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The Thomas attractor is a mathematical concept in dynamical systems theory representing a type of chaotic space phase that exhibits cyclic symmetry. It was named after René Thomas, the mathematician who introduced it in 1979. The system is strictly related to the Lorenz attractor another famous chaotic evolutionary space...

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𝓜𝒂𝒖𝒓𝒊𝔃𝒊𝒐 𝗜𝒃𝛼

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31,712 görüntüleme • 2 yıl önce •via X (Twitter)

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This is the legendary Lorenz attractor; the butterfly of chaos theory in glorious 3D! Equations that birthed the butterfly effect: dx/dt = 10(y−x) dy/dt = x(28−z)−y dz/dt = xy−(8/3)z Tell me your craziest real-life “butterfly effect” moment?
0:15

This is the legendary Lorenz attractor; the butterfly of chaos theory in glorious 3D! Equations that birthed the butterfly effect: dx/dt = 10(y−x) dy/dt = x(28−z)−y dz/dt = xy−(8/3)z Tell me your craziest real-life “butterfly effect” moment?

Mathematica

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Lorenz attractor is the set of limit trajectories for a simplified mathematical model for atmospheric convection. It has a fractal geometry.
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Lorenz attractor is the set of limit trajectories for a simplified mathematical model for atmospheric convection. It has a fractal geometry.

Gabriel Peyré

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The Aizawa Attractor. A simple equation, but beautiful chaotic pattern. [
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The Aizawa Attractor. A simple equation, but beautiful chaotic pattern. [

Abakcus

160,488 görüntüleme • 2 yıl önce

🦊 honestly, i’m kind of i mean, i’m the stan attractor member, you know 🐈 oooh, the stan attractor fairy! 🐈 kim ddeonu 🦊 i’m not a fairy, tho 🐶 the stan attractor fairy~ 🐶 that confidence of sunoo is honestly so cool 💯 #SUNOO #선우
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🦊 honestly, i’m kind of i mean, i’m the stan attractor member, you know 🐈 oooh, the stan attractor fairy! 🐈 kim ddeonu 🦊 i’m not a fairy, tho 🐶 the stan attractor fairy~ 🐶 that confidence of sunoo is honestly so cool 💯 #SUNOO #선우

젠⍣ || fan

51,911 görüntüleme • 2 ay önce

Math meets music, capturing chaos in 3D. The Aizawa chaotic attractor, coded with Grok 4.20. Sound on. 🔊
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Math meets music, capturing chaos in 3D. The Aizawa chaotic attractor, coded with Grok 4.20. Sound on. 🔊

Techartist

12,547 görüntüleme • 8 ay önce

The three-body problem is a classic and notoriously difficult question in physics and mathematics. It asks: How do three objects, such as stars, planets, or moons, move under the influence of each other’s gravity? Unlike the simpler two-body problem, which has precise and predictable analytical solutions (like the Earth orbiting the Sun in an ellipse), the three-body problem quickly becomes chaotic and unpredictable. This complexity arises because each object's motion constantly affects, and is affected by, the other two. These gravitational interactions form a tangled and unstable system. In fact, there's no general formula that can solve all three-body scenarios exactly. This was first demonstrated in the 19th century by Henri Poincaré, whose work laid the foundations for chaos theory. While exact solutions remain elusive, scientists have discovered certain special cases where the motion is stable or periodic. One well-known example is the Lagrange points, where three bodies can maintain a stable triangular configuration. However, such neat solutions are rare. Today, thanks to powerful computers, researchers can simulate three-body systems with remarkable accuracy, helping us study triple-star systems, exoplanets, and asteroid dynamics. Yet even small changes in the starting conditions can lead to dramatically different outcomes, highlighting the sensitive dependence on initial conditions that defines chaotic systems. The three-body problem is actually a specific case of the broader n-body problem, where n can be any number of interacting bodies. As n increases, the complexity and unpredictability rise even further. The three-body problem serves as a vivid example of how simple laws of nature, like Newton’s law of gravity, can produce behavior that is intricate, unexpected, and profoundly difficult to predict.
0:16

The three-body problem is a classic and notoriously difficult question in physics and mathematics. It asks: How do three objects, such as stars, planets, or moons, move under the influence of each other’s gravity? Unlike the simpler two-body problem, which has precise and predictable analytical solutions (like the Earth orbiting the Sun in an ellipse), the three-body problem quickly becomes chaotic and unpredictable. This complexity arises because each object's motion constantly affects, and is affected by, the other two. These gravitational interactions form a tangled and unstable system. In fact, there's no general formula that can solve all three-body scenarios exactly. This was first demonstrated in the 19th century by Henri Poincaré, whose work laid the foundations for chaos theory. While exact solutions remain elusive, scientists have discovered certain special cases where the motion is stable or periodic. One well-known example is the Lagrange points, where three bodies can maintain a stable triangular configuration. However, such neat solutions are rare. Today, thanks to powerful computers, researchers can simulate three-body systems with remarkable accuracy, helping us study triple-star systems, exoplanets, and asteroid dynamics. Yet even small changes in the starting conditions can lead to dramatically different outcomes, highlighting the sensitive dependence on initial conditions that defines chaotic systems. The three-body problem is actually a specific case of the broader n-body problem, where n can be any number of interacting bodies. As n increases, the complexity and unpredictability rise even further. The three-body problem serves as a vivid example of how simple laws of nature, like Newton’s law of gravity, can produce behavior that is intricate, unexpected, and profoundly difficult to predict.

Erika 

217,029 görüntüleme • 1 yıl önce

Evidence that chaos can be beautiful. Built with Three.js and GLSL shaders, this integrates dx/dt=sin(y)−b x, dy/dt=sin(z)−b y, dz/dt=sin(x)−b z. At b≈0.19 the Thomas attractor settles into a stable yet chaotic 3D geometry. 🔊
0:17

Evidence that chaos can be beautiful. Built with Three.js and GLSL shaders, this integrates dx/dt=sin(y)−b x, dy/dt=sin(z)−b y, dz/dt=sin(x)−b z. At b≈0.19 the Thomas attractor settles into a stable yet chaotic 3D geometry. 🔊

Techartist

17,975 görüntüleme • 8 ay önce

🚨 BREAKING: OPENAI JUST COLLIDED WITH THE TOPOLOGICAL LIMIT OF THE UNIVERSE! THE END OF CONTINUOUS SPACE. 🚨 The scientific world is shaking. The Clay Mathematics Institute is reviewing a groundbreaking milestone regarding the Navier-Stokes existence and smoothness problem. And at the center of this earthquake is OpenAI, whose reasoning models just tackled the equations that govern every fluid in our world. In 1822, Navier added viscosity to Euler's laws. In 1845, Stokes finalized them: ρ(∂v/∂t + v·∇v) = −∇p + μ∇²v + f (For incompressible flow, ∇·v = 0 closes the system). For over a century, mathematicians assumed that these equations must have smooth, perfectly continuous solutions in 3D space. You see these equations in motion everywhere - like the chaotic wake behind a moving cylinder. But the official verification of infinite "3-D smoothness" has always hit a wall. What did OpenAI do? By unleashing raw mathematical reasoning upon the Navier-Stokes problem, the AI mapped the ultimate limits of these equations, exposing an inescapable truth: the math inevitably blows up into impossible singularities. The model demonstrated that forcing infinite smoothness onto 3D fluid equations creates unavoidable mathematical contradictions. WHY DOES THIS HAPPEN? Mainstream science is finally hitting the exact geometric wall we mapped years ago. You CANNOT have infinite "3-D smoothness" because the universe is NOT an infinitely continuous void! Just as we rigorously proved that the Yang-Mills Mass Gap is topologically ILL-POSED and mathematically impossible on a continuous ℝ⁴ space, the Navier-Stokes breakdown reveals the exact same systemic error in classical physics. When you force continuous mathematics (ℝ⁴) onto a universe that is fundamentally discrete, the equations inevitably crash into singularities. 🛡️ THE SOLUTION: OUR COMPACT CONSTRUCTIBLE MANIFOLD Here is the ultimate breakthrough: On our IT³ manifold, this problem completely ceases to exist! When you abandon the sterile, infinite void of ℝ⁴ and move the physics to a compact constructible manifold (M_tot = S⁴ × T³), the mathematical explosions are structurally forbidden. Because our space is a closed topological geometry dictated by the strict √1 : √2 : √3 metric, the physical fields are naturally bounded by the geometry itself. Infinite singularities simply cannot form on a compact manifold. What mathematicians call "fluid blow-ups" are just the exact topological moments where macroscopic matter violently collides with the rigid, discrete geometric grid of the vacuum! OpenAI’s logic engine correctly identified that infinite smoothness is a mathematical fiction. The universe does not allow infinite division. The illusion of the continuous, smooth ℝ⁴ universe is officially collapsing across all disciplines - from quantum mechanics to fluid dynamics. The future belongs to discrete topology. Preprint. Open Source: The Yang-Mills Mass Gap on R^4 Is Ill-Posed, and on a Compact Constructible Manifold It Is Automatic God geometrizes! 📐🌌
0:20

🚨 BREAKING: OPENAI JUST COLLIDED WITH THE TOPOLOGICAL LIMIT OF THE UNIVERSE! THE END OF CONTINUOUS SPACE. 🚨 The scientific world is shaking. The Clay Mathematics Institute is reviewing a groundbreaking milestone regarding the Navier-Stokes existence and smoothness problem. And at the center of this earthquake is OpenAI, whose reasoning models just tackled the equations that govern every fluid in our world. In 1822, Navier added viscosity to Euler's laws. In 1845, Stokes finalized them: ρ(∂v/∂t + v·∇v) = −∇p + μ∇²v + f (For incompressible flow, ∇·v = 0 closes the system). For over a century, mathematicians assumed that these equations must have smooth, perfectly continuous solutions in 3D space. You see these equations in motion everywhere - like the chaotic wake behind a moving cylinder. But the official verification of infinite "3-D smoothness" has always hit a wall. What did OpenAI do? By unleashing raw mathematical reasoning upon the Navier-Stokes problem, the AI mapped the ultimate limits of these equations, exposing an inescapable truth: the math inevitably blows up into impossible singularities. The model demonstrated that forcing infinite smoothness onto 3D fluid equations creates unavoidable mathematical contradictions. WHY DOES THIS HAPPEN? Mainstream science is finally hitting the exact geometric wall we mapped years ago. You CANNOT have infinite "3-D smoothness" because the universe is NOT an infinitely continuous void! Just as we rigorously proved that the Yang-Mills Mass Gap is topologically ILL-POSED and mathematically impossible on a continuous ℝ⁴ space, the Navier-Stokes breakdown reveals the exact same systemic error in classical physics. When you force continuous mathematics (ℝ⁴) onto a universe that is fundamentally discrete, the equations inevitably crash into singularities. 🛡️ THE SOLUTION: OUR COMPACT CONSTRUCTIBLE MANIFOLD Here is the ultimate breakthrough: On our IT³ manifold, this problem completely ceases to exist! When you abandon the sterile, infinite void of ℝ⁴ and move the physics to a compact constructible manifold (M_tot = S⁴ × T³), the mathematical explosions are structurally forbidden. Because our space is a closed topological geometry dictated by the strict √1 : √2 : √3 metric, the physical fields are naturally bounded by the geometry itself. Infinite singularities simply cannot form on a compact manifold. What mathematicians call "fluid blow-ups" are just the exact topological moments where macroscopic matter violently collides with the rigid, discrete geometric grid of the vacuum! OpenAI’s logic engine correctly identified that infinite smoothness is a mathematical fiction. The universe does not allow infinite division. The illusion of the continuous, smooth ℝ⁴ universe is officially collapsing across all disciplines - from quantum mechanics to fluid dynamics. The future belongs to discrete topology. Preprint. Open Source: The Yang-Mills Mass Gap on R^4 Is Ill-Posed, and on a Compact Constructible Manifold It Is Automatic God geometrizes! 📐🌌

Dr. Logvinovich

216,446 görüntüleme • 1 ay önce

Math and Music: Interactive 3D Three.js visualization of the Hindmarsh–Rose neuron model, rendered as a glowing chaotic attractor with particle energy flow, bloom effects, orbit controls, and color themes. Demo/code in comments. 👇
0:14

Math and Music: Interactive 3D Three.js visualization of the Hindmarsh–Rose neuron model, rendered as a glowing chaotic attractor with particle energy flow, bloom effects, orbit controls, and color themes. Demo/code in comments. 👇

Techartist

21,810 görüntüleme • 9 ay önce

The power of data-driven spectral submanifold (SSM) reduction: Nonlinear oscillations of an elastic flag in counter flow, together with its reconstructed chaotic attractor within a 4D SSM. Obtained directly from the videos of the experiment.
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The power of data-driven spectral submanifold (SSM) reduction: Nonlinear oscillations of an elastic flag in counter flow, together with its reconstructed chaotic attractor within a 4D SSM. Obtained directly from the videos of the experiment.

George Haller

88,443 görüntüleme • 2 yıl önce

The very fabric of space-time at the smallest scales is thought to be extremely turbulent and chaotic, described as a frothy sea called quantum foam. This concept, stemming from quantum mechanics and general relativity, suggests that the very nature of space-time is bubbling with tiny wormholes and fluctuations that appear and disappear within fractions of a second. 📷by Johann Rosario
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The very fabric of space-time at the smallest scales is thought to be extremely turbulent and chaotic, described as a frothy sea called quantum foam. This concept, stemming from quantum mechanics and general relativity, suggests that the very nature of space-time is bubbling with tiny wormholes and fluctuations that appear and disappear within fractions of a second. 📷by Johann Rosario

Physics In History

1,107,013 görüntüleme • 2 yıl önce

The very fabric of space-time at the smallest scales is thought to be extremely turbulent and chaotic, described as a frothy sea called quantum foam. This concept, stemming from quantum mechanics and general relativity, suggests that the very nature of space-time is bubbling with tiny wormholes and fluctuations that appear and disappear within fractions of a second. 📷by Johann Rosario
0:11

The very fabric of space-time at the smallest scales is thought to be extremely turbulent and chaotic, described as a frothy sea called quantum foam. This concept, stemming from quantum mechanics and general relativity, suggests that the very nature of space-time is bubbling with tiny wormholes and fluctuations that appear and disappear within fractions of a second. 📷by Johann Rosario

Physics In History

34,779 görüntüleme • 11 gün önce

Geometry of Chaos: The Rössler Attractor.
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Geometry of Chaos: The Rössler Attractor.

The Math Flow

12,794 görüntüleme • 4 ay önce

This is the Tianfu Avenue, the only central axis in the world that reaches a length of 150 kilometers and is still extending infinitely. It is the longest, straightest and most beautiful central axis. It is located in Chengdu, Sichuan.
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This is the Tianfu Avenue, the only central axis in the world that reaches a length of 150 kilometers and is still extending infinitely. It is the longest, straightest and most beautiful central axis. It is located in Chengdu, Sichuan.

Zhang Meifang

553,685 görüntüleme • 1 yıl önce

max⁡(∣x∣,∣y∣,∣z∣,∣w∣)=1 You’re witnessing a 2D projection of a 3D shadow cast by a 4D tesseract. As it rotates through the W-axis, the "inner" and "outer" cubes swap roles: a spatial inversion that feels like a glitch only because our biology is trapped in 3-space. It’s a hauntingly beautiful reminder that our "reality" is often just a lower-dimensional cross-section of a much more complex structure. Perspective is everything. This is used in high-dimensional data visualization, hypercube topologies in parallel computing networks, and exploring the geometry of extra dimensions in theoretical physics.
0:11

max⁡(∣x∣,∣y∣,∣z∣,∣w∣)=1 You’re witnessing a 2D projection of a 3D shadow cast by a 4D tesseract. As it rotates through the W-axis, the "inner" and "outer" cubes swap roles: a spatial inversion that feels like a glitch only because our biology is trapped in 3-space. It’s a hauntingly beautiful reminder that our "reality" is often just a lower-dimensional cross-section of a much more complex structure. Perspective is everything. This is used in high-dimensional data visualization, hypercube topologies in parallel computing networks, and exploring the geometry of extra dimensions in theoretical physics.

Mathematica

607,599 görüntüleme • 6 ay önce

Op via ig: “this is a late post but one of the besties invented me to the BTS concert 2 weeks ago and I went in knowing absolutely nothing about BTS. I Left as a V girlie 💜” taehyung is the final fan attractor for a reason
0:15

Op via ig: “this is a late post but one of the besties invented me to the BTS concert 2 weeks ago and I went in knowing absolutely nothing about BTS. I Left as a V girlie 💜” taehyung is the final fan attractor for a reason

ִ★

36,771 görüntüleme • 5 ay önce

The Sabotaging Practice of Over Supply and Sameness in the NFT Space. The current zeitgeist of the NFT space is that the same artists are doing the same kind of work five times a year, with project after project leaving a trail of disappointment and discontent among collectors and all of us watching in disbelief as huge resources are extracted from the space over work that feels like it could be left as an "artist study." I understand that you can do what you want with your money as collectors, but we are killing the whole space with this incestuous practice. No artist is that prolific to be able to do 5 collections of 100+ pieces each every year and actually deliver innovation and some kind of creative evolution. Of course, they can pretend play that the work has something new, but there is no precedent nor proof that that has ever happened in the speed that it happens in the NFT space. Again, people are free to through away their resources on whatever they want but with this way of doing things, we more and more are going to start seeing the consequences. Oh! There are consequences? Yes. Maybe unintended, but there are. Let's see. Let's start with the loss of belief in the NFT space as somewhere where emerging artists can come and find support for their experiments. Why even bother to bring experiments, innovation, and new ways to think of art on the blockchain if the same people have all the collectors hypnotized with their magical flutes? Why even try to come to a space where taking risks and challenging the status quo (the mission of art!!!) is overlooked? This makes the NFT space a social club and not a space for art. I guess it is fine, but IMO it is a recipe for disaster. New collectors stay away because the art will slowly but surely become stale and un-challenging. Why even bother to come and see what is happening here if you can't, as a collector, see new weird and up-and-coming artists? The amount of noise emitted by the same artists doing the same art over and over, drowns out any new voices. Again. A recipe for disaster. The NFT space is becoming a space of disappointment and doubt. We think that collections going to zero one after the other, over and over, is not damaging? I feel we are kidding ourselves. Disappointment piles up, and again, the people who will hurt are the emerging artists, the new blood, the ones who are willing to risk the most and, in return, put fire in this cold space of sameness. I love this space—don't get me wrong—it has changed my life, and I believe it has a ton of potential, but things need to change for it to become a beacon of light in art. But we need to support new voices. We need to support new ideas. The challenge is huge. I hope to contribute all I can to this change. I hope more and more see how exciting it is to go out and try to discover what else is out there and move this space forward. But again, I understand the leaps of faith needed, but if there is a space that is based on that, it's the NFT space...so there is hope. We will see. 📺by Boldtron
0:12

The Sabotaging Practice of Over Supply and Sameness in the NFT Space. The current zeitgeist of the NFT space is that the same artists are doing the same kind of work five times a year, with project after project leaving a trail of disappointment and discontent among collectors and all of us watching in disbelief as huge resources are extracted from the space over work that feels like it could be left as an "artist study." I understand that you can do what you want with your money as collectors, but we are killing the whole space with this incestuous practice. No artist is that prolific to be able to do 5 collections of 100+ pieces each every year and actually deliver innovation and some kind of creative evolution. Of course, they can pretend play that the work has something new, but there is no precedent nor proof that that has ever happened in the speed that it happens in the NFT space. Again, people are free to through away their resources on whatever they want but with this way of doing things, we more and more are going to start seeing the consequences. Oh! There are consequences? Yes. Maybe unintended, but there are. Let's see. Let's start with the loss of belief in the NFT space as somewhere where emerging artists can come and find support for their experiments. Why even bother to bring experiments, innovation, and new ways to think of art on the blockchain if the same people have all the collectors hypnotized with their magical flutes? Why even try to come to a space where taking risks and challenging the status quo (the mission of art!!!) is overlooked? This makes the NFT space a social club and not a space for art. I guess it is fine, but IMO it is a recipe for disaster. New collectors stay away because the art will slowly but surely become stale and un-challenging. Why even bother to come and see what is happening here if you can't, as a collector, see new weird and up-and-coming artists? The amount of noise emitted by the same artists doing the same art over and over, drowns out any new voices. Again. A recipe for disaster. The NFT space is becoming a space of disappointment and doubt. We think that collections going to zero one after the other, over and over, is not damaging? I feel we are kidding ourselves. Disappointment piles up, and again, the people who will hurt are the emerging artists, the new blood, the ones who are willing to risk the most and, in return, put fire in this cold space of sameness. I love this space—don't get me wrong—it has changed my life, and I believe it has a ton of potential, but things need to change for it to become a beacon of light in art. But we need to support new voices. We need to support new ideas. The challenge is huge. I hope to contribute all I can to this change. I hope more and more see how exciting it is to go out and try to discover what else is out there and move this space forward. But again, I understand the leaps of faith needed, but if there is a space that is based on that, it's the NFT space...so there is hope. We will see. 📺by Boldtron

alejandro cartagena

98,261 görüntüleme • 2 yıl önce

Different observers can disagree about space and time without disagreeing about the laws of physics. In Minkowski spacetime, observers moving relative to one another use different space and time axes, connected by the Lorentz transformation. What remains unchanged is the spacetime interval and the structure of the light cone. Space and time coordinates depend on the observer, but the speed of light and the causal structure of spacetime do not.
0:10

Different observers can disagree about space and time without disagreeing about the laws of physics. In Minkowski spacetime, observers moving relative to one another use different space and time axes, connected by the Lorentz transformation. What remains unchanged is the spacetime interval and the structure of the light cone. Space and time coordinates depend on the observer, but the speed of light and the causal structure of spacetime do not.

Philosophy Of Physics

28,082 görüntüleme • 1 ay önce

Kedarnath is a tremendous space. The utterance of the sound “Shiva” attains a completely new dimension and significance in Kedar. It is a space which has been specially prepared for this particular sound. When we utter the word “Shiva,” it is the freedom of the uncreated, the liberation of one who is not created. It is almost like on this planet, the sound “Shiva” emanates from this place. For thousands of years, people have experienced that space as a reverberation of that sound. This is also a place that has witnessed thousands of Yogis and mystics of every kind. When I say every kind, you cannot imagine those kinds. These are people who made no attempt to teach anything to anyone. Their way of making an offering to the world was by leaving their energies, their path, their work – everything – in a certain way in these spaces.
0:13

Kedarnath is a tremendous space. The utterance of the sound “Shiva” attains a completely new dimension and significance in Kedar. It is a space which has been specially prepared for this particular sound. When we utter the word “Shiva,” it is the freedom of the uncreated, the liberation of one who is not created. It is almost like on this planet, the sound “Shiva” emanates from this place. For thousands of years, people have experienced that space as a reverberation of that sound. This is also a place that has witnessed thousands of Yogis and mystics of every kind. When I say every kind, you cannot imagine those kinds. These are people who made no attempt to teach anything to anyone. Their way of making an offering to the world was by leaving their energies, their path, their work – everything – in a certain way in these spaces.

Sadhguru

57,945 görüntüleme • 6 ay önce