Forwarded: Why do prime numbers make these spirals?
It’s a humid summer day back at the university and I’m sitting in for the semester examination of Modern Algebra in the 3rd year of my Masters in Mathematics. There is a question about Dirichlet’s Theorem and I totally blank on what I had crammed with my friends a day before the examination.
This video brings about an intuition for the relation between geometric representation of numbers with some analytical validations.
Residue classes mod n (n.k + m): everything m above multiples of n.
Coprimes: relatively prime pair of numbers.
Euler’s Totient Function φ(n) : no of integers from 1 to n which are coprime to n.
Relation between primes and π.
Dirichlet’s Theorem.
How pretty but pointless patterns in polar plots of primes prompt pretty important ponderings on properties of those primes. Such arbitrary geometric exploration of numbers led me to understand something so beautiful in math.
Cue back to the university examination.
Most of the concepts and theorems I came across in my formal mathematical education I approached them as arbitrary, rigorous definitions that served an immediate purpose rather than learning fundamental constructs in their depth - much like what you would feel if you read the above list and not watch the video. Grant Sanderson’s simple explanations to such complex mathematical concepts definitely help me understand better and build the intuition I was lacking.
I find rediscovering mathematical topics on my own out of genuine curiosity is so much more fulfilling than the obligation to study them for the sake of formal education. Fascinating this!