# Borsuk-Ulam Theorem

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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 $f: S^n \to \mathbb{R}^n$ is continuous then there exists an $x\in S^n$ such that:  $f(-x)=f(x)$.

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, $f$, with an interval, $[a, b]$, as its domain, takes values $f(a)$ and $f(b)$ at each end of the interval, then it also takes any value between $f(a)$ and $f(b)$ 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:

Life is never fair. So I made my life a mathematical fair.

### 6 responses »

1. Found this very interesting. Great post!

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• All credit goes to James Grime, Marc Chamberland, Grant Sanderson and Michael Stevens.

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2. Yes, this is one of the most profound insights of Stanislaw Ulam (and Borsuk). Refer to “Adventures of a mathematician” by Stanislaw Ulam. I think some applications have been found of this theorem in Combinatorics. There is a book on it by Jiri Matousek. It is available in Amazon India….I am not sure…but please do check…Or, your teacher, Dr. Ritwik Mukherjee might already be aware of some deeper literature in this area.

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• Yes, there are applications of this theorem to combinatorics. For example, Sperner’s Lemma (and its application to Hex game..). Yesterday, Burkard Polster (aka The Mathologer) uploaded a video discussing its application to fair division problem: https://youtu.be/7s-YM-kcKME
I tried to read Ulam’s biography 3 years ago, but wasn’t able to finish it since I didn’t enjoyed his writing style.

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• thanks for the pointers to its connections to Combinatorics!

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