**Things to Make and Do in the Fourth Dimension**

Matt Parker

Farrar, Straus and Giroux (2014)

Physicist Stanislaw Ulam was bored. Stuck in the middle of an interminable lecture, he started doodling on a piece of graph paper. Starting with “1” and going outwards, he made a spiral pattern of all the positive integers. Then, he marked off all the prime numbers in the spiral. Something odd popped out. Prime numbers weren’t distributed randomly, as one might think. They tended to lie in clusters along diagonal lines. It turns out that this is NOT random, but why it is so is still a puzzle.

Self-described “stand-up mathematician” Matt Parker has turned his videos at Numberphile into book form (and added a few more fascinating topics). Like those videos, the book covers the entire world of mathematics. From counting in different base systems to packing coins into squares to untangling knots to the many different types of numbers, it’s all presented in a delightful and easy-to-follow manner.

Parker avoids a historical approach to math (or “maths”, using the British way) which can often take the fun out of a subject. When he does get into history, it’s only to illustrate how mathematical advances can come from purely mundane considerations. Sir Walter Raleigh wants to know the best way to stack cannonballs, and how to tell how many are in a given pile. That simple question led to an entire field of geometry known as “packing problems”, and there’s still a heck of a lot of active work being done in that area.

With regards to the “make and do” part, Parker has a lot of little activities that the reader can try – provided they are willing to mark up, copy, or cut out their copy of the book. If you don’t want to make that sacrifice, there’s an associated website – makeanddo4D.com – where you can download and print out everything. The website also has a lot of gadgets that are referred to in the book, as well as links to sites for further exploration.

The book is a lot of fun to read. Parker never talks down to the reader, or burdens him with too much detail. For example, rather than delve into a detailed explanation of “surreal” numbers, he just mentions them in a few paragraphs. That’s all you need to understand just how weird they are. There’s plenty to whet the appetite of someone looking for more. Except an index. That’s a bit of an oversight on the part of the publisher.

One might suspect that all this advanced mathematics has little or no real world application. But just as stacking cannonballs can lead to advanced geometry, so to can things go the other way. Prime numbers are vitally important in computer security. DNA and some proteins exist in closed loops – so knot theory may help lead to new advances in medicine. Colleagues of Parker’s used graph theory to predict World Cup results. And I note that when baseball fans talk about “wins above replacement”, they are actually referring to a complex calculation done in multi-dimensional space.

It’s obvious that Parker is enjoying the heck out of himself – and all these mathematical explorations. The enjoyment is contagious. Give the book a read, and the next time you find yourself trapped in an interminable lecture, you’ll have some interesting ideas to explore in your doodles.