compound) operator. So are all these operators "symmetries"? I find it confusing. - Symmetries are also explained as permutations, but the relationship remains vague.
- A second problem is that the level of explanation is very uneven: the root sign is explained, but the j-function is written out without any explanation.
- We learn a lot about the people around the Monster but next to nothing about the Monster itself, except that it is 196,884-dimensional, but that's already on the cover of the book. Does it have a geometric representation, like a cube? Or is it just a network of symbols? (Does a network of symbols have symmetries?) If it can be geometric, it must have sides. Are all sides the same length like in a cube or a dodecahedron? How big is it if the length of the shortest side is 1 unit? Answers to such questions would have made the Monster much more accessible.
- Perhaps the subject is too complicated to allow a popularized treatment, in which case sticking to just the history is OK. But it would have been nice to see an example or two of representatives of the simpler symmetry groups. Some examples are given, but they are not assigned to groups. And it would have been nice to be told to what position in the periodic table of symmetries Rubik's cube occupies, probably the most complicated symmetric object any of us can relate to.

- Popularisations of mathematics are difficult to do well because you need to have a fair amount of the language of maths under your belt before you can follow the arguments. To that end, putting across the ideas in a non-technical manner needs a skill that few possess.
- Ronan does a sparkling job here. The
__basic concepts of group theory___{7}are glossed over without going into tedious detail (and despite my affection for this particular branch of maths, I consider a lot of the detail *extremely* tedious), and once the story gets under way, the ideas are brought forward in a flowing, almost breathlessly excited, style which is infectious. - The author himself was involved in this stupendous quest of classification, so he knows what he's talking about.
- One of the aspects of such a popular account is the bringing to life of the people behind the name, many of whom I'd never heard, quite a few of whom I'd already encountered in my travels through an undergrad degree in mathematics. Neither does the author shrink from confronting the political circumstances in which certain of the mathematicians were working, which adds a further dimension of interest to the tale.
- The first thing one wants to do having read this book is to go and find out the mathematics behind it all. Be warned: it is difficult area to get to grips with. The basics are simple but the detail is diabolical.

- See Wikipedia: Monster group.
- Is this relatively recent ‘quest’ really up there with the others?
- What about
- See also Wikipedia: Hilbert's problems

- See Wikipedia: Rubik's Cube.
- This is on p. 49.

- As an incompetent undergraduate at Cambridge, I was struck by the wonders – especially in number theory – that could be discovered from a mathematical system with such simple axioms.
- however, he doesn’t give the axioms.
- Nor does Wikipedia: Group theory.
- See UCL: Definition of a Group.

- Oxford University Press, USA; Revised edition (4 Sept. 2007)
- Cited in "Hand (David) - The Improbability Principle: Why coincidences, miracles and rare events happen all the time", as an example of a meaningful, rather than chance, coincidence.

- Blue: Text by me; © Theo Todman, 2024
- Mauve: Text by correspondent(s) or other author(s); © the author(s)

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