This open problem taught me what topology is
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it as a clear worked example of a concrete problem solved with topological language, and as a reference when discussing the Möbius strip and surface constructions.
Recommended by 1 people across 1 different months, 2025 to 2025.
What it is
A 3Blue1Brown video on the inscribed square and rectangle problem, showing how the rectangle case can be solved by mapping it to questions about surfaces such as the Möbius strip and Klein bottle. It ends by explaining what topology is, using the argument's origins in Vaughan's work and the 2020 Greene and Lobb paper.
Cited in 2 Wikipedia articles, including Inscribed square problem, Joshua Evan Greene.
What people said
Recommends D.R. · 2025-03 · Mathematics Stack Exchange · on “Simplest 'proof of concept' that scheme theory can solve significant concrete questions in number theory”
@AlexYoucis thanks for your comments; they could very well be an answer! For the last comment: it's more like "what's the easiest simple sounding/simply phrased questions can be solved by taking the perspective of manifolds & answering abstract manifold questions". And I have an answer: [this video] 3b1b shows how the inscribed rectangle problem can be phrased naturally as a topological question about a certain surface. It is immediately clear that having a language to talk about this surface rigorously+small intuitive lemma (1 slide at 22:30) directly solve the problem.
Cites TLss · 2025-01 · Mathematics Stack Exchange · on “Intuitive and detailed explanation for why a Möbius strip is homeomorphic to two unordered points on a circle”
A recent 3Blue1Brown video [this video] [14:56] noted that a Möbius strip was homeomorphic to the space of two unordered points on a circle. The homeomorphism however was not explicit and not particularly intuitive. Is there a clear and direct map from the space of two unordered points points on a circle to a Möbius strip in 3-space that elucidates what is going on? What does the Möbius strip having just one side have to do with the space of pairs of unordered points? This issue was discussed in this stackexchange discussion: Various proofs: pairs of points in a circle = Möbius strip [LINK] . However, that’s over my head mathematically. I was hoping for something more concrete.
Cites D.R. · 2025-03 · Mathematics Stack Exchange · on “Simplest 'proof of concept' that scheme theory can solve significant concrete questions in number theory”
In this MO answer [LINK] , Felipe Voloch makes the claim Actually, this is my only serious complaint about Hartshorne. He doesn't do any Number Theory and $\operatorname{Spec}\mathbb Z$ is where schemes really shine. Matt Emerton writes an amazing answer [LINK] discussing the use of scheme theory in Mazur's theorem on torsion points of elliptic curves, but that result is of course highly highly nontrivial. EDIT: Actually, maybe a small piece of Mazur's proof can work for my question: Emerton sketches the proof that the elliptic curve with equation $$y^2 +y = x^3 - x^2 - 10 x - 20$$ (the explicit equation whose projectivization is $X_0(11)$ ) has only finitely many rational solutions. Reading
Cites Emil · 2025-05 · Mathematics Stack Exchange · on “Given a 2-manifold $M$,the space obtained by "attaching a handle" to $M$ is homeomorphic to the connected sum $M\#\mathbb T^2$.”
Reminds me of this video where he needed to rephrase things back and forth between seemingly unrelated things a while until he found some other mathematical area where there existed a proof that solved his issue :-) [this video] I mean I understand the gray one better than the connected sum of a torus, how does he even connect the torus without reshaping it into the three holed blob with a cylinder between two holes? Does he add glue to the side of the torus or what?