# How come you can fold a paper, regardless of thickness and size, up to 8 times?

For example, when you fold an A4 sheet (each time double) you can fold a maximum of 8 times. Now I’ve tried the same with a tissue paper of almost 1m², and I could still only fold 8 times, despite the larger surface and smaller thickness. How come?

That number 8 is not quite right, on the internet I find numbers from 6 to 11, although that 11 was a (successful) record attempt.

There’s part materials science and part math in the answer.

Whether something folds well or not (feels soft or stiff) depends on the bending stiffness of a material. The bending stiffness depends on the elastic modulus (this is a fixed property of the material) and the thickness of the material to the third power. You can see that the thickness will play a crucial role. A plate that is twice as thick is 8 times stiffer. Aluminum foil is very easy to bend, although aluminum is a very hard material.

When you fold something, the thickness doubles with every fold. So the thickness increases like 2N, where N is the number of times you folded the paper. Just write down those numbers for N from 1 to eg 15 and you will see that the series increases very quickly! See e.g.: http://www.frankdeboosere.be/getallen/getal%20folding.html

All paper types presumably have an elastic modulus that is in the same order of magnitude (I mean that they will not differ by a factor of 10). For the thicknesses of common types of paper I find values ​​between 0.02 and 0.1 milliliters on the internet. This is a factor of 5 difference. Every time you bend, the bundle becomes stiffer by a factor of 8. The difference in thickness between soft and stiff paper is therefore insufficient to be able to fold it an extra time. That is why you end up with the same number for all paper types.

Also see:
http://www.een.be/television1_master/programma/e_hoez_zozitdat_9xpapierpluwen/index.shtml

#### Professor Walter Lauriks

Physics Acoustics

Catholic University of Leuven
Old Market 13 3000 Leuven
https://www.kuleuven.be/

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