**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
-"Mmmm…. 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 it’s 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.

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