This week, we’re investigating the Millennium Prize Problems – a set of mathematical equations that, if solved, will not only nab the lucky winner a million, but also revolutionise the world. Plus, the headlines from the world of science and technology, including why screams are so alarming, how fat fish help the human fight against flab, and what’s the future of money?

Better yet, the episode includes a contribution from our very own Katie Steckles talking topology, Poincaré and Perelman.

The episode is available to listen or download as a podcast or, less conveniently, at 5am tomorrow on Radio 5 Live (or later on iPlayer). Not a listener? Read a transcript.

]]>One of her favourite modes of attack is the “30 Second Challenge” from the Daily Mail. It looks like this:

You start with the number on the left, then follow the instructions reading right until you get to the answer at the end. It’s one of Grandma’s favourites because it’s very hard to do in your head when she’s just reading it out!

I decided it would be a fun Sunday morning mental excursion to make a random 30 second challenge generator.

Making a random challenge generator involves thinking about what the space of possible challenges is, and how to pick fairly from them corresponding to different difficulty levels. The strategy I came up with is for each difficulty level to have a pool of possible operations, and to pick at random from those for each step. An operation can involve more than one step, and each operation has a function which looks at the current state of the puzzle to decide if it can be applied.

As far as I know, each operation must leave you with a whole number. That means that a “divide by $N$” instruction can only appear when your number is divisible by $N$. Since some divisors are much more common than others, but I wanted to have a good distribution of numbers to divide by, I made the divide by” operation pick a number $N$, and then add a step to add or subtract the right amount to get to a multiple of $N$, before adding the step to divide.

Some pairs of operations shouldn’t appear next to each other – you shouldn’t get an “add” followed by a “subtract”, or a “halve it” followed by a “double it”.

With a rough system of making valid challenges in place, I needed to make three difficulty levels. My rough rule of thumb was that dividing is really hard, cubing is hard because it leads to big numbers, and adding and subtracting are quite easy. I could probably split the additions or multiplications into easier or harder versions – there’s some evidence that the 6 and 8 times tables are hardest – and add some more complicated operations like “square root of this”. At the moment, I don’t feel like the difficulty levels are consistent enough: sometimes you’ll get a really easy “hard” challenge, and sometimes you’ll get a pretty tricky “easy” one.

Finally, I decided to look at accessibility. Grandma continues to be unsatisfied with me because her son-in-law, who’s partially sighted, always solves the challenges much quicker than I can. She reads the steps out and he does the calculations in his head. With that in mind, I made sure the challenge is usable when you can’t see it. Thanks to modern web standards, that was easy – I set `tabindex=1`

on the step elements, so that you can navigate between them by pressing tab, and made sure all of the instructions make sense when read out by a screen reader: I had to add an `aria-label`

attribute to the fraction instruction with some alternate text to read out instead of the fancy formatting I use in the visual version. I tested it all with the ChromeVox extension, which works pretty well and is very easy to set up (arguably too easy – it went a bit mad reading every other tab I had open while I was testing).

In the end, this is how the finished game looks:

It was a fun excursion, and the game is pretty addictive. You can play it at christianp.github.io/30secondchallenge. Post your average times and record streaks in the comments!

]]>The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.

]]>On 3rd July it was announced that both men have received the Hirst Prize, and Edmund Robertson has been been invited to give the associated Hirst Lectureship, all part of LMS 150th Anniversary celebrations.

As noted in the University of St. Andrews press release, the MacTutor site has grown from its humble origins two decades ago, and now has

“over 2800 detailed biographies of mathematicians and related scientists with bibliographies (primary and secondary sources) accompanying each article, around 150 historical articles on mathematical topics, over 2000 other pages of essays on specific topics and further resources, such as an interactive page on historical curves. The web site has become a hugely successful resource for school-children, undergraduates, graduate students and their teachers all over the world, receiving 10,000,000 hits per month during the academic year, with around 2,000,000 distinct users. It is the first port of call for those interested in the historical side of the mathematical sciences, giving mathematicians direct links to their profession’s past. It bridges the gap between old books and modern journals, and its biographies give lives to names otherwise known only for the theorems to which they are attached. It is the most widely used and influential web-based resource in history of mathematics.”

The site went live in 1994, and its two curators have devoted more time to it in recent years since retiring from their university positions. Edmund writes the great bulk of the biographies and John does the web implementation. (A few biographies have been submitted by others, and published modulo some editorial work to bring them into house style.) John has written his own tools to handle the database, which now contains about 30,000 files, and modestly says, “I am only an amateur programmer/website maker, so I am pleased that our efforts have been approved by so many professionals.”

What is remarkable, as Edmund points out, is that “this has all been done with minimal support from the university, other than our department providing the computer on which the archive runs and our scientific officer doing the continual work to keep that running and the software up-to-date.” Most of the biographies are for mathematicians from earlier eras, from ancient times to the recent past, although they make exceptions for exemplary living practicioners like Andrew Wiles and Barry Mazur.

“We try to make it as international as possible but it certainly has a USA, UK bias simply because obituaries of people from these countries are easier to handle (written in English in journals we have easy access to), Edmund explains. “Also sometimes a particular project leads me to write biographies of many people from one country. For example, we did a project for the 125th anniversary of the Edinburgh Mathematical Society which involved trying to find details of all the members of that Society at a particular time. That produced many biographies of Scottish mathematicians which we added to our Archive. Again I was at a conference in Romania a year ago and found out about a number of Romanian mathematicians who I added.” Indeed, correspondence over the last six months about Irish mathematicians has led to new additions, e.g., Sheila Tinney, the first Irish woman to get a PhD in the mathematical sciences, and Bletchley codebreaker John Herivel, with more such biographies in the pipeline.

In addition to mathematicians both well known and obscure, there are other surprising appearances. Edmund: “You might not expect to find artists like Albrecht Dürer, Leonardo da Vinci, Maurits Escher and Filippo Brunelleschi, or the famous architect Sir Christopher Wren, or the children’s author Lewis Carroll, or Irish President Éamon de Valera, or the famous Florence Nightingale in a mathematics database, but they are all in our Archive and deserve their places!”

Why the name MacTutor? Edmund recalls, “The name came from our MacTutor teaching software which is where our web history archive started. Our MacTutor software was so named because it ran on Macintosh computers using Hypercard (at the time we started a PC wouldn’t have had the facilities for the interactive software we produced).”

Edmund and John won the European Academic Software award in Heidelberg in November 1994 for the MacTutor teaching software. “Before we went to Heidelberg we put the history from our system onto the web which, at that time, was quite new,” Edmund adds. John remembered a problem with another prize from many years ago, “We won an earlier prize for the mathematical teaching version of MacTutor that was sponsored by the US Department of Defence. Then they discovered that we weren’t Americans (the fact that we came from St Andrews should have given them a clue) and they weren’t allowed to give us the money. We should be OK this time with the London Mathematical Society.”

London Mathematical Society Hirst Prize and Lectureship – press release from the University of St Andrews

Announcement of 2015 LMS Prizes (we normally do a round-up of these but haven’t had the time this year)

]]>Desmos is the web-based interactive geometry program that isn’t GeoGebra. It’s very popular with teachers.

Someone’s made a nifty tool to turn a Desmos construction into an animated gif. It’s called – you guessed it – GIFsmos. They’ve got a blog containing a few nice animations, but it doesn’t seem to have been updated since I discovered it in March. Anyway, the tool still exists, so go and see what you can create!

]]>

*Genius at Play* is a biography of John Conway, the mathematician. Look, that’s his face on the cover, surrounded by doodles of some of the maths he’s done.

I first encountered John Conway’s name in the book *Surreal Numbers*, which I found on Amazon back when all it sold was books. (OK, I saw the Game of Life before I saw surreal numbers, but I wasn’t aware of the human being “Conway” who invented it.) That was my introduction to real maths – I was in sixth form at the time and stuck in the world of crank-the-handle algebra – and I suppose it gave me an overly optimistic impression of what grown-up maths might be like. It took me quite a few years after reading maths at uni to rediscover the deliberately unserious strain of maths that Conway champions.

*Genius at Play* is a hybrid biography/autobiography – while Siobhan Roberts is the nominal author, verbatim quotes from Conway are so plentiful and sometimes lengthy that they get their own font, and even that’s not enough – quite often I got the impression that stories covered by Roberts could only have come directly from Conway.

That’s something Roberts clearly grapples with throughout the book – one of the main themes is the unreliability of Conway’s narrative, coupled with the multitude of implausible things that really did happen. When she can, she tries to corroborate facts with other people who were there – a good chunk of the leading mathematicians of our time pop up throughout the book to back up or knock down anecdotes. I’m reminded of the film *Big Fish*, where the line between exaggeration for dramatic effect and implausible reality is distinctly fuzzy.

Going into the book, one of my main concerns was how it would stack up against the other recent popular biographies of great mathematicians. Clearly the publisher thinks that concern will be widely held amongst the readership, as they’ve put a quote from Sylvia Nasar, author of the John Nash biography *A Beautiful Mind*, on the front cover, calling Roberts’ book “Absolutely brilliant”. I can’t really comment on how it compares to the likes of *A Beautiful Mind* or *The Imitation Game*, because I don’t bother reading most books unless the author sends me a copy, but I can agree with that assessment. A significant chunk of the mathematicians interviewed or quoted in this book have had their own biographies written, and at times I began to wonder if Conway was so special after all, when geniuses seem to abound so plentifully. So a Conway biography was definitely overdue.

Going back to films, there have been a significant number of mathematical biopics based on popular biographies recently, so I wondered what the big-budget Hollywood adaptation of this book would be like. Nash, Turing and Hawking all got recast as buff young hunks, so why not Conway? Maybe Michael Sheen could grow a beard to play Conway (is Sheen a buff young hunk? I reckon he bears a decent resemblance to Conway, anyway), and it could be retitled “The Game of Life”. The problem is that *Genius at Play* doesn’t have a big, emotive struggle of the sort faced by Nash and Turing to pin its narrative on – Conway gets his own way more often than a third-act redemption story can support. That didn’t have to be the case, because there are lots of difficult themes that Roberts could have dwelt on: Conway’s suicide attempt is the most dramatic, but there’s also his serial philandering and absolute rejection of all responsibility for his personal affairs. None of that sticks though. The book is written with great affection for Conway, and that seems to be shared by just about everyone interviewed (apart from Stephen Wolfram, who sounds almost like a Bond villain). It comes up again and again that he’s a particularly charismatic man.

Maybe the correct film reference is Willy Wonka. I found myself humming this song on a few occasions:

I’d pay good money to watch Michael Sheen dancing around the halls of the Institute for Advanced Study while singing that.

By the way, “The Game of Life” would’ve been such a perfect title for this book. It’s got everything – the title of the thing everyone knows him for, the book’s about Conway’s life, he treats life as a game. It’s just a shame that Conway really doesn’t want to be “the Life guy”: he says “I HATE LIFE!” with varying levels of sincerity on several occasions throughout the book. That’s understandable: nobody wants to be a one-hit wonder.

It’s good that the book isn’t about John Conway, Inventor of the Game of Life and one-hit wonder, because he’s done enough maths for a twelve-volume *Greatest Hits* collection, with each volume covering a completely different area of inquiry. He’s *The Beatles* to Evariste Galois’ *Ace of Base*, if you will. “Genius at Play” is a good enough title, in the end – what marks Conway apart from his peers is his ideological devotion to playfulness. And there’s no doubt Conway is a genius, by whatever standard.

There’s got to be a lot of maths in a biography of a mathematician, and there’s a lot of it in this book. If it’s going to be a popular book, the maths has to be explained (or abbreviated) in such a way that the general reader can get the gist of it. Consequently, my other main concern going into the book was that the maths would be simplified into incorrectness or so familiar and uninteresting I’d gloss over it. That’s not the case, probably helped in large part by Conway’s more-direct-than-usual input into the text. And frankly, you’d do extremely well to already know all of the maths mentioned in the book. He provides diagrams when necessary, and Roberts lets him take over and explain things when it’s obvious she wouldn’t add anything by paraphrasing him. This engaging style could have been made easier by the common principle in Conway’s work of starting at very simple precepts and finding deep mathematical concepts within while bypassing the intermediate prove-a-ton-of-theorems-and-see-what-sticks step, but even the section about the ATLAS of Finite Groups – definitely not a simple topic – gave me a feeling of understanding what was achieved and a little bit of what it all meant. There are three appendices which go into greater detail about things it’s just not convenient to explain in the main text.

The book starts and ends with a visit to the neuroscientist Sandra Witelson, who wanted to examine Conway’s brain to see if it revealed anything about his genius. I could’ve done without this bit – I went into it sort of disagreeing with the premise, and I wasn’t dissuaded of that. No conclusions were drawn, and I didn’t learn anything new about Conway along the way.

I would’ve liked to see some more thoughtful investigation of Conway’s psychology – the depression and the zany public persona, which are surely linked. There are a couple of allusions to the possibility that Conway’s extroversion is a defence mechanism, but Roberts doesn’t probe too deeply at that, and we’re left to draw our own conclusions. I suppose I just want some opinions spelt out straightforwardly.

Throughout the book, a few different versions of Conway are presented: the conventionally successful mathematician with important theorems and prestigious awards to his name; the scatter-brained eccentric who becomes the subject of urban legends; the charismatic, effervescent trickster god; the irresponsible manchild who exploits those around him; the depressive who worries he hasn’t fulfilled his potential.

Finally, I’ve got a gripe about the typesetting that I need to get off my chest. I don’t know if this is an American thing, but just about every number in the book is printed as a numeral, instead of in words. In a few instances, that’s particularly maddening – when “half of it” was printed as “½ of it”, I winced. It doesn’t really matter though.

*Genius at Play* is a good book, I enjoyed reading it and I learnt a lot.

*Genius at Play* is published by Bloomsbury, priced \$30 / £20. It’s out now in the US, and from the 10th of September in the UK.

Friend of The Aperiodical Colm Mulcahy is also a friend of John Conway, and he’s written his own review of this book for The Huffington Post. Make sure you read it!

]]>It’s easy to forget about MathML, because unless you’re a publisher or doing complicated things with data flows, you never need to see it.

I’m not doing a great job of selling this story, am I? I couldn’t even find a picture to illustrate it.

After a lengthy lull in which MathML was deeply unpopular, mainly due to browser makers not supporting it but mainly due to it being extremely hard for the average mathematician to work with, the format which aimed to be able to represent all maths is having a bit of a resurgence these days. The web is catching up – MathJax uses MathML to represent mathematical notation internally, and that is adding pressure to browser makers to implement support for rendering MathML without any additional library.

The W3C have announced that MathML 3.0 is now an ISO/IEC international standard. That doesn’t have much impact on anything other than giving it a stamp of approval, so you can carry on with your day.

See, I told you this wasn’t interesting.

]]>Puzzlebomb is a monthly puzzle compendium. Issue 43 of Puzzlebomb, for July 2015, can be found here:

Puzzlebomb – Issue 43 – July 2015

The solutions to Issue 43 will be posted at the same time as Issue 44.

Previous issues of Puzzlebomb, and their solutions, can be found here.

]]>There’s a lovely function in mathematics called the factorial function, which involves multiplying the input number by every number smaller than it. For example: $\operatorname{factorial}(5) = 5 \times 4 \times 3 \times 2 \times 1 = 120$. The values of factorials get alarmingly big so, conveniently, the function is written in shorthand as an exclamation mark. So when a mathematician writes things like $5! = 120$ and $13! = 6,\!227,\!020,\!800$ the exclamation mark represents both factorial and pure excitement. Factorials are mathematically interesting for several reasons, possibly the most common being that they represent the ways objects can be shuffled. If you have thirteen cards to shuffle, then there are thirteen possible cards you could put down first. You then have the remaining twelve cards as options for the second one, eleven for the next, and so on – giving just over 6 billion possibilities for arranging a mere thirteen cards.

For a full deck of fifty-two cards, the number is much bigger. Calculating $52!$ manually would take a long time, so it’s a perfect thing to get a computer to do for us. But in order to ask a computer to do something, you need to state it as an algorithm for the computer to follow. So here’s a set of instructions I’ve written to take an input number and progressively multiply it by every smaller number:

Step 1:Remember the starting value of $n$ as your running total.

Step 2:Subtract $1$ from $n$.

Step 3:Multiply the running total by this new $n$.

Step 4:Repeat steps 2 and 3 until $n$ reaches $1$.

Step 5:Return the running total.

We can try running this for the first few laps of calculating $52!$:

We’re only six laps in, and the running total has already gone past 14 billion! (No that’s not a factorial, I just wanted to emphasize how quickly these numbers grow.) This is definitely the sort of long calculation we want a computer to do for us. The last step is to translate our algorithm steps into a language a computer can understand. So what does an algorithm look like to a computer? Well, much as humans can speak different languages, computers can understand different programming languages. I’ve selected one called Python, because it has a very simple grammar and is one of the closest computer languages to being readable English. I’ll also include some comments to the right of each line to clarify

what the code is doing.

Here is the factorial algorithm, coded in Python. If you are so inclined, you could run this on a computer. Much in the way we were naming functions before, I can give each algorithm a name then write its inputs inside brackets next to it.

def factorial(n): # I am defining the algorithm called # ‘factorial’ and it starts with a number ‘n’ running_total = n # Remembers n as the starting value of # the running total while n > 1: # Repeats the next bit until n is no longer # bigger than 1 n = n - 1 # Subtracts 1 from n running_total = # Multiplies the running total running_total * n # by the new n return running_total # Returns the running total

I’ve just double-checked this program to make sure it works. Once loaded up, I can enter `factorial(13)`

into my computer to double-check $13!$. And, sure enough, $6,\!227,\!020,\!800$ appears

back on the screen. So then I let it loose on $52!$ and here is the exact output from my computer:

>>> factorial(52) 80658175170943878571660636856403766975289505440883277824000000000000

That’s a number with sixty-eight digits. A truly huge number. To say it the long way, there are 800,000 billion billion billion billion billion billion billion ways a pack of cards could be

shuffled. There are only about 1 million billion billion stars in the observable universe. And the universe is only about 4 hundred million billion seconds old. So this means that if every star in the universe had a billion planets, each with a population of a billion hypothetical aliens, all shuffling a billion packs a second since the dawn of the universe, we would now be only halfway through every possible arrangement. And that’s for a pack of only fifty-two cards! Thank goodness we didn’t include the jokers.

Anyway, now we know what the answer is, we can try to calculate it in a different way: using a recursive algorithm. We simply need to state the algorithm in terms of itself. Something like: ‘The factorial of $n$ is $n$ times the factorial of $n – 1$.’ The only extra thing we need to know is that the factorial of $1$ is $1$. This is the ‘get-out clause’ in the algorithm which stops the recursion going on for ever.

Step 1:Remember that the factorial of $1$ is $1$.

Step 2:Multiply $n$ by the factorial of $n – 1$.

Translated into Python:

def factorial(n): # I am calling the algorithm ‘factorial’ # and it starts with a number ‘n’ if n == 1: return 1 # The factorial of 1 is 1 return n * factorial(n - 1) # Calculate n multiplied by the factorial # of n - 1

Presented with `factorial(13)`

, my computer now runs down the chain of recursions until it hits `factorial(1)`

, then comes racing back up to give $6,\!227,\!020,\!800$ once more. Likewise, putting in `factorial(52)`

gives back the same 68-digit monster. Yet nowhere have I actually told the program how to calculate a factorial, merely how one factorial calculation relates to a smaller one. Once again, recursive algorithms are magic: conjuring answers out of seemingly empty code.

We now have two different computer programs which compute the same answer two different ways. Which leaves us with only one option: make them fight! I just took a quick break to race both factorial programs head to head. I decided that the winner would be the first to compute the factorial of $100$. Computing the full 158-digit answer to $100!$ took the recursive Python program 0.000068 seconds, whereas the normal one came in at only 0.000046 seconds.^{1} A decisive win for the non-recursive program!

*If you liked that, “Things to Make and Do in the Fourth Dimension” is out now in paperback and contains plenty more playful mathematical investigations. There’s also a fantastic website to accompany the book, with loads of interactive doodads, at makeanddo4d.com*

- 93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000, for the record.

We’ve often mentioned category theorist and occasional media-equation-provider Eugenia Cheng on the site, and she’s now produced a book, Cakes, Custard and Category Theory, which we thought we’d review. In a stupid way.

The book is in two halves: the first is a general introduction to the idea of mathematics and mathematical thinking, and a tour of some fundamental ideas in mathematics such as how to interpret real-world problems, as well as an introduction to group theory and topology. The second half is about category theory. Cheng describes it as being ‘the mathematics of mathematics’, and anyone who knows a bit of category theory will understand what a challenge it is to develop and communicate the necessary ideas in a form understandable to a book audience.

Cheng is well-known for her love of baking and cakes, and is often called on to produce equations relating to foodstuffs (including scones, pizza and donuts), so naturally her book ties this hobby in with her other passion, mathematics. As well as using baking as an analogy for the process of doing maths, Cheng goes one step further and actually includes full recipes for several different foods in her book. The start of each chapter features a recipe, which is then used as an example to illustrate a point she’s making about maths.

Since we’re idiots, we spent an afternoon actually baking and trying out the recipes. I invited round two mathematicians called Sam (you may remember them from our π approximation challenge), along with Aperiodical regular Paul, to have a go at baking the recipes, tasting the results and not in any way letting our ineptitude at making edible food influence our opinion of the book.

The first thing we tried making were Cheng’s gluten free chocolate brownies, from Chapter 1. The idea behind including this recipe is that you need both the ingredients and the method to make a recipe work (in this case, it won’t work if you bake it in one big pan, since it’s gluten-free flour, you have to do small ones), in the same way that maths is about not just the ingredients but also the method. I think. We found it pretty easy, although the baking time given in the recipe wasn’t nearly enough, and we ended up putting them back in the oven twice, because they were basically liquid in the middle, but then the third time we put them back in we forgot because we were busy baking something else, so they ended up horribly overbaked. They’re still quite nice though. 8/10.

Next we tackled the so-called ‘Conference chocolate pudding': baked one evening after a boozy conference social, these illustrate that if you have the principles of how to bake cakes you can do it, even if you’re wasted, without a recipe. This echoes the way that a mathematician who knows basic principles can derive any mathematical result. It turned out that we didn’t have enough cocoa powder to do this properly, since we already weighed out everything for the other recipes, and so we used vanilla instead for these. They turned out pretty presentable, if a little plain-looking. 7/10.

The book covers an impressive range of maths – not shying away from difficult topics, and jumping from topic to topic with impressive enthusiasm. As well as being split into chapters, practically each paragraph has its own heading, with different topics connecting from one to the other. It’s basically the stuff I’d like someone to have written a book about – from the way mathematicians think, to some of the really interesting pure mathematical concepts that people don’t even find out exist unless they study maths at university.

Our next baking adventure was the Olive Oil Plum Cake, from Chapter 5. While not containing many of the traditional things you’d expect a cake to, like flour, or sugar, this illustrated the chapter on generalisation – once you know what a cake is, you can imagine things which are basically the same as a cake, but aren’t a cake. We managed to successfully bake what we all agreed was not a cake, and while we followed the instructions to the letter and it looked like a cake on exit from the oven, once inverted it quickly became a soggy mass which looked hugely unappetising, despite still being a bit tasty (mostly due to how awesome plums are). Its popularity among the group varied, and we give it 5/10.

We then followed the recipe for Jaffa Cakes, from Chapter 7. The chapter is about axiomatisation, and the recipe simply calls for ‘small round cakes’, ‘marmalade’ and ‘chocolate’. Each of these could have been broken down into more detailed recipes, but the point is that you have to decide what’s an axiom and what can be derived (using a recipe from simpler ingredients). We bought a cuboid cake, and cut it into small round pieces; we also tried, since they do technically count as ‘small round cakes’, using some Eccles cakes, to make Eccles Jaffa cakes. If nothing else good comes from this whole exercise, we at least have that. The main conclusion we came to here is that bought Jaffa cakes are better, although you can’t go far wrong combining cake, marmalade and chocolate. 6/10.

Cheng uses a lot of analogies in the book. Imagine a hedgehog, but where the spines of the hedgehog are analogies. That’s what the book is like. This is good, because they’re well chosen analogies (One of my favourites: pure maths is building things out of basic 2×1 Lego bricks, while in applied maths you get to use the wheels and doors and hinge pieces, so you can do more interesting stuff, but isn’t the pure maths better somehow?), although it does sometimes make for a slightly tangled web (imagine if analogies were woven by spiders). As a result, the amount of time dedicated to explaining the actual mathematics is sometimes slightly less than you’d like. It’s important to get where the maths sits in context, and the analogies are great for that, but sometimes the maths explanations might need reading and re-reading to get an understanding of the way it actually works.

By this stage our energy was flagging and our attention waning, but we pushed on to bake (or rather, not bake) the Raw Chocolate Cookies from Chapter 13. By now the book is starting its explanation of how category theory works, and dealing with the concept of equality and sameness – what things are equal to each other? Numbers can be equal, but there’s a sense in which ‘2+4′ is not the same thing as ‘6’; is homotopy equivalence in topology a form of equality? The analogy here is that the cookies, which might have started as regular cookies, have been made gluten-free, then vegan, then sugar-free, then low-fat, and each step doesn’t take you too far away from the previous cookie, but overall when you’ve finished they’re very different.

Here we struggled with a few things – firstly, some of the ingredients are pretty rare. I now own a jar of coconut oil, which I didn’t before, but struggled to get hold of cocoa butter, which we substituted with… some butter, and we also ran out of cocoa powder halfway through, so we had to add some other stuff, including a bit of flour and some white hot chocolate powder. We basically undid the veganness and the gluten-freeness, and the low-fatness. All of the things we put into these cookies were ingredients you’d normally find in a nice baked product, but somehow all our substitutions and ineptitude culminated in biscuits which looked, and tasted, like dog biscuits. Not even my funky maths cookie cutters could save these. They’re very much not equal to biscuits. 2/10.

The book has a lovely tone – it’s relentlessly positive about maths, the ways in which it can be useful and fascinating, and people’s ability to do and engage with mathematics. It’s also full of friendly anecdotes about Cheng’s own failures and successes, in and out of the kitchen and maths department, and draws the reader in with her enthusiasm and charm. Cake-based metaphors are used even away from the recipes in each chapter – had you noticed that the multiplication table for $\mathbb{Z}_2$ was a Battenberg cake?

We did have one final success story – and since we’d been doing it in amongst everything else and leaving things to soak, we almost forgot – which was that we’d also made the chocolate and prune bread-and-butter pudding from Chapter 6, which came with one of my favourite cake/maths analogies. The recipe had been developed to use up some leftover bread, and prunes, and the chapter discusses motivation for doing and learning maths – the motivation can be external (people telling you maths is useful, or you having a particular problem to solve) or internal (you just want to do it, or you enjoy it, or you want to know the answer) and you can’t expect people to learn maths without both types of motivation. The leftover food had been an internal motivation for making the pudding, which probably wouldn’t have happened otherwise.

We found this the easiest of the recipes to follow – it uses shredded up breadcrumbed bread instead of whole pieces, which is unusual, but works – and the result was a lovely, moist, fruity cake which would go well on its own or warm with custard or ice cream, in our opinion. We ended up with one big square cake, which we cut into squares and put each in a paper case, and they were well tasty. 8/10.

Our forays into mathematical baking were probably hindered by a combination of the fact that a) this isn’t actually a cookbook, so you’re probably not meant to bake these things; b) we didn’t have all of the right ingredients, despite a frantic afternoon in a huge ASDA trying to find cocoa butter and potato flour (which I had to swap for just regular gluten-free flour), and c) trying to bake 6 different things at the same time probably doesn’t result in the best focus/concentration. But we had fun! And that’s what maths is all about.

I’d imagine this book would be great for someone studying maths at uni, or a keen A-level student, although it’d probably take a decent amount of concentration to make sense of it all – it’s definitely not for the casual reader. As someone who already understands most of the concepts covered, I found it a lovely round-up of some brilliant kinds of mathematics (and I never did quite *get* category theory first time round, so it was nice to have it laid out in simple terms) – although I’d imagine if you don’t already know all of the maths, you’d need to be careful navigating the complex web of analogies and jumping from topic to topic.

We’re terrible at baking, but this book was fun and covered some cool maths, using some nice analogies, and would serve as a good intro for someone getting into category theory. Maybe not for complete beginners, but there’s already plenty of books out there for them, right?

Cakes, Custard and Category Theory, on Amazon

Noel-Ann Bradshaw’s review, in the Times Higher Education

Cakes, Custard and Category Theory, on the Profile Books website

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