Magical Patterns

Roger Antonsen

March 14, 2016

The number π is rightly a well-known, famous number surrounded by myths. Many have an almost religious relationship to this number, and many believe that it is by far the most beautiful and unique number that exists. However, if we think about it, is it really *that* special? It's just one of many other numbers. And it is nicely located on the number line, side by side with the other numbers:

If you ask a random person about what π is, you will almost certainly get the answer "three point fourteen". But this association is at best misleading. It is true that π can be *rounded* to 3.14, but really π is just a *ratio*.

I can not stress it enough; it is a *ratio*. That is, it is defined as a *ratio* between two magnitudes.

In exactly the same way as the relationship between the length and width of a sheet of A4 paper always equals the square root of two, the ratio between the diameter and circumference of a circle always equals π. This is the *definition* of π. So, no one just sat down defined π as the number 3.14. (Although the state of Indiana, USA came dangerously close to defining π as 3.2 in 1897.)

The symbol π represents a *ratio of sizes*, and it has been used like this since the mathematician Leonard Euler made it popular in the mid-1700s. If you take a circle, no matter how small or large it is, and measure how long the circumference is in relation to how long the diameter is, you always get π.

If you really insist on redefining π, for example to 3, you may do that, but then you must also agree that all circles look like regular hexagons.

If you have a hexagon like here, the ratio between the circumference and the diameter, that is π, is equal to 3.

I hear now and then people say that "π is infinite". That is somewhat strange, because π is as finite as any other number between 3 and 4. What is meant is that if you write out all the digits, *you can go on endlessly* without a repeating pattern occurring.

If we begin, π looks like this:

```
3,
1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999...
```

Starting with decimal number 762, we find six ninths in a row! A familiar π joke is that you can memorize all the numbers until this points, and then continue with "nine, nine, nine, nine, nine, nine, and so on". There is story about the physicist Richard Feynman (1918–1988), who commented on this in one of his lectures, and these six digits are often referred to as *the Feynman point*.

The reason why the numbers do not repeat or form a pattern, is that π is not a so-called *rational* number. This was proved by mathematician Johann Heinrich Lambert in 1761. In short, it means that it is impossible to write π as a fraction.

There are no integers A and B such that π = A/B. The fractions 22/7 and 355/113 are good estimates, and even though we can come arbitrarily close with fractions, we will never be able to come all the way there.

But, it's quite irrelevant just *how* we write down π. Just as there are many ways to write the number 4 on (FIRE, four, 4, 2+2, IV, 1111, 100, 11, 10, ...), there are infinitely may ways of writing down the number π. The important thing is how π is defined.

The obsession surroundingthe number π has also resulted in many records. For example, Rajveer Meena managed to recite 70,000 digits in March 2015. This is the current world record. And how many decimal places have we managed to calculate? In October 2014, after 208 days of calculations, a new record was set: π has now been calculated to 13,3 trillion – that is 13 300 000 000 000 - digits.

How many digits do you really need? Probably never more than ten. And if you have 39 digits, you can calculate the volume of the universe to the size of an atom.

I enjoy programming, and a while ago I made a program that replaced all the digits in π with colors. Then, I got the following. Here you see 1160 digits. (Do you see the Feynman point?)

Here you see 7500 digits:

No one has managed to find any pattern in this picture, and nobody knows if it's equally much of each color, if you keep drawing infinitely long.

Happy π-day, March 14th!