Paul P. Mealing

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Saturday, 29 November 2014

A book for geeks

Matt Parker is a mathematical entertainer, as oxymoronic as that sounds, because apparently, in the UK, he does stand-up comedic mathematics and mathematical-based magic tricks with cards. Originally a school teacher from Oz, he has the official title of Public Engagement in Mathematics Fellow at Queen Mary University of London.

He’s written a very accessible book called Things to Make and Do in the Fourth Dimension, where he attempts to introduce the reader to more obscure areas of mathematics by wooing them with games and little-known intriguing mathematical facts.

For example: if you square any prime number greater than 3 and take off 1 you’ll find it’s divisible by 24.  As he says: ‘That sentence can freak out even the most balanced mathematician.’ In a section at the back, called The Answers in the Back of the Book, he provides an easy-to-follow proof that shows this applies to any number that is not factored by 2 or 3 – so not just prime numbers. Obviously, any prime number above 2 or 3 fits that category as well. So the converse is not true: a number divisible by 24 plus 1 is not necessarily a squared prime. Otherwise, as Parker points out, we would have a very easy ready-made method of finding all primes, which we haven’t.

Basically, he is a mathematical enthusiast and he wants to share his enthusiasm. As anyone who reads my blog would know, I’m familiar with a fair sample of mathematical concepts and esoterica, so I don’t believe I’m the audience that Parker is seeking. Having said that, he managed to augment my knowledge considerably, like in the previous paragraph. Another example is his description of how to make binary computer logic gates by just using dominoes that actually can perform a calculation. In fact, he and a team of mathematicians spent 6  hours setting up a 10,000 domino ‘computer’ that took 48 seconds to compute 6 + 4 = 10, performed at the Manchester Science Festival in October 2012.

The title of this post is apt: geeks would love this book; yet Parker’s objective, one feels, is to make mathematics attractive to a wider audience. In particular, those who were turned off maths in their high school years, if not before. One of the virtues I found in this book is his selective use of visual representation, even of the simplest kind. I’m not just talking about graphs of exotic equations like Zeta functions and perspective drawings of Platonic solids or even 2D renderings of tesseracts (4D cubes), but rough hand-drawn sketches and sometimes just a list of numbers to demonstrate a series or sequence. I found these most helpful in understanding a tricky concept.

We are visual creatures because sight is our prime medium for comprehending the world. It should be no surprise that visualising an abstract concept, mathematical or otherwise, is the shortest way to understanding it. I work a lot with engineers and when they want to explain something they invariably draw a picture.

The problem with maths in education is that it’s a cumulative subject. More esoteric topics are dependent on lesser ones. If a student falls behind, the gap between what they’re expected to know and what they can actually achieve grows over the years of schooling.

Books like Parker’s attempt to short-circuit this process. He tries to introduce the reader to the more ‘sexy’ aspects of mathematics without grinding them into the ground with mind-bending exercises. His Answers in the Back of the Book allows the more adventurous and less intimidated reader to understand a topic more fully, whilst not burdening a less experienced reader with mind-expanding exercises. It is possible to read this book and come away with both a sense of awe at its magisterial wonder and an appreciation of how maths literally drives our digital world without having to do a lot of mental gymnastics. On the other hand, Parker is letting you into some of the secrets of the priesthood without feeling like you’ve done a PhD.

Although it is divided chapter by chapter into separate topics, this is a book that should be read in the order it is presented. Parker often references material already covered, partly to demonstrate how the mathematical world is so interconnected. To give an example, he sneaks up on the famous Zeta function in a way that makes it appear less intimidating then it really is, yet still manages to explain its relationship to Riemann’s famous hypothesis and the distribution of primes. I was disappointed that he didn’t explain that the non-trivial zeros, which are both the core mystery and ultimate unsolved puzzle, are in fact complex numbers involving the imaginary axis. However, he explains this in a later chapter when he introduces the reader to imaginary numbers and the ‘complex plane’.

Pythagoras famously said (or so we are led to believe, as he never wrote anything down) that everything is numbers. In the digital world this is literally true, and one of Parker’s most illuminating chapters explains how everything you do on your smart-phone from pictures to texting to music are all rendered by 0s and 1s.

Parker is very clever in that he discusses highly esoteric mathematical topics like the Zeta function (already mentioned), quarternions (imaginary numbers in 4D),  the so-called Monster or Friendly Giant in 196,833 dimensions, computer-generated self-correcting algorithms using binary arithmetic, multiple infinities, knot theory’s relevance to DNA not getting tangled and Klein bottles (4D bottles in 3D); without discussing more fundamental topics like logarithms, trigonometry or calculus. He doesn’t even explain the fundamental relationship between polar co-ordinates and Cartesian co-ordinates that makes imaginary numbers such a widely used tool.

He doesn’t get philosophical until the very end of the book, when he discusses the relevance of Godel’s Incompleteness Theorem to the study of mathematics for ever (quite literally). As I’m sure I’ve mentioned in previous posts, implicit in Godel’s Theorem is the fact that mathematics is never-ending, therefore it’s a human activity that will never stop. Also Parker points out that there could be other universes with other dimensions to ours, but any hypothetical residents (he calls them ‘hypertheticals’) would still discover the same mathematics as us, assuming they have the intellect to do so.