The Paper Folding Paradox
Let’s imagine a large, square sheet of paper.
Fold it along one of its diagonals. You now have a right triangle.
Fold it again so that the two triangles perfectly overlap — that is, along the bisector of the right angle.
Repeat this process again and again, as many times as you can.
As unbelievable as it sounds, you can’t do this 64 times.
There are two reasons for that: one physical and one mathematical.
The Paper-Folding Paradox
It’s impossible to fold a sheet of paper more than 7 or 8 times, no matter how large it is, because the thickness of the material doubles with every fold.
If the fold is always along the angle bisector, the paper becomes smaller and smaller while the layers pile up — after just a few dozen folds, the thickness would reach unimaginable proportions.
A quick calculation:
If an average sheet of paper is 0.1 mm thick, then
- After 10 folds: 0.1 × 2¹⁰ = 102.4 mm
- After 20 folds: about 100 meters
- After 30 folds: about 100 kilometers
- After 50 folds: about 100 million kilometers — roughly the distance from the Earth to the Sun
- After 64 folds: more than 170 billion kilometers — far beyond the Solar System
And if we look at the shape…
Each fold doubles the number of small triangles created.
The original square becomes one triangle, then two, four, eight, sixteen…
With each fold, the number doubles again.
- After the 1st fold: 1 triangle
- After the 2nd fold: 2 triangles
- After the 3rd fold: 4 triangles
- After the 4th fold: 8 triangles
- After 10 folds: 1,024 triangles
- After 20 folds: 1,048,576 triangles
- After 30 folds: more than one billion triangles
If we could somehow fold it 64 times, the number of triangles would reach
2⁶⁴ = 18,446,744,073,709,551,616 — over 18 trillion.
That’s more tiny triangles than there are grains of sand on all the beaches of Earth.
Summary
This problem is reminiscent of the classic puzzles of the Towers of Hanoi or the grains of wheat on a chessboard (one on the first square, two on the second, four on the third, and so on).