Happy Pi Day!

In the American date format, today is 3.14 - a pretty good approximation to $\pi$, which (as any fule kno) is the ratio between a circle’s diameter and circumference. It’s also the focus of most of the jokes about maths, because - get this - it sounds just like pie! So, you can get T-shirts about apple $\pi$, shepherd’s $\pi$, $\pi$rates and so on - the only limit is the size of your dictionary and/or recipe book.

So, in celebration, I thought I’d come up with a few interesting things about $\pi$. That may not be on a t-shirt.

### Approximations of pi

The usual approximations for $\pi$ are 3.14 or - if you’re feeling really fancy - 3.142. With decimals (and even fractions), you can only approximate $\pi$, because it’s an irrational number - but you can get pretty close. Here are some of the best approximations, with their percentage error:

Approximation

Error

3

-4.51%

3.1

-1.32%

3.14

-0.05%

$\frac{22}{7}$

+0.04%

3.142

+0.01%

$\frac{355}{113}$

0.0000085%

As you can see, $\frac{355}{113}$ is as close as you’ll ever need. At that scale, you could estimate the circumference of the earth to within about 3 metres. It’s not bad.

### Mnemonics for pi

There are all manner of mnemonics - silly phrases to help you to remember the decimal expansion of $\pi$. My favourite:

“May I have a large container of coffee? Thank you. Beans! Espresso! Brazilian! Roasted Bahitalia for my pot. Mornings need coffee, or sleepy time all day.”

The numbers of letters in each word give $\pi$ to 24 decimal places: 3.141 592 653 589 793 236 846 243 3. You can find more digits here and more mnemonics here.

### Unusual places to find pi

One of the most famous equations in maths is Euler’s identity - just like you can’t walk through a town in America without seeing something named after George Washington, there’s barely any maths you can do without finding something Euler had a hand in. Euler’s identity is $e^{\pi i} + 1 = 0$, which involves the imaginary square root of -1 ($i$). It’s one of the places where, if you don’t know anything about complex numbers, you do a double-take and say “what’s that $\pi$ doing there? This is nothing to do with circles!” … only for you to realise later that it’s everything to do with circles. And all of your trig identities, too.

It also appears - and this is my favourite - in the Mandelbrot set. This is another area of complex analysis that you probably don’t care too much about, except in a “look at the pretty pictures” way. It’s generated by applying an algorithm to all of the points in an area and plotting the ones that converge to a value. It turns out that in certain areas (such as the very thin ‘neck’ stretching out to the left), the number of iterations you need is almost exactly a multiple of $\pi$. See here for more details!

Lastly, you find $\pi$ in rivers. Einstein (all hail!) was the first to think about the difference between the actual length of a river and the distance as the crow flies, from the source to the mouth. The ratio between the two? It’s $\pi$.

I think that’s enough $\pi$ for now. I’m going to go and celebrate - as is traditional in the maths world - by having some pie. (Here in the UK, we get to celebrate both the American $\pi$ day - 3.14 - and the British $\pi$ day, which is $\frac{22}{7}$. As we established earlier, the British version is a better approximation.)