Using topology for discrete problems | The Borsuk-Ulam theorem and stolen necklaces

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Strong rec Kaligule · 2020-07 · Mathematics Stack Exchange · on “Do infinitely many points on earth have the same temperature as their antipodal?”

Take a point $p \in X \setminus A$ and call it northpole, call $-p$ southpole. Now we look at the longitudes, the circles (longitudes) through these poles on the sphere. Take one longitude and call it $L$ . $f$ is defined on $L$ and it is nonzero on $p$ . As $f(-p) = -f(p)$ we know that $f$ has both negative and positive points on $L$ . As $L$ is connected and $f$ is continous this means that there is a point $z_L$ on $L$ where $f (z_L)=0$ . So $ z_L \in L \cap A$ . $p$ and $-p$ are not in $L \cap A$ , though, so $z_L$ is not one of the poles. As this is true for each of the infinite longitudes $L$ this means that there is a point in $A$ for every longitude. These points are distinct as the

Recommends Calvin Lin · 2020-04 · Mathematics Stack Exchange · on “Stolen Necklace problem - Borsuk Ulam - function continuity”

In the video, Grant explains how to consider the " Continuous Necklace Problem [this video] ", and why that resolves the discrete case.

Recommends Sammy Black · 2022-06 · Mathematics Stack Exchange · on “What is the big picture of topology”

Sometimes the best way to get a feel for a subject is to see some of its theorems and unsolved problems in action. You might enjoy this [LINK] , this [this video] , or perhaps this [LINK] .

Recommends Azur · 2023-02 · Mathematics Stack Exchange · on “Ham sandwich theorem.”

I can only recommend the 3b1b video on the Stolen Necklace Problem [this video] , which explains the visual intuition behind a very similar problem (though you'd need to formalize it a bit and adapt it to your situation). (You can skip everything from 3:45 to 10:45, which is just explaining the Borsuk-Ulam theorem. The part you really care about is around the 16:00 timecode, but it's good to see the statement of the problem first)

Recommends user21820 · 2026-01 · Mathematics Stack Exchange · on “What are the most engaging consequences of the Intermediate Value Theorem?”

For mathematicians, another interesting thing is that it has some purely discrete consequences (e.g. necklace splitting problem [this video] ) that are not easy to prove by purely discrete means. Note that this is not because the problem is fundamentally continuous, but actually because it is fundamentally as hard as the combinatorial core of the multi-dimensional IVT, in the same way that Sperner's lemma is as hard as the combinatorial core of Brouwer's fixed-point theorem [LINK] .

Cites Tyson Che · 2022-07 · Mathematics Stack Exchange · on “Does the Hairy Ball theorem imply the Borsuk-Ulam for even dimensions?”

WOW! I've learnt the hairy ball theorem several weeks ago and I've just been watching 3blue1brown's video on the Borsuk-Ulam theorem [this video] and the idea just hit me! I googled and found this post! Now here're my naive thoughts on that: We can cover the sphere $S^{2n}$ with patches $U_k:= S^{2n}-(0,\dots,0,1)$ whose chart map is the stereographic projection, and $ V_k:= S^{2n}\cap B^3_{\frac{1}{k}}(1,0,\dots,0) $ whose chart map is the projection to $B^2_{\frac{1}{k}}(0,\dots,0)$ . The charts with them give two continuous frame bundles $\{ \frac{\partial }{\partial x^i}\},\{\frac{\partial }{\partial y^i} \}$ over $U_k,V_k$ respectively. Since $S^{2n}$ is a compact manifold, the open cover ad