Physicists create formula for how many times you can fold a crêpe

If you gently fold a disc made of some flexible and possibly tasty material, what makes it stay folded? And how many times can you fold it before it puts up a fight and flips back?

A physicist from France, home of the crêpe, decided to find out. He discovered that just one number tells you all you need to know.

Tom Marzin at Cornell University in Ithaca, New York, wondered about crêpe-folding when he was on holiday in his home region of Brittany, France, where this thin pancake is especially popular. Just folding a tip of it would result in it flipping back, but with a larger fold, friction and gravity would conspire to keep it still. What rules could govern this behaviour?

Marzin turned it into a research project, the results of which he will present on 20 March at a meeting of the American Physical Society in Denver, Colorado.

His work is different from the origami-like folds some physicists study, which are permanent. “What we’re dealing with here is what I call a soft or smooth fold. And it is just a competition between gravity and elasticity,” says Marzin.

One way to observe this competition is to stick part of a pancake to a tabletop, let the other end hang over the edge and measure how much it sags. Marzin worked out that the answer can be predicted with one number, dubbed the elasto-gravity length, which combines the material’s density, its stiffness and the force of gravity. He suspected that this number would also govern the behaviour of flexible materials in other situations, and in a computer model this turned out to be the case.

To check his simulations in the real world, Marzin experimented with plastic discs, store-bought tortillas and, of course, crêpes. He started out making the latter himself, but scientifically they weren’t fit for purpose.

“I didn’t control the thickness well,” he says. “So I asked my mom to perform the experiments over in France. I asked her to buy the callipers and rulers and a bunch of crêpes from a commercial brand. Those were probably made by a machine, [so] that guarantees a good uniform thickness. And she did it really correctly.”

Marzin’s experiments confirmed that all aspects of crêpe-folding depend on the elasto-gravity length. For instance, it governs how much of the area of a sheet that’s folded will go into the part that loops over. This determines if there will be enough flat area left for another fold.

His equations correctly predict that a crêpe 26 centimetres in diameter and 0.9 millimetres thick can be folded up to four times, whereas a 1.5-mm-thick tortilla of the same size, with an elasto-gravity length 3.4 times as large, will allow only two folds. “This length captures all the physics underneath,” Marzin says.