How to have a thickness equal to the Earth-Moon distance by folding a simple sheet of paper?

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According to various publications on the Web, folding a single sheet of paper several dozen times would have a thickness approximately equal to the Earth-Moon distance. This is theoretically possible, but in practice we would obviously be unable to achieve it.

Fold a sheet 42 times in a row

In a publication, the platform Maths Live explains an amazing theory. Consider a simple sheet of classic A4 paper, the thickness of which is 0.1 mm. By folding the latter 42 times in a row, it would be possible to obtain a thickness exceeding the Earth-Moon distance. Remember that the distance separating our planet from its satellite is 384,400 kilometers. This is obviously an average distance, because in reality, the latter constantly oscillates between 356,700 and 406,300 km.

It should be noted in passing that this is indeed the lunar distance, not to be confused with the astronomical unit (AU). The latter corresponds approximately to the distance between the Earth and the Sun, that is to say approximately 150 million kilometers.

Credit: Abdouj / Pixabay

What calculation?

First of all, remember that with each folding, we double the number of thickness. At the end of the third folding, the thickness of the sheet is equal to 0.1 × 2 × 2 × 2 = 0.1 × 23 = 0.1 × 8 = 0.8 mm. Following this logic, we get 0.1 × 242 = 439804651110.4 mm after 42 bends. In a publication Quora, an engineer presents the calculation otherwise: 2 ^ 42 (2 to the power of 42) paper thicknesses, or 4,398,046,511,104 thicknesses, or 4.4 trillion (or 4,398 billion) thicknesses. By multiplying by a thickness of 0.1 mm, we get 439804651.1104 m and so over 439,804 km.

If this theory is completely valid, its implementation remains completely impossible. Indeed, the surface obtained would be infinitely reduced. In addition, another much simpler reason explains this impossibility: it is usually very difficult to fold A4 sheet more than 8 times. Despite the infeasible side of the thing, this kind of reasoning remains interesting. On the platform forum Secouchermoinsbete, user Touko wrote:

“It’s true that this type of reasoning is infinite: if I crush a Carambar, it goes as far as Saturn and if I unfold the atoms of a couscous seed, I go around the earth a million times. … We don’t do it anymore! ”

Here is an explanatory video published by TED-Ed in 2021:

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