Why You Can’t Fold a Piece of Paper 8 Times

You may have heard the adage “you can’t fold a piece of paper eight times,” and I guarantee many of you have at least once tried to do it only to discover after the seventh fold that you couldn’t get to the eighth fold. I started imagining a mathematical approach to explain why.

We first need to know the measurements of an A4 piece of paper. A4 paper measures typically 297mm x 210mm x 0.075mm. The initial fold is along the 297mm side because I’ll presume that every time we fold the paper, the length of the side that is the longest will change.

Fold 1

Let’s see what occurs during the initial fold. As there are now two layers of paper and the 297mm side is cut in half, the thickness doubles to 0.1mm. However, we must take into account the fold’s own modest amount of length when computing the new dimensions. I’ll suppose that each fold forms a semicircle and that there is very little space between the layers. Let’s examine the initial fold by focusing on the crease where each paper layer is visible.

Fold 2

We’ll see a pattern for the sizes of each fold after looking at one more fold.

Now that the crease has a 0.15 circumference and a 0.3mm thickness, the new measurements are (297–0.075)/2mm x (210–0.15)/2mm x 0.3mm.

We can notice the pattern for each dimension if we table our dimensions after each fold.
The fact that you are stacking layers on top of each other explains why the thickness doubles every time. The dimensions are getting narrower and shorter. Every even number fold has its width changed by halving the new crease circumference and every odd number fold has its length changed by doing the same.