This year sees the 50th anniversary of the UK’s Institute of Mathematics and its Applications (IMA). As part of the celebrations, the IMA has published a book called 50 Visions of Mathematics.
Edited by Sam Parc, the book contains 50 mini chapters covering many aspects of mathematics written for the general reader. Topics include subjects such as ‘Whats the use of a quadratic equation’, ‘The mathematics of obesity’, ‘A glass of bubbly’ and ‘Motorway Mathematics’. The full list can be found under the ‘Table of Contents’ tab at http://ukcatalogue.oup.com/product/9780198701811.do
The level of mathematics required to understand this book isn’t high at all which makes it suitable for anyone with even a passing interest in the subject.
Along with the 50 essays, there are also 50 full colour mathematical images with number 6 being contributed by yours truly — based on some Mathematica code I wrote a few years ago.
Back in the good old days when I was a freshly minted postgraduate student I had big plans– In short, I was going to change the world. Along with a couple of my friends I was going to revolutionize the field I was working in, win the Nobel prize and transform the way science and mathematics is taught at University. Fast forward four years and it pains me to say that my actual achievements fell rather short of these lofty ideals. I considered myself lucky to simply pass my PhD and land a job that didn’t involve querying members of the public on their preferences regarding potato based products. The four subjects of Laura Snyder’s latest book, The Philosophical Breakfast Club had broadly similar aims to my younger self but they actually delivered the goods and they did so in spades.
In this sweeping history of nineteenth century science, Snyder gives us not one biography but four — those of Charles Babbage, John Herschel, William Whewell and Richard Jones. You may not have heard of all of them but I’d be surprised if you didn’t know of some of their work. Between them they invented computing, modern economics, produced the most detailed astronomical maps of their age, co-invented photography, made important advances in tidology, invented the term scientist (among many other neologisms) and they are just the headliners! Under-achievers they were not.
These four men met while studying at Cambridge University way back in 1812 where they held weekly meetings which they called The Philosophical Breakfast Club. They took a look at how science was practiced in their day, found it wanting and decided to do something it. Remarkably, they succeeded!
I found Snyder’s combination of biography, history and science to be utterly compelling…so much so that during my time reading it, my beloved iPad stayed at home, lonely and forgotten, while I undertook my daily commute. This is no dry treatise on nineteenth century science; instead it is a living, breathing page-turner about a group of very colourful individuals who lived in a time where science was done rather differently from how it is practiced today. This was a time where ‘computer’ meant ‘a person who was good at arithmetic’ and professors would share afternoon champagne with their students after giving them advice. Who would have thought that a group of nineteenth century geeks could form the basis of one of the best books I’ve read all year?
Because I spend so much time talking about and helping people with MATLAB, I often get asked to recommend a good MATLAB book. I actually find this a rather difficult question to answer because, contrary to what people seem to expect, I have a relatively small library of MATLAB books.
However, one could argue the fact that I only have a small library of books suggests that I hit upon the good ones straight away. So, for those in the market for an introductory MATLAB book, here are my recommendations
MATLAB Guide by Desmond and Nicholas Higham
If you only ever buy one MATLAB book then this should be it. It starts off with a relatively fast paced mini-tutorial that could be considered a sight-seeing tour of MATLAB and its functionality. Once this is done, the authors get down to the business of teaching you the fundamentals of MATLAB in a systematic,thorough and enjoyable manner.
Unlike some books I have seen, this one doesn’t just show you MATLAB syntax; instead it shows you how to be a good MATLAB programmer from the beginning. Your programs will be efficient, robust and well documented because you will know how to leverage MATLAB’s particular strengths.
One aspect of the Higham’s book that I particularly like is that they include many mathematical examples that are intrinsically interesting in their own right. This was where I first learned of Maurer roses, Viswanath’s constant and eigenvalue roulette for example. Systems such as MATLAB are ideal for demonstrating cool little mathematical ideas and so it’s great to see so many of them sprinkled throughout an introductory textbook such as this one.
The only downside to the current edition is that the chapter on the symbolic toolbox is out of date since it refers to the old Maple based one rather than the current Mupad based system (See my post here for more details on this transition). This is only a minor gripe, however, and I only really mention it at all in an attempt to give a review that looks more balanced.
Full disclosure: One of the authors, Nicholas Higham, works at the same university as me. However, we are in different departments and I paid for my own copy of his book in full.
MATLAB: A Practical Introduction to Programming and Problem Solving by Stormy Attaway
I haven’t had this book as long as I’ve had the Higham book and I haven’t even completely finished it yet. I am, however, very impressed with it (as an aside, it’s also the first technical book I ever bought using the iPad and Android Kindle apps).
One of the key things that I like most about this one is that the text is liberally sprinkled with ‘Quick Questions’ that give you a little scenario and ask you how you’d deal with it in MATLAB. This is quickly followed by a model answer that explains the concepts. These help break up the text, make you stop and think and ultimately lead to you thinking in a more MATLABy way. There is also a good amount of exercises at the end of each chapter (no model solutions provided though).
The book is split into two main parts with the first half concentrating on MATLAB fundamentals such as matrices, vectorization, strings, functions etc while the second half covers mathematical applications such as statistics, curve fitting and numerical integration. So, it’ll take you from being a complete novice to someone who knows their way around a reasonable portion of the system.
Since I first bought this book using the Kindle app on my iPad I thought I’d quickly mention how that worked out for me. In short, I hated the Kindle presentation so much that I went out and bought the physical version of the book as well. The paper version is beautifully formatted and presented whereas the Kindle version is just awful. It’s hard to navigate, looks awful and basically makes one wish that they had just given you a pdf file instead!
Within every field of human endeavor there is a collection of Holy Grail-like accomplishments; achievements so great that the attainment of any one of them would instantly guarantee a place in the history books. In Physics, for example, there is the discovery of the Higgs boson or room temperature superconductivity; Medicine has the cure for cancer and how to control the ageing process whereas Astronomy has dark matter and the discovery of extra-solar planets that are capable of supporting life.
Of course, mathematics is no exception and its Holy Grail set of exploits include problems such as the proof of the Riemann hypothesis, the Goldbach conjecture and the Collatz Problem. In the year 2000, The Clay Mathematics Institute chose seven of what it considered to be the most important unsolved problems in mathematics and offered $1 million for the solution of any one of them. These problems have since been referred to as The Millenium Prize Problems and many mathematicians thought that it would take several decades before even one was solved.
Just three years later, Grigori Perelman solved the first of them - The Poincaré conjecture. Stated over 100 years ago (in 1904 to be exact) by Henri Poincaré, the conjecture says that ‘Every simply connected closed three-manifold is homeomorphic to the three-sphere’. If, like me, you struggle to understand what that actually means then an alternative statement provided by Wolfram Alpha might help – ‘The three-sphere is the only type of bounded three-dimensional space possible that contains no holes.’ 99 years later, after generations of mathematicians tried and failed to prove this statement, Perelman published a proof on the Internet that has since been verified as correct by several teams of mathematicians. For this work he was awarded the Field’s Medal, one of the highest awards in mathematics, which he refused. According to Wolfram Alpha, Perelman also refused the $1 million from the Clay Institute but, as far as I know at least, he has not yet been offered it (can anyone shed light on this matter?).
Yep, Girgori Perelman is clearly rather different from most of us. Not only is he obviously one of the most gifted mathematicians in the world but he also sees awards such as the Field’s Medal very differently from many of us (after all, would YOU refuse such an award – I know I wouldn’t!). So, what kind of a man is he? How did he become so good at mathematics and why did he turn down such prestigious prizes?
In her book, Perfect Rigor, Masha Gessen attempts to answer these questions and more besides by writing a biography of Perelman. Starting before he was even born, Gessen tells Perelman’s story in the words of those who know him best – his friends, colleagues and competitors. Unfortunately, we never get to hear from the man himself because he cut off all communications with journalists before Gessen started researching the book. Despite this handicap, I think that she has done an admirable job and by the end of it I have a feeling that I understand Perelman and his motives a little better than before.
This is not a book about mathematics, it is a book about people who DO mathematics and gives an insight into the pressures, joys and politics that surround the subject along with what it was like to be a Jewish mathematician in Soviet-era Russia. What’s more, it is absolutely fascinating and I devoured it in just a few commutes to and from work. With hardly an equation in sight, you don’t need pencil and paper to follow the story (unlike many of the books I read on mathematics), all you need is a few hours and somewhere to relax.
The only problem I have with this book is that by the end of it I didn’t feel like I knew much more about the Poincaré conjecture itself despite getting to know its conqueror a whole lot more. Since I didn’t know much (Oh Ok…anything) about it to start with this is a bit of a shame. Near the end of writing this review, I took a look at what reviewers on Amazon.com thought of it and it seems that some of them are also disappointed at the mathematical content of the book and they come from a position of some authority on the subject. The best I can say is that if you want to learn about the Poincaré conjecture then this probably isn’t the book for you.
If, on the other hand, you want to learn more about the human being who slayed one of the most difficult mathematical problems of the millennium then I recommend this book wholeheartedly.
It’s that time of year again – a time when your thoughts naturally turn to what Christmas presents you might buy for the geek in your life. If you are a geek yourself then this is an easy exercise – just think what you might want yourself and buy that. Geeks know what other geeks like you see!
What if you are not a geek though? How do you work out what your nerdy friend would like most for Christmas? What you need, dear reader, is a geek guide – someone of the nerdy persuasion who can help you separate the geek wheat from the nerd chaff.
Now if you have read much of my blog you will have probably come to the conclusion that I am a geek (or possibly a nerd – the difference is subtle) and so maybe you are thinking that I can help you. Well, maybe I can – but only for a certain type of geek.
You see, there are many varieties of geek and each one has different needs and wants, thus making it impossible to write a post entitled “Christmas gifts for geeks” which will please everyone. So, I am going to concentrate on gifts for the mathematically inclined which includes (but is not limited to) mathematicians, scientists, engineers and, most importantly…..me! Many of my friends read this blog and so this post is mainly a shameless hint dropping exercise but it is possible that it will be of use to other people as well.
In addition, if you actually buy any of these books using the links in this post then I will earn some commission from Amazon without it costing you a penny extra. Doing this helps support Walking Randomly and is greatly appreciated but I really won’t mind if you choose not to.
For the mathematician in your life you almost certainly cannot go wrong by buying them a book – just make sure that they haven’t already got it! Another tip is ‘don’t try to be too specialised’ – advanced textbooks may well be useful but they are not (often) much fun and Christmas presents are supposed to be fun!
With these thoughts in mind I will separate this section into two parts – books I own and books I wish I owned. In addition, I will only consider the lighter side of the mathematical spectrum (for a given value of ‘lighter’) so these books should be of interest to mathematicians of any level – from high school students to research scientists. The large number of equations that some of them contain may make them look like text books in some cases, but let me assure you that they are (mostly) easy reading.
Books I own – and highly recommend
- “e”, The Story of a Number by Eli Maor. Some numbers are so important that they get whole books written about them and e (sometimes known as Euler’s number) is one of them. It’s a constant that certainly gets around as it appears in all manner of places from compound interest to calculus with detours through subjects such as complex analysis and trigonometry. This book is easy to read and contains a mixture of mathematics, history and biography.
- An Imaginary Tale: The Story of “i” by Paul Nahin. I’ll never forget the look on my dad’s face when he asked what I was learning at University and I told him ‘they are teaching us about imaginary numbers.’ It didn’t exactly strengthen his faith in further education I can tell you! It turns out that the term ‘imaginary’ is an unfortunate byproduct of history and if you delve into the mathematics then you’ll soon learn that not only do imaginary numbers exist but that they form the basis of one of the most beautiful and powerful areas of mathematics there is.
- Gamma: Exploring Euler’s Constant by Julian Havil. Everyone has heard of Pi, quite a few people know about e but you’ll be hard pressed to find a non-mathematician who knows about Gamma. Impress the mathmo in your life by giving them a book about a mathematical constant that has been seriously undersold by its PR team. It contains some heavier mathematics than the books mentioned above but it is still accessible to good high school students and undergraduates. I’m still working my way through it to be honest and loving every minute.
- Flatland: A Romance of Many Dimensions by Edwin Abbott. We live in a three dimensional world world (some say 4, some say 11 but for the sake of this note I am saying 3) but what would it be like if we inhabited a world of only 2 dimensions. Imagine how life would be in such a world and how we would react to a mysterious visitor from the 3rd dimension. Edwin Abbott did exactly this and in the process wrote a satire on Victorian England (the book was written in 1884, making this a very early example of science fiction). This is a very charming (and very cheap) book that was first recommended to me by a fellow physicist.
- The Music of the Primes by Marcus du Sautoy. Prime numbers fascinate us, there can be no denying that, and in this book Marcus takes us to meet some of the people and mathematics behind them. Some reviewers complain about the fact that the mathematics isn’t detailed enough but then others may well say that it is too mathematical – writing popular maths books is a difficult game. Personally I think he had it just right and told a great story with just enough maths to keep it from being a book on history rather than a book on maths . Whenever I wanted more detail I looked to other sources and this got me reading more books on number theory. This is precisely what popular maths books should do in my opinion – invite the reader in….show them enough to whet their appetite but not so much that it scares them off, point them in the direction of further study and leave them wanting more….
- The Man Who Loved Only Numbers by Paul Hoffman. This is a book about the life of an extremely eccentric mathematician called Paul Erdős – one of the most prolific writers in mathematics apparently. Erdős was a strange character but an extremely well respected mathematician. This book is serious easy reading and contains a lot less actual math than most of the other books I mention here. I have a copy in my office and many people have borrowed and enjoyed it – only one of them was a mathematician by training. This is a great book. In fact I am going to read it again on the train home this evening.
- Inside Your Calculator: From Simple Programs to Significant Insights by Gerald Rising. “Sir, how does the calculator know the sine of a number?” I innocently asked my teacher at the end of a maths lesson back when I was far too young to have a scientific calculator. He blustered for a bit before answering ‘It stores them in memory – it’s just a big look up table’. I was deeply suspicious. That would take a lot of memory I thought! A lot more than my cheapo calculator had that’s for sure. So how do calculators do this stuff? This book makes a good job of explaining the detail (hint…CORDIC).
- Surely You’re Joking, Mr.Feynman! by Leighton, Feynman and Hutchings. Richard Feynman is an all time hero of mine and this book is a collection of anecdotes about his life. If memory serves me there is not a single equation in this book but I would be surprised to meet a mathememtican who doesn’t enjoy it. Wanna see the human side of a genius? Buy this book then.
Books I wish I owned (feel free to buy one for me if your mood takes you that way.)
- The Princeton Companion to Mathematics Edited by Timothy Gowers (click here for his blog) – This beautiful looking book is for the more serious mathematician but it is the sort of thing that will remain on their bookshelf for years to come. Essentially it is a guide to as much pure mathematics as you can fit into a single volume and would be a perfect addition to any mathematicians library. It’s a bit expensive but looks like it’s worth every penny.
- Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi by Martin Gardner. Martin is something of a legend among recreational mathematicians and has many books and articles to his name. He wrote the ‘mathematical games’ column for Scientific American for many years where he picked up a large following among mathematicians at all levels. This book is a collection of some of the best of his Scientific American articles that have been expanded with updates and new material. Even if you only have a passing interest in mathematics – this looks like a good book to get.
- Bad Science I feel like I know Ben Goldacre well and yet I have never met him but his blog and column in the Guardian have kept me entertained and informed for years. An expert in refuting dodgy statistics and sham science, Ben takes no prisoners. I find his writing extremely entertaining as well as providing much food for thought so naturally I would like his book. Where is the maths connection? Well, maths is often abused by the media and it’s often statistics that gets abused. Ben tends to have a lot to say about that.
- An Adventurer’s Guide to Number Theory by Richard Friedberg. I have no idea what this book might be like but I enjoy number theory, I like the title, it gets good reviews and it’s reasonably priced. Sounds good to me.
- Digital Dice Computational Solutions to Practical Probability Problems by Paul Nahin. I have a couple of Nahin’s books and they are both great so I am guessing this one will be just as good. From what I have seen, he includes lots of simulations in MATLAB. Randomness? MATLAB? Nahin? Of course I want this book.
- Nonplussed: Mathematical Proof of Implausible Ideas by Julian Havil. There are a lot of true facts in Mathematics that make you say ‘no way – that can’t be true’ when you first hear them. I have said this to myself several times over the years and it usually takes a good, solid proof (along with several concrete examples) before I concede the point. By the sound of it this book is choc full of this sort of thing.
- Euler’s Gem: The Polyhedron Formula and the Birth of Topology by David Richeson. Euler’s name seems to be everywhere in mathematics – you only need to look at this list from Wikipedia to get an idea of just how pervasive his ideas have become. I have never seen this book and I don’t know much about his polyhedron formula but I do intend to find out.
- The Drunkard’s Walk: How Randomness Rules Our Lives by Leonard Mlodinow. When you write a blog called ‘Walking Randomly’ you really should have some books about randomness on your shelf. That’s partly why this one is here. The good reviews don’t do any harm either.