Paper Folding And Constructing Primes

Suppose you had a rod of steel and wanted to cut it into perfect thirds.

One way to do it is to get a piece of paper that is as long as the rod, fold it in thirds, then cut the steel at the crests.

If you are worried that your folding skills are not good enough for a perfect cut, this neat method for folding the paper should make your life easier:

Now, this piece of paper has actually become a ruler. This ruler is of Base 3, because each crest marks 0.1 and 0.2 in Base 3 (i.e., 1/3 and 2/3). The pattern on the ruler would go on to be like this: 0.1, 0.2, 1, 1.1, 1.2, 2, ...

But what if you folded the piece of paper into ten parts instead? Then, what you would have is a decimal ruler. This decimal ruler, although the most common of all rulers, has a problem for us here.

The first problem is that our ruler only has the marks 0.1, 0.2, 0.3, 0.4, 0.5..., and we want to cut at 0.333... (a third of the steel rod).

We can however cut at 0.4 (because we know we want more than 0.3), then use one division of the decimal-paper-ruler and fold that in 10 parts, so that in effect we have 0.01, 0.02, 0.03,... marks.

And so on and so on until we perfectly cut at 0.333...

This is obviously an infinite task. In fact, this is an inherent limitation of the decimal point system.

However, the Base 3 ruler suffers the same problem if you want to cut the rod into fifths. Luckily though, there is also a way to fold a paper into fifths:

From this we can say: for every prime number, there is a unique ruler. Unique meaning, without it, some divisions would be infinite tasks. And since such rulers can be constructed (or at least the ones we discussed so far!) through paper folding, one can say that if there is a general method for folding paper into unique rulers, one can use the same method for constructing primes.