Yesterday, I was fortunate enough to attend a lecture delivered by Dr. Ritwik Mukherjee, one of my professors, to motivate the study of algebraic topology. Instead of using the “soft targets” like Möbius strip etc. he used the following profound theorem for motivation:
If is continuous then there exists an such that: .
This is known as Borsuk-Ulam Theorem. To appreciate this theorem, one need to know a fundamental theorem about continuous functions known as Intermediate Value Theorem:
If a continuous function, , with an interval, , as its domain, takes values and at each end of the interval, then it also takes any value between and at some point within the interval.
Here is a video by James Grime illustrating Borsuk-Ulam Theorem in 3D:
Though the implications of the theorem itself are beautiful, following corollary known as Ham sandwich theorem is even more interesting. Here is a video by Marc Chamberland explaining this theorem:
Also, yesterday Grant Sanderson uploaded a video exploring the relation of Borsuk-Ulam Theorem with a fair division problem known as Necklace splitting problem:
But, to my amazement, this theorem is related to one of the other most astonishing theorem of algebraic topology called Brouwer fixed-point theorem:
Every continuous function from a closed ball of a Euclidean space into itself has a fixed point.
Here is a video by Michael Stevens illustrating Brouwer fixed-point theorem in some interesting cases:
Now the applications of this theorem are numerous, and there is a book dedicated to this theorem: “Fixed Points” by Yu. A. Shashkin. But my favourite application of this fixed point theorem is to the board game called Hex, explained by Marc Chamberland here:
If you come across some other video/article discussing the coolness of “Borsuk-Ulam Theorem” please let me know.
Edit (18 May 2018): Proof of Brouwer’s Fixed Point Theorem by Tai-Danae Bradley: