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What if I told you any shape can be broken down into circles? This visualization demonstrates the stunning power of the Fourier Transform. By using a series of rotating vectors—known as epicycles—we can reconstruct complex, continuous paths from simple circular motions. Each circle represents a specific frequency, phase, and... show more

Mathematica

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26,160 次观看 • 7 个月前 •via X (Twitter)

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A closed outline shaped like the Greek letter pi can be rebuilt as the path of a point carried by a chain of rotating circles, each circle matching one frequency component through its radius and phase, so that successive circles refine the match to the original curve.

A closed outline shaped like the Greek letter pi can be rebuilt as the path of a point carried by a chain of rotating circles, each circle matching one frequency component through its radius and phase, so that successive circles refine the match to the original curve.

Matheorems

12,280 次观看 • 1 个月前

The Fourier series is a way to represent any periodic function as a sum of simple sine and cosine waves. In essence, it breaks down complex signals into a combination of basic trigonometric functions, making it a powerful tool in mathematics, physics, and engineering for analyzing waveforms, heat transfer, vibrations, and more. 📹 Credit : mathswithmuza

The Fourier series is a way to represent any periodic function as a sum of simple sine and cosine waves. In essence, it breaks down complex signals into a combination of basic trigonometric functions, making it a powerful tool in mathematics, physics, and engineering for analyzing waveforms, heat transfer, vibrations, and more. 📹 Credit : mathswithmuza

Physics In History

312,532 次观看 • 1 年前

Canada is the hardest country to draw with circles. Fourier epicycles can only trace one closed loop: a chain of spinning circles, each attached to the tip of the one before. Canada isn't one loop. It has the longest coastline of any country and tens of thousands of islands, including three of the ten largest on Earth (Baffin, Victoria and Ellesmere). So we cheated, mathematically. The mainland and Canada's 30 largest islands are stitched into a single continuous line. The pen travels out to each island and back along a thin bridge, so the whole country is drawn without ever lifting the pen. 🔁 N = 1: one circle per axis, just a wobbly loop 🔁 N = 25: the shape of Canada appears 🔁 N = 100: Hudson Bay and the Arctic islands come into focus 🔁 N = 500: 1,000 circles, within 0.17% of the real outline Italy and France needed about 100 circles per axis. Canada needed 500, because every island and inlet adds another twist the circles have to learn. At the end you'll see the full equation: x(t) and y(t) are each just a sum of sines and cosines. Add enough of them and they draw a country. Outline: Natural Earth / geo-countries dataset #canada #fourier #fourierseries #epicycles #math

Canada is the hardest country to draw with circles. Fourier epicycles can only trace one closed loop: a chain of spinning circles, each attached to the tip of the one before. Canada isn't one loop. It has the longest coastline of any country and tens of thousands of islands, including three of the ten largest on Earth (Baffin, Victoria and Ellesmere). So we cheated, mathematically. The mainland and Canada's 30 largest islands are stitched into a single continuous line. The pen travels out to each island and back along a thin bridge, so the whole country is drawn without ever lifting the pen. 🔁 N = 1: one circle per axis, just a wobbly loop 🔁 N = 25: the shape of Canada appears 🔁 N = 100: Hudson Bay and the Arctic islands come into focus 🔁 N = 500: 1,000 circles, within 0.17% of the real outline Italy and France needed about 100 circles per axis. Canada needed 500, because every island and inlet adds another twist the circles have to learn. At the end you'll see the full equation: x(t) and y(t) are each just a sum of sines and cosines. Add enough of them and they draw a country. Outline: Natural Earth / geo-countries dataset #canada #fourier #fourierseries #epicycles #math

Everything Math

425,300 次观看 • 28 天前

$The Lévy C curve is a fascinating fractal created by repeatedly replacing every straight line segment with two shorter segments that meet at a right angle. With each iteration, the curve becomes more intricate, folding over itself in a pattern that is surprisingly orderly despite its complexity. Even though the construction follows a very simple rule, the resulting shape has an infinitely detailed boundary and displays self-similarity, meaning that small sections resemble the whole curve when viewed at different scales. One of the most striking features of the Lévy C curve is how a simple line gradually transforms into what looks like a crumpled ribbon while still remaining a continuous path. It has applications in mathematics, computer graphics, and procedural art, demonstrating how complex structures can emerge from repeated, straightforward operations. The curve is a beautiful example of how recursion and iteration can produce patterns that are both mathematically rich and visually captivating. #math #art #fractal #fractalart #visual$

The Lévy C curve is a fascinating fractal created by repeatedly replacing every straight line segment with two shorter segments that meet at a right angle. With each iteration, the curve becomes more intricate, folding over itself in a pattern that is surprisingly orderly despite its complexity. Even though the construction follows a very simple rule, the resulting shape has an infinitely detailed boundary and displays self-similarity, meaning that small sections resemble the whole curve when viewed at different scales. One of the most striking features of the Lévy C curve is how a simple line gradually transforms into what looks like a crumpled ribbon while still remaining a continuous path. It has applications in mathematics, computer graphics, and procedural art, demonstrating how complex structures can emerge from repeated, straightforward operations. The curve is a beautiful example of how recursion and iteration can produce patterns that are both mathematically rich and visually captivating. #math #art #fractal #fractalart #visual

mathswithmuza

923,600 次观看 • 3 个月前

One of the odd things about learning mathematics is how long you can manipulate an equation without having much of a picture of it in your head. You learn the notation, follow the derivation, get the answer, and move on. Then somebody shows you the same idea visually and you realize there was a geometric intuition hiding underneath the symbols the whole time. A Fourier sum becomes rotating vectors added head-to-tail. Attention becomes a weighted combination you can watch being formed. SVD becomes a sequence of transformations you can follow one step at a time. Even the equation behind a diffusion process feels different when you can see a point following a changing vector field. None of this replaces the mathematics. The equations are still where the precision lives. But once you have a picture in your head, the notation has something to attach itself to. I think this is why some mathematical ideas suddenly become much easier years after we first learned them. We finally stop seeing the equation as a line of symbols to manipulate and start seeing the process it describes. I wish more mathematics had been taught that way.

One of the odd things about learning mathematics is how long you can manipulate an equation without having much of a picture of it in your head. You learn the notation, follow the derivation, get the answer, and move on. Then somebody shows you the same idea visually and you realize there was a geometric intuition hiding underneath the symbols the whole time. A Fourier sum becomes rotating vectors added head-to-tail. Attention becomes a weighted combination you can watch being formed. SVD becomes a sequence of transformations you can follow one step at a time. Even the equation behind a diffusion process feels different when you can see a point following a changing vector field. None of this replaces the mathematics. The equations are still where the precision lives. But once you have a picture in your head, the notation has something to attach itself to. I think this is why some mathematical ideas suddenly become much easier years after we first learned them. We finally stop seeing the equation as a line of symbols to manipulate and start seeing the process it describes. I wish more mathematics had been taught that way.

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57,649 次观看 • 1 个月前

The Google Gemini can transform your questions and complex topics into custom and interactive visualizations — directly within your chat. Whether you’re rotating a molecule or simulating a complex physics system, you create 3D simulations with just one prompt in Gemini. Try starting your prompt with “show me...,” to get your interactive visualization. For the best results, try using Gemini’s Flash model.

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Today I took Farokh and Sergito ❚ ❚ to the gallery we are building! Heres a video update. This space will truly be a hub for art and a high end experience for collecting art at all levels. Whats inside? 4000 square feets Café/cocktail bars Sculptures, painting, digital art Collection graded cars will be rotating immersive projection mapping rooms You can rent the immersive room at the back by burning Fvck_Crystals// and Fvck_Avatars// We already have a line up, if you are interested in curating that room drop a ⛓️ in the comments! See you at the end of the summer.

Today I took Farokh and Sergito ❚ ❚ to the gallery we are building! Heres a video update. This space will truly be a hub for art and a high end experience for collecting art at all levels. Whats inside? 4000 square feets Café/cocktail bars Sculptures, painting, digital art Collection graded cars will be rotating immersive projection mapping rooms You can rent the immersive room at the back by burning Fvck_Crystals// and Fvck_Avatars// We already have a line up, if you are interested in curating that room drop a ⛓️ in the comments! See you at the end of the summer.

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This is Chaos Theory in a nutshell. What looks like a simple circle is actually a complex engine of divergence. We are all just results of the first few bounces.

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Richard Feynman called Euler's formula "the most remarkable formula in mathematics." And there's a fascinating engineering reason why this equation is so important. e^(iθ) = cos(θ) + i·sin(θ) It connects exponential functions, imaginary numbers, trigonometry, and rotation through a single mathematical relationship. But its usefulness goes much further. Consider a sinusoidal signal: x(t) = A cos(ωt + φ) Using Euler's formula, we can represent it as the real part of a complex exponential: A e^(i(ωt + φ)) Why would engineers use this representation? Because differentiating e^(iωt) simply multiplies it by iω. Phase shifts can be represented through complex multiplication. For linear, time-invariant electrical circuits in sinusoidal steady state, this allows differential equations to be reduced to algebraic equations. Engineers use this mathematics in phasor analysis, AC circuit calculations, impedance, and frequency-response analysis. The same complex-exponential representation is fundamental to Fourier analysis, signal processing, communication systems, and the study of mechanical vibrations. Feynman also described the complex-plane representation of numbers as the unification of algebra and geometry. And that's what makes Euler's formula so remarkable. An equation connecting imaginary numbers and circular motion provides a practical way to analyze real electrical signals, oscillations, and physical systems.

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The subsidy question is a straightforward one. Except you are Peter Obi. If a person/politician cannot be trusted to take a stand on the simple questions, how can they be trusted with the complex ones or even the complicated world of governance itself?

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23,102 次观看 • 7 个月前

Smart crystals can now perform complex motions... Check out our most recent paper in J. Am. Chem. Soc., where we demonstrate that adaptive crystals with colossal thermal expansion are capable of complex locomotion, such as rolling and even climbing up a slope! New York University NYU Abu Dhabi

Smart crystals can now perform complex motions... Check out our most recent paper in J. Am. Chem. Soc., where we demonstrate that adaptive crystals with colossal thermal expansion are capable of complex locomotion, such as rolling and even climbing up a slope! New York University NYU Abu Dhabi

Panče Naumov, PhD ACSF FRSC

14,838 次观看 • 1 年前

WalkTheDog Cross-Morphology Motion Alignment via Phase Manifolds We present a new approach for understanding the periodicity structure and semantics of motion datasets, independently of the morphology and skeletal structure of characters. Unlike existing methods using an overly sparse high-dimensional latent, we propose a phase manifold consisting of multiple closed curves, each corresponding to a latent amplitude. With our proposed vector quantized periodic autoencoder, we learn a shared phase manifold for multiple characters, such as a human and a dog, without any supervision. This is achieved by exploiting the discrete structure and a shallow network as bottlenecks, such that semantically similar motions are clustered into the same curve of the manifold, and the motions within the same component are aligned temporally by the phase variable. In combination with an improved motion matching framework, we demonstrate the manifold's capability of timing and semantics alignment in several applications, including motion retrieval, transfer and stylization.

WalkTheDog Cross-Morphology Motion Alignment via Phase Manifolds We present a new approach for understanding the periodicity structure and semantics of motion datasets, independently of the morphology and skeletal structure of characters. Unlike existing methods using an overly sparse high-dimensional latent, we propose a phase manifold consisting of multiple closed curves, each corresponding to a latent amplitude. With our proposed vector quantized periodic autoencoder, we learn a shared phase manifold for multiple characters, such as a human and a dog, without any supervision. This is achieved by exploiting the discrete structure and a shallow network as bottlenecks, such that semantically similar motions are clustered into the same curve of the manifold, and the motions within the same component are aligned temporally by the phase variable. In combination with an improved motion matching framework, we demonstrate the manifold's capability of timing and semantics alignment in several applications, including motion retrieval, transfer and stylization.

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27,569 次观看 • 2 年前

When mathematics draws Homer Simpson Fourier series turned into epicycles From equations to animation #math #animation #geometry #circle #simpson

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