# The croissant equation

Each time you eat a croissant you might be biting more than 500 layers of dough!

If you have a sweet tooth, then perhaps you enjoy just standing outside a fancy bakery and observing the many cakes and bakes from the shop, from their indulgent red velvet cupcakes, creamy sponges or decadent brownies. Cakes, cookies and cupcakes are complicated pieces of baking engineering which require sophisticated techniques to get the many flavours and textures into the single bite that you enjoy so much.
From the many options in a bakery, one of my favourites has to be the croissant. Layer after layer of spongy crunchiness mixed with a creamy buttery flavour: this flaky pastry is simply the best. Each time I eat one (which inevitably happens quite frequently) I cannot avoid thinking about the complex technique to get those many layers of dough I am eating. There are dozens of them!
If we go into the baking process for a croissant, we can actually obtain an equation for the number of layers in a croissant. Firstly, the baker prepares a mix of flour, sugar, milk, yeast and a bit of salt into a dough. It gets kneaded into a roll and then wraps a rectangle of chilled butter in it. The baker produces a layer of butter trapped between two layers of dough.

Then, using a rolling pin, the baker stretches the two layers of dough with a layer of butter inside into a large rectangle and then folds it in equal thirds. Since the butter remains cold after this stretching and folding, it divides what will eventually be different layers on a croissant. Dough in contact with dough eventually merges, but not when there is a layer of chilled butter in-between. After the first folding process, there are three layers of butter dividing four layers of dough.

The recipe to create the best croissant requires a cooling step after the first folding, so that the butter inside gets chilled again, ready for the next folding. The baker then stretches the dough into a rectangle and folds it into thirds, pretty much like he or she did in the previous step. By the time of the second folding, there are 9 layers of butter between 10 layers of dough… and then again, cooling down, stretching and folding, and again (and for some bakers, again). The recipe for a croissant usually takes hours!

After each step of stretching and folding, there are three times the number of layers of butter than on the previous step, since each time the baker folds the dough into thirds. Hence, if $s$ is the number of times that the baker has stretched and folded the dough, then we get that the layers of butter are $3^{s}$. For example, when $s=0$, i.e., no folding and stretching, we get the $3^{0}=1$, the initial layer of butter sandwiched between the two layers of dough.

Counting the layers of dough at this step is easy if we know the number of layers of butter, which is $3^{s} + 1,$ because of the top and bottom layers of dough. Most of the croissant recipes require 3 or 4 folds, so this wonderful creation already has 27 or 81 thin layers of butter trapped between layers of dough (called a laminated dough).

The baker then gives the dough a final stretch to make a long rectangle, cuts it into triangles and rolls each triangle from the base to the tip. Each rolled triangle will eventually become, after the baking process, a crunchy croissant. Now, let’s focus on the centre of the piece, the one fantastic bite that has all the layers. If the baker rolls once, then there are twice as many layers of butter; with two rolls, there are 4 times the layers of butter, and, if the baker rolls $r$ times, then there will be $2r$ times the layers of butter in the central part of the croissant. Sometimes they do a small cut on the base of the triangle, just to stretch the base a bit more, in which case you can just count one less roll.

Combining the two parts of the recipe, the stretching and the rolling of a croissant, we get that the number of layers of butter is $2r ( 3^{s} )$ which means that the number of layers of dough is
$$\text{Layers of dough} = 2r ( 3^{s} ) + 1,$$
which then will go into the oven to form one wonderful piece of baked goodie.

Different bakeries make their croissants with different rolls and folds, but most commonly, $s=4$ and $r = 4$, so that the number of layers of dough in a regular croissant is 649.

Gordon Ramsay prepares his croissants with $s = 3$ and $r = 4$, so when you eat one of his croissants, you will go through 217 layers of crunchy dough, whereas Laura Vitale makes her croissants with $s = 4$ and $r = 3$, so you go through 487 layers of crunchy dough on her fantastic pastries.

So remember, the next time you bite one of these succulent pieces of bread, at the centre you might actually be biting more than 500 layers of dough!

[Pictures: 1 – adapted from Flickr.com – croissant proff by Anthony Gergeff, CC-BY 2.0; 2 – adapted from Flickr.com – croissants by VV Nincic, CC-BY 2.0; other pictures by Chalkdust]

Rafael Prieto Curiel is doing a PhD in mathematics and crime. He is interested in mathematical modelling of any social issues, such as road accidents, migration, crime, fear and gossip.
@rafaelprietoc    rafaelprietocuriel.wordpress.com    + More articles by Rafael

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