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Explore Every Shape Hides a Square No Matter What. We Can't Prove It. with 97,315 views, 3,820 likes, and 332 comments. Experience the impact of this video content that has captured audience attention.

Video source code (manim): https://quantiayt.gumroad.com/ Draw any closed loop on a page. Any loop at all — smooth, jagged, fractal, a coastline, the outline of a leaf. Otto Toeplitz conjectured in 1911 that somewhere on that curve, four points always form a perfect square. Same side lengths. Right angles. Every time. It should be obvious. It isn't. One hundred and fifteen years later, it's still open. In this video we walk through the strangest unsolved problem in elementary geometry: - Arnold Emch's beautiful 1913 proof for *convex* curves — pure midpoints and continuity. - Why the jump from "there's a rectangle" to "there's a square" breaks every tool we have. - The Möbius strip argument that proves every smooth loop inscribes a rectangle — and how a non-orientable surface ends up controlling four-point configurations in the plane. - Schnirelmann 1929, Stromquist 1989, Meyerson's 1981 dense-set result, Tao's 2016 integration approach. - The May 2020 breakthrough by Joshua Greene and Andrew Lobb — lockdown math, symplectic geometry, Klein bottles embedded in Lagrangian ℝ⁴, and Shevchishin's theorem about non-orientable Lagrangians. - Why *every* rectangle aspect ratio is now known to be inscribable in any smooth Jordan curve. - And why the original problem — allowing the wildest fractals — is still wide, wide open. The hook isn't the conjecture. It's that after a century of the finest mathematicians in the world, we still cannot prove the one thing that should be easiest: the square. Chapters: 00:00 — Cold open 00:10 — The intro 00:58 — What is a Jordan curve 01:48 — Toeplitz's claim 02:39 — The convex case (Emch, 1913) 03:42 — The Möbius strip proof (rectangles) 05:43 — Why squares are so much harder 06:27 — A century of partial results 07:22 — Tao's integration approach (2016) 08:21 — Greene–Lobb, May 2020 10:12 — Koch snowflakes and still-open wilderness 11:07 — What is your loop hiding? References: - Otto Toeplitz, Jahresbericht DMV 20 (1911) — the original conjecture - Arnold Emch (1913, 1915, 1916) — convex and smooth cases - Lev Schnirelmann (1929, corrected 1944) — C² curves - Walter Stromquist (1989) — locally-monotone curves - Mark Meyerson (1981) — all but at most two points on any Jordan curve lie on some inscribed square - Terence Tao (2017), "An integration approach to the Toeplitz square peg problem" - Joshua Greene & Andrew Lobb (2020), "The rectangular peg problem" — Annals of Mathematics - Vaughan's Möbius strip argument (1977, unpublished; popularized later) Music, animations, narration: all original. Animations rendered in Manim Community Edition. #math #topology #inscribedsquare #toeplitz #mathvideo #mathanimation YouTube tags (comma-separated): inscribed square problem, toeplitz conjecture, square peg problem, jordan curve, manim math, derivia, math animation, topology, unsolved math problem, open problem, emch 1913, greene lobb, symplectic geometry, mobius strip proof, klein bottle, lagrangian, rectangular peg problem, terence tao, schnirelmann, stromquist, meyerson, fractal math, koch snowflake, 3blue1brown style, pure math, 115 years open

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