The Mathematically Perfect Doughnut

16th December 2014

Ok, lets get one thing out of the way. Every so often we see items in the news that usually say the same thing - "'Boffins' at such and such university have done the maths on the perfect (insert food or festive element here) and have come up with a mathematical formula to get it right every time." Many mathematicians groan at this, because they see, like most of us, that the study is usually commissioned by some corporation, and they complain that it is just a gimmick and trivialises proper mathematical research. This is probably very true.

But what studies like this also do is make maths seem more alive for the average non 'boffin', they make it more fun, more accessible, more applicable - and more delicious! People love their food, and realising you can, in theory, apply maths to something you love and make it more awesome? That's a win-win situation. With that in mind, we will allow the trivialisation of our favourite subject and bring you - the formula for the perfect doughnut! Apparently the hole is the key element...



By the way, did you know that both the Americans and the Dutch claim to have invented the doughnut, but just last year a recipe for doughnuts was found in an recipe book written in the UK in 1800 by Baroness Dimsdale, the wife of smallpox pioneer Baron Thomas Dimsdale? This pre-dates the American and Dutch claims by at least 37 years! Which makes this formula all the sweeter.

The formula has been developed by Professor Eugenia Cheng, a senior lecturer of Pure Mathematics at Sheffield University - admittedly in association with Dominos who are currently rolling out a range of doughnuts!

We have encountered Professor Eugenia Cheng before though, and we can honestly admit that this mathematician loves her food as well as her maths.

She likes to call herself the 'Maths-ster Chef' and she has already developed the formula for the 'perfect cream tea' in a study commissioned by Rodda's Cornish Clotted Cream, which we loved so much we featured it in a previous newsletter article. But on her personal staff page at the Sheffield University website she proudly shows off other mathematical food projects that she has done for her own satisfaction and fun such as these mathematical bagels cut into interlocking halves. She must like food with a hole in it!


Professor Cheng says she likes to "bring personality, fun and hilarity to maths." She also likes "baking, and using food to explain maths whenever possible" We're all for that!

So why doughnuts? As Homer Simpson would say -"MmmmÖ. Doughnut" they are pretty yummy and they are currently becoming much more popular in this country. Professor Cheng explains that 'The doughnut, aka torus, is an important mathematical object, as well as being delicious' plus Domino's need for a new dessert may have had something to do with it as well.

Incidentally Homer suggested his theory of a Doughnut shaped universe to Stephen Hawking in an episode of The Simpsons to which Hawking replied he found the theory "intriguing".


This was a reference to a genuine theory about the nature of the universe - the three-torus model of the universe, known informally as the doughnut theory of the universe. It's really pretty interesting if you would like to know more. There is an article here that explains it quite well.



So what makes the perfect doughnut then? Apparently itís all about finding the volume and surface area of doughnuts, the sugar to doughnut ratio, the mass of sugar and the 'squidge to crisp ratio.' Professor Cheng identified that the 'squidge to crisp' ratio is defined by a very simple factor - the size of the hole in the middle. The bigger the hole, the crispier the doughnut.

They did the maths and discovered that the perfect level of 'squidginess to crispiness', a ring doughnut should have an average hole size of 0.4- inches (11mm). This gives it a ratio of 3.5 to 1. This means the doughnut's diameter should lie somewhere between 2.8 inches and 3.2 inches (72mm - 82mm).


Dr Cheng explains that "This relatively small hole means that the doughnuts are 78 per cent squidge and 22 per cent crisp. You imagine that as the doughnut grows, it has to keep adding on an infinitely thin surface area amount of doughnut, like putting on extra layers of clothing. Of course, there's no such thing as an infinitely thin layer of doughnut around the outside - in reality it has some thickness. This is the crispy part around the outside."

"The hole is integral to the 'whole' doughnut experience, so it makes complete sense that it affects the texture and taste." said Simon Wallis, sales and marketing director at Domino's. Domino's are bringing out a range of doughnuts, and we are pleased that they will be applying this yummy mathematics to their new treats. We are determined to test some as soon as possible - in the name of science and mathematics of course!

Here is the formula itself, in it's entirety...



In the formula, R means the radius of the doughnut, measured from the centre of the hole to the middle of the dough, while the smaller radius of the dough inside is shown as 'r', measuring the thickness of the dough. Dr Cheng tried testing 5g of sugar, and found it covered a radius of 70mm. She then worked out how much sugar she needed to cover an average doughnut with an R of 30mm and a r of 15mm, before using the ratio to calculate that 5.8g of sugar is sufficient to adequately cover the perfect doughnut.

Dr Cheng found that the sugar to doughnut ratio was established as two over r. She admits that "If we fix the volume of dough, the amount of sugar we get is proportional to the square root of the size of the hole."

However, Dr Cheng is such a lover of food, that she is happy to advise that we take her formula with a pinch of salt (or sugar) saying "It's easy to get carried away messing around with calculus." Here are the final results - and they do look pretty good.



Ultimately her advice is "Go ahead and eat your doughnuts however you like them." That's a relief but we still can't wait to try the new Domino's doughnuts - we guess the marketing exercise worked, but we enjoyed the maths - and we're sure we will enjoy the results too! Professor Cheng has also done research on pizza, which we might feature next month if we are impressed with the doughnuts.