Hey, hey! So what’s so cool about pi? Probably an infinite number of things! Here I want to give you a few fun facts about pi that aren’t directly related to geometry and circles.

So let’s begin!

#1 Buffon!

One of the first recorded uses of the Monte Carlo method (before it was even called that!) in a geoemetric setting is now known as “Buffon’s Needle Problem”. Take a rectangular surface with evenly spaced stripes and throw a bunch of needles on it at random. The needles ought to be a specific size. Do this with enough needles and you can approximate $\pi$! Here’s one sample:

For further reading, check out: Buffon.

#2 The “Most Beautiful Equation”

Beauty is in the eye of the beholder. However, a 2014 paper concluded that $e^{i\pi }+1=0$ (Euler’s identity) was indeed the most beautiful equation. This equation uses the mathematical constant $e$, $\pi$, the imaginary number $i$, the number $1$, and the number $0$. You decide, most beautiful?

If you want to read about the research, check out this paper.

#3 Infinite Series

Well, we’ve got Euler’s identity, but there are some cool looking infinite series that add up a quantity involving $\pi$. Here is one particular one that is often a standard problem to prove in a Fourier series course or by use of Parseval’s theorem (or a bunch of other ways).

$$\sum _{k=0}^{\mathrm{\infty }}\frac{1}{k^2}=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots =\frac{{\pi }^2}{6}$$

#4 Actually, my version of beauty

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-x^2}dx=\sqrt{\pi }$$

’nuff said.

#5 OEIS Sequence A283247

Guess who is credited with this $\pi$-prime sequence? Yours truly!

It begins like this: $3,31,13147,73141,314159,314159,131415923,\dots$ — the description of this sequence is “$a(n)$ is the smallest prime number whose representation contains as a substring the first $n$ digits of $\pi$ in base 10.” It’s a $\pi$ containing sequence of prime numbers!

#6 Speaking of prime numbers

David Radcliffe finds this beautiful (probably) prime number using only the digits 3, 1, and 4!

A 314 prime for Pi Day: A 3141 digit probable prime, with random digits chosen from 3, 1, and 4, starting with 314. https://t.co/rRhv1Ptf2c

— David Radcliffe (@daveinstpaul) March 3, 2019

I had found a measly 463-digit number …
3141143341443144133134133414411314111313431343331433433334311443134444443334411143134144133143343311113144131431144414314114434341444114411144331444113313414341411141314344441344331131143134133343341134333144311444434333413131114414313313441431313131344413411111434331441333134333414431331433444344114341333331334341411434444341431334444114141441131343133314333333334414344411444443111443114141434133414314444133143431111343111313314141314114143331143434131411441

Scavenger hunt question for you! Reading left to right only, how many “314” sequences can you find?

#7 Continued Fraction!

$$\frac{4}{\pi }=1+\frac{1^2}{2+\frac{3^2}{2+\frac{5^2}{2+\frac{7^2}{2+\cdots }}}}$$

Learn more about continued fractions.

#8 A Fun Comparison

Which is larger? $e^{\pi }$ or ${\pi }^e$. But let’s add a twist!

Order these four numbers from least to greatest:
$$e^{\pi },{\pi }^e,e^{\frac{22}{7}},(\frac{22}{7}{)}^e$$

#9 It’s a little complex

$$i^i=e^{-\frac{\pi }{2}}$$
Maybe this the most beauitful equation!

#10 It’s not just irrational, it’s transcendental!

$\pi$ is a transcendental number! And it is unknown if $\pi +e$ is an irrational number! But I’m putting my money on that that sum is irrational. Learn more about transcendental numbers.