Why is Pi Everywhere? 5 Levels from Basics to the Unexpected

Why does pi show up everywhere, even when there are no circles in sight? Let's explore pi in 5 levels, ranging from geometry to its surprise appearances in complex numbers, calculus, and probability!

This video is sponsored by Squarespace.

00:00 Introduction
00:26 Level 1: Circles & Distance
01:18 Level 2: Trigonometry
03:24 Sponsor Message
04:15 Level 3: Complex Numbers
07:01 Level 4: Gaussian Integral
09:52 Level 5: Central Limit Theorem

[Steindium]

Realistically π deserves at least 10 levels. I mean surely we can't miss out on the sum of all the inverse squares of natural numbers (Basel problem) and the problem of two blocks colliding against a wall!

[MarcosOliveira-vlun]

it is strange how satisfying this video is! thank you!!!

[Bob]

Making a video that covers mathematics from primary school to advanced university level is basically a bit strange, but this guy is actually able to make it work 🙂

[isavenewspapers]

0:33 Welp, I'm glad it at least showed up.

[aeonturnip]

1: Circles. 2: Circles. 3: Circles. 4: Circles. 5: Circles.

[EJBWraithlin]

More succinctly:
Because many problems have the symmetry of a circle.

-A circle is circular.
-Trigonometric functions sit on the radius of a circle.
-Imaginary numbers sit on the radii of circles.
-The area under the normal distribution is derived from a volume swept out by a circular rotation.
-The central limit theorum takes distributions back to the Gaussian.

[marcodamota]

The question I've always had...

Thanks!

[joe_lwrnc]

Basil problem not being on here is criminal

[niccolopaganinifranzliszt]

It's still fascinating me the fact that the result of the series from 0 to infinity of (-1)^(n)/(2n+1) equals π/4. I know the proof for that, doesn't makes it less fascinating.

[PaulTeska]

Love and conflict in humans and animals is 1 over 2 pi for conflict and love is 2 pi -1 over 2 pi. Birds are the easiest and simplest to study and prove this.
Basically show love 84% of the time and conflict show 16% of the time (obviously rounded).

[subbotin]

Thank you so much for making this type of content! You are explaining very well 👍

[mox]

Imagine two bodies initially at rest with masses m1 and m2 separated by distance r. How long will it take until they collide solely under the force of their gravitational attraction? Hint: the closed-form solution involves pi!

[firashadjtaieb]

Hi there, thanks for the video, some other places: Sum of 1/k^2, also sterling approximation for n! Has a Pi. Gamma(1/2) has pi

[JirivandenAssem]

pi is just a partition of X :D where pi is a subset of the powerset of X and X is the union of pi

[isavenewspapers]

2:44 Wouldn't this have been a good time to show those labels?

[wavephasemusic]

Because circles and triangles are everywhere … approximately.

[Visualizingmathandphysics]

Hello, nice video.
Could you please share as to which software you used to create these Animations, especially at 8:39
.
Would be really helpful for me.

[Phi]

When defining Pi, that's probably the first and last time you'll ever hear mathematicians talk about the diameter of a circle, which is why it's insane to me that they don't use the radius instead.

[Sithisine]

no coulomb's constant being in terms of pi and vacuum permittivity :(

[bobnewman]

Cool video, but doesn’t really seem to answer the question in the title of why pi appears everywhere. Instead it just shows examples of pi appearing in different areas of mathematics without explaining a connection. I think explaining the utility of pi is nearly the same as explaining the utility of circles, and the real important connections seems to be radial symmetry. That’s the common thread between trigonometry, complex numbers, and Gaussian statistics is the concept of radial symmetry, and why you see pi appearing everywhere