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Every Curve Hiding Inside a Circle.

The Math Flow

89,527 subscribers

105,147 görüntüleme • 3 ay önce •via X (Twitter)

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Cumulus T. Lump profil fotoğrafı

Cumulus T. Lump3 ay önce

Kind of astonishing that a circle could hide inside a circle.

The Math Flow profil fotoğrafı

The Math Flow3 ay önce

Mathematics is full of such wonders.

Frank Tariquet profil fotoğrafı

Frank Tariquet3 ay önce

Fourier enters the room: "More circles !"

Space Walker profil fotoğrafı

Space Walker3 ay önce

Infinity inside infinity

Rahul P Sharma profil fotoğrafı

Rahul P Sharma3 ay önce

Are these different shapes possible due to the different size of circle each time?

The Math Flow profil fotoğrafı

The Math Flow3 ay önce

No, i think you can draw that circle at any radius, and the traced curve would be identical.

DLLME- Construction Command Center profil fotoğrafı

DLLME- Construction Command Center3 ay önce

Awesome

Art Is Groovy profil fotoğrafı

Art Is Groovy3 ay önce

Need to break out my Spirograph

Jagdish profil fotoğrafı

Jagdish3 ay önce

Spirograph

CryptoPatrick profil fotoğrafı

CryptoPatrick3 ay önce

@grok can you provide the mathematical formula for each of these curves?

swflboater profil fotoğrafı

swflboater3 ay önce

Spirograph.

Analizo a fondo busca lo mejor para 🇲🇽 profil fotoğrafı

Analizo a fondo busca lo mejor para 🇲🇽3 ay önce

Beautiful work When math becomes beauty

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In 1830, Liouville proved the area under the most important curve in mathematics has no formula. Not because nobody was clever enough - because the formula was never there to be found. Then somebody squared the problem, lifted it into a second dimension, and a circle walked into the exponent entirely on its own. The area is the square root of π - the same √π that normalizes every probability distribution inside every LLM and every diffusion model on every GPU cluster today. The curve is e to the minus x squared - the bell curve De Moivre discovered in 1733 solving a coin-flip problem. One dimension and every integration method slides off it. Square the integral, rename one variable to y, and x² + y² appears in the exponent - Pythagoras hiding inside a function with no circles anywhere. Switch to polar coordinates and geometry hands you a free factor of r. The infinite hill collapses to π. That factor now sits inside every softmax normalization, every Gaussian noise layer in stable diffusion training, every loss function a transformer optimizes during backpropagation. 20 minutes. Bookmark & watch today. The proof that a circle was hiding inside a straight line the entire time.

In 1830, Liouville proved the area under the most important curve in mathematics has no formula. Not because nobody was clever enough - because the formula was never there to be found. Then somebody squared the problem, lifted it into a second dimension, and a circle walked into the exponent entirely on its own. The area is the square root of π - the same √π that normalizes every probability distribution inside every LLM and every diffusion model on every GPU cluster today. The curve is e to the minus x squared - the bell curve De Moivre discovered in 1733 solving a coin-flip problem. One dimension and every integration method slides off it. Square the integral, rename one variable to y, and x² + y² appears in the exponent - Pythagoras hiding inside a function with no circles anywhere. Switch to polar coordinates and geometry hands you a free factor of r. The infinite hill collapses to π. That factor now sits inside every softmax normalization, every Gaussian noise layer in stable diffusion training, every loss function a transformer optimizes during backpropagation. 20 minutes. Bookmark & watch today. The proof that a circle was hiding inside a straight line the entire time.

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jungkook is ROUND. the roundiest roundie. a bunch of circles. every part of him a circle. eyes? circle. nose? circle. cheeks? circle. mouth? circle. ears? circle. just a community of circles existing within jungkook. the main council of round circledom.

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The area under the bell curve has no formula. Mathematicians proved that in 1830 — not that nobody found it yet, that it literally cannot exist. Then someone squared the integral, switched to polar coordinates, and got √π in four minutes. The same curve LLMs assume for weight noise and gradient errors. The trick: squaring turns one line integral into a surface over a plane. Suddenly x²+y² appears in the exponent — that's a circle, hiding inside e^(-x²) the whole time. Polar coordinates hand you a free factor of r, the chain rule clicks, and the "impossible" integral collapses to π. Every GPU training run assumes this result thousands of times per second. 14 minutes. Bookmark & watch today.

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Keep the ball inside the circle 😅 A tough ask 🧐

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