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A Blaschke-Quotient Flow on the Poincaré Disk Inside the unit disk, zeros and poles perform a Möbius ballet, twisting under SU(1,1) symmetries. Each point follows the gradient of an inner function, its path tracing folds of velvety hyperbolic fabric. Phase interference becomes light bending like thought itself. 🤯🤯🤩🤩🤗🫠

Mathelirium

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26,418 次观看 • 10 个月前 •via X (Twitter)

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Mathematics is beautiful🤩❤️ My favourite from 2025. Merry Christmas guys! 🎄 Zeros and poles of a Blaschke-Quotient spinning under SU(1,1) Möbius symmetries on the Poincaré disk with the phase shaded by the Fubini-Study metric.

Mathematics is beautiful🤩❤️ My favourite from 2025. Merry Christmas guys! 🎄 Zeros and poles of a Blaschke-Quotient spinning under SU(1,1) Möbius symmetries on the Poincaré disk with the phase shaded by the Fubini-Study metric.

Mathelirium

20,051 次观看 • 7 个月前

Much better visually. We now visualize the 3D version of our stunning Blaschke-Quotient Flow from our previous post.🤗🤩 What follows is a mesmerizing break of dawn in the some realm of complex space with zeros and poles still spinning under SU(1,1) Möbius symmetries on the Poincaré disk…but phase is now shaded by the Fubini-Study metric. 🤯🤯🤯

Much better visually. We now visualize the 3D version of our stunning Blaschke-Quotient Flow from our previous post.🤗🤩 What follows is a mesmerizing break of dawn in the some realm of complex space with zeros and poles still spinning under SU(1,1) Möbius symmetries on the Poincaré disk…but phase is now shaded by the Fubini-Study metric. 🤯🤯🤯

Mathelirium

15,826 次观看 • 10 个月前

Hyperbolic waves collide on the Poincaré disk. Bright blooms are constructive peaks, dark bands are cancellations. White tracers ride the phase-gradient, threading filaments. SU(1,1)...the set of Möbius moves that preserve the disk’s geometry...steers the beacons. Multiple tempos modulate the spatial frequency.

Hyperbolic waves collide on the Poincaré disk. Bright blooms are constructive peaks, dark bands are cancellations. White tracers ride the phase-gradient, threading filaments. SU(1,1)...the set of Möbius moves that preserve the disk’s geometry...steers the beacons. Multiple tempos modulate the spatial frequency.

Mathelirium

30,034 次观看 • 11 个月前

You're watching a Quadratic-Differential Teichmüller flow unfold. We drive the scene with a time-varying quadratic differential built from moving zeros and poles. Its log-magnitude sets the color, and its phase defines the horizontal foliation (the direction is half the phase). Then a Teichmüller stretch biases things along vs. against that foliation...so the surface deforms in a geodesic-like way. 🤩🤗🫠

You're watching a Quadratic-Differential Teichmüller flow unfold. We drive the scene with a time-varying quadratic differential built from moving zeros and poles. Its log-magnitude sets the color, and its phase defines the horizontal foliation (the direction is half the phase). Then a Teichmüller stretch biases things along vs. against that foliation...so the surface deforms in a geodesic-like way. 🤩🤗🫠

Mathelirium

11,938 次观看 • 9 个月前

This is a living map of a complex rational field: cool regions are root-dominant (|R| 1), where R is the complex rational function. Phase stripes come from arg S (the logarithmic derivative of our complex field R) and particles surf the Newton flow. 🤩🤗😍

This is a living map of a complex rational field: cool regions are root-dominant (|R| 1), where R is the complex rational function. Phase stripes come from arg S (the logarithmic derivative of our complex field R) and particles surf the Newton flow. 🤩🤗😍

Mathelirium

29,237 次观看 • 10 个月前

Lanterns on the Logarithmic Sea Zeros and poles of a complex rational function R(z,t) drift across a 3D landscape carved from the logarithmic potential U(z,t)=log|R(z,t)|. Every ridge, valley, and glowing thread comes from the moving R(z,t), while the particles follow the tangent field of its level curves. The golden lantern ovals mark the singularities directly. So, the whole scene becomes a living map of where the function vanishes, where it blows up, and how the surrounding geometry bends in response.

Lanterns on the Logarithmic Sea Zeros and poles of a complex rational function R(z,t) drift across a 3D landscape carved from the logarithmic potential U(z,t)=log|R(z,t)|. Every ridge, valley, and glowing thread comes from the moving R(z,t), while the particles follow the tangent field of its level curves. The golden lantern ovals mark the singularities directly. So, the whole scene becomes a living map of where the function vanishes, where it blows up, and how the surrounding geometry bends in response.

Mathelirium

14,941 次观看 • 1 个月前

A physical system, its phase-space state, and the energy function that governs the flow, all linked together. The point of the animation is that these are not three different objects. They are three views of the same dynamics. The pendulum on the left is the physical motion. The curve on the floor is the evolution of its state. The translucent surface shows the energy landscape that organizes that motion.

A physical system, its phase-space state, and the energy function that governs the flow, all linked together. The point of the animation is that these are not three different objects. They are three views of the same dynamics. The pendulum on the left is the physical motion. The curve on the floor is the evolution of its state. The translucent surface shows the energy landscape that organizes that motion.

Mathelirium

19,865 次观看 • 3 个月前

The Torus Becomes the Canvas A Jacobi Theta Function lives naturally on a complex torus. Instead of drawing it on a flat plane, we wrapped the mathematics onto the torus. The surface is driven by θ₁(z|τ), with its moving zeros, phase winding, and logarithmic derivative shaping the colour, seams, and raised divisor points across the geometry. What you are see is a periodic quantum-like field painted onto the space where it actually belongs. #JacobiTheta #ComplexTorus #MathematicalArt #ComplexAnalysis #RiemannSurfaces #MathAnimation

The Torus Becomes the Canvas A Jacobi Theta Function lives naturally on a complex torus. Instead of drawing it on a flat plane, we wrapped the mathematics onto the torus. The surface is driven by θ₁(z|τ), with its moving zeros, phase winding, and logarithmic derivative shaping the colour, seams, and raised divisor points across the geometry. What you are see is a periodic quantum-like field painted onto the space where it actually belongs. #JacobiTheta #ComplexTorus #MathematicalArt #ComplexAnalysis #RiemannSurfaces #MathAnimation

Mathelirium

22,197 次观看 • 2 个月前

We now visualize the complex phase of our earlier self-gravitating quantum fluid twisting into smooth helicoids. Think of phase as the wave's "angle" at each point...mapping it to color lets you see how wavefronts curl and shear. Brightness tracks density. The faint moving bands are phase ribbons...contours of equal phase sliding as the field evolves. The tiny white dots are vortices: spots where density dips and the phase wraps by one full turn (2pi). Same Schrödinger-Poisson physics, revealed in phase...live!🫠❤️🫠

We now visualize the complex phase of our earlier self-gravitating quantum fluid twisting into smooth helicoids. Think of phase as the wave's "angle" at each point...mapping it to color lets you see how wavefronts curl and shear. Brightness tracks density. The faint moving bands are phase ribbons...contours of equal phase sliding as the field evolves. The tiny white dots are vortices: spots where density dips and the phase wraps by one full turn (2pi). Same Schrödinger-Poisson physics, revealed in phase...live!🫠❤️🫠

Mathelirium

14,200 次观看 • 11 个月前

In 1998, Japanese Mathematician, Engineer and Machine Learning pioneer Shun-ichi Amari published "Natural Gradient Works Efficiently in Learning (722 Cites in Papers and 9 Cites in Patents)" and made a point that still feels fresh even today If your loss is L(θ), then the usual update θ̇ = −∇ L(θ) quietly assumes your parameter space is flat. Amari asked what happens when the model itself has geometry, described by G(θ), the Fisher information. Then the more natural direction becomes θ̇ = −G(θ)⁻¹∇ L(θ). This is the same loss, but with a different notion of distance. As the animation shows, ordinary Gradient Descent follows the raw slope and gets dragged around by distortion, while the natural gradient moves in a way that respects the geometry of the statistical model itself.

In 1998, Japanese Mathematician, Engineer and Machine Learning pioneer Shun-ichi Amari published "Natural Gradient Works Efficiently in Learning (722 Cites in Papers and 9 Cites in Patents)" and made a point that still feels fresh even today If your loss is L(θ), then the usual update θ̇ = −∇ L(θ) quietly assumes your parameter space is flat. Amari asked what happens when the model itself has geometry, described by G(θ), the Fisher information. Then the more natural direction becomes θ̇ = −G(θ)⁻¹∇ L(θ). This is the same loss, but with a different notion of distance. As the animation shows, ordinary Gradient Descent follows the raw slope and gets dragged around by distortion, while the natural gradient moves in a way that respects the geometry of the statistical model itself.

Mathelirium

45,802 次观看 • 4 个月前

When Light Becomes Heavy Enough to Trap Itself Can light collapse into a black hole? In General Relativity, light is not weightless in the geometric sense. It has energy, it carries momentum, and energy curves Spacetime. A Vaidya-type collapse describes a mass that grows because radiation is falling inward. The mathematics is the same brutal idea behind Einstein’s equation Gᵤᵥ = 8πTᵤᵥ If the stress-energy Tᵤᵥ is made from radiation, then light itself becomes a source of curvature. As more radiation falls inward, the mass aspect grows, the Spacetime fabric tightens, and a clean black horizon appears where escaping paths begin to fail. #GeneralRelativity #EinsteinEquations #BlackHole #Spacetime #Astrophysics #TheoreticalPhysics #Light #Mathematics #Science

When Light Becomes Heavy Enough to Trap Itself Can light collapse into a black hole? In General Relativity, light is not weightless in the geometric sense. It has energy, it carries momentum, and energy curves Spacetime. A Vaidya-type collapse describes a mass that grows because radiation is falling inward. The mathematics is the same brutal idea behind Einstein’s equation Gᵤᵥ = 8πTᵤᵥ If the stress-energy Tᵤᵥ is made from radiation, then light itself becomes a source of curvature. As more radiation falls inward, the mass aspect grows, the Spacetime fabric tightens, and a clean black horizon appears where escaping paths begin to fail. #GeneralRelativity #EinsteinEquations #BlackHole #Spacetime #Astrophysics #TheoreticalPhysics #Light #Mathematics #Science

Mathelirium

27,635 次观看 • 1 个月前

Speech Carries Its Own Hidden Geometry in Phase Space Speech is usually shown as a waveform of amplitude changing in time. Phase-space reconstruction gives a different view by pairing the signal with a delayed version of itself, so the voice is seen as a path rather than just a trace. On the left is the raw sound signal. On the right is the geometric structure formed by rhythm, syllables, and emphasis as the voice unfolds. * Real human speech data from the public GitHub repository voxserv/audio_quality_testing_samples, using the 16 kHz file test01_20s.wav.

Speech Carries Its Own Hidden Geometry in Phase Space Speech is usually shown as a waveform of amplitude changing in time. Phase-space reconstruction gives a different view by pairing the signal with a delayed version of itself, so the voice is seen as a path rather than just a trace. On the left is the raw sound signal. On the right is the geometric structure formed by rhythm, syllables, and emphasis as the voice unfolds. * Real human speech data from the public GitHub repository voxserv/audio_quality_testing_samples, using the 16 kHz file test01_20s.wav.

Mathelirium

18,725 次观看 • 4 个月前

This isn't VFX visual effects...it's actual freakin physics! 🤯🤩🤗🙌🏾 You're watching the physics of space and matter coevolve...a small window into the kind of world Wolfram Physics Wolfram predicts. Every ripple, collision, and shimmer you see is a causal event in a living hypergraph. Space itself is an active network of nodes and links...springs and diagonals...constantly stretching, relaxing, and rewriting as the system evolves. Each interaction you see...a nutrient diffusing through the medium, a burrower tunneling, a tentacle feeling drag...is a computation of spacetime itself. Every connection that forms or breaks is an update to the causal structure: a new link, a new moment in the unfolding hypergraph universe. Our environment behaves like an adaptive fabric...the lattice tightens where organisms stir it, loosens where they pass. Chemical fields spread through those same links, feeding back into motion and growth. When the cluster pulsates, it's literally reorganizing local causal geometry...matter and space changing together!👌🏾 We even test causal invariance frame by frame, swapping operation orders to ensure that spacetime remains consistent no matter the update path. That's not animation logic...that's a computational physics experiment running live!🥳 Tentacles feel friction, burrowers dig through gradients, and behind them the lattice stiffens, storing history as structure. #Mathematics #Physics #WolframPhysics #Mathelirium

This isn't VFX visual effects...it's actual freakin physics! 🤯🤩🤗🙌🏾 You're watching the physics of space and matter coevolve...a small window into the kind of world Wolfram Physics Wolfram predicts. Every ripple, collision, and shimmer you see is a causal event in a living hypergraph. Space itself is an active network of nodes and links...springs and diagonals...constantly stretching, relaxing, and rewriting as the system evolves. Each interaction you see...a nutrient diffusing through the medium, a burrower tunneling, a tentacle feeling drag...is a computation of spacetime itself. Every connection that forms or breaks is an update to the causal structure: a new link, a new moment in the unfolding hypergraph universe. Our environment behaves like an adaptive fabric...the lattice tightens where organisms stir it, loosens where they pass. Chemical fields spread through those same links, feeding back into motion and growth. When the cluster pulsates, it's literally reorganizing local causal geometry...matter and space changing together!👌🏾 We even test causal invariance frame by frame, swapping operation orders to ensure that spacetime remains consistent no matter the update path. That's not animation logic...that's a computational physics experiment running live!🥳 Tentacles feel friction, burrowers dig through gradients, and behind them the lattice stiffens, storing history as structure. #Mathematics #Physics #WolframPhysics #Mathelirium

Mathelirium

20,090 次观看 • 9 个月前

You're watching a quantum fluid pull on itself. Bright clouds = high-density clumps ("proto-stars"); dark ripples = low-density filaments. The white dots are tracer particles...their motion draws the fine threads. It's a Schrödinger-Poisson self-gravity, in real time. 🤩🤗🫠

You're watching a quantum fluid pull on itself. Bright clouds = high-density clumps ("proto-stars"); dark ripples = low-density filaments. The white dots are tracer particles...their motion draws the fine threads. It's a Schrödinger-Poisson self-gravity, in real time. 🤩🤗🫠

Mathelirium

109,155 次观看 • 11 个月前

A coherent state wave packet oscillates in a 1D harmonic trap. - |ψ|² determines helix radius. - Phase determines helix twist. - The fabric deforms under the packet's pressure and current.

A coherent state wave packet oscillates in a 1D harmonic trap. - |ψ|² determines helix radius. - Phase determines helix twist. - The fabric deforms under the packet's pressure and current.

Mathelirium

12,365 次观看 • 2 个月前

You start with a plain old complex polynomial p(z) = a₀ + a₁z + … + aₙzⁿ. Instead of fixing the coefficients, you treat them as variables in a controlled experiment and modulate them. Maybe you draw each aₖ from some distribution, or you let them drift along a parameter path. Do that 50,000 times, say, and you no longer have a polynomial. You have an ensemble. For each realization, the goal is the same Find its roots. Instead of attacking p(z)=0 directly with something like fsolve, you rewrite the polynomial as a matrix problem. Specifically, you build its Companion Matrix. For a monic polynomial zⁿ + bₙ₋₁zⁿ⁻¹ + … + b₀ the companion matrix is an n×n matrix whose last column contains the coefficients (with minus signs), and whose subdiagonal is filled with ones. Everything else is zero. It’s sparse, structured and deterministic. The theorem that makes this worth doing is simple and beautiful The eigenvalues of the companion matrix are exactly the roots of the polynomial. So each polynomial sample becomes a matrix. Each matrix becomes an eigenvalue problem. And eigenvalues are something numerical linear algebra is very, very good at computing.

You start with a plain old complex polynomial p(z) = a₀ + a₁z + … + aₙzⁿ. Instead of fixing the coefficients, you treat them as variables in a controlled experiment and modulate them. Maybe you draw each aₖ from some distribution, or you let them drift along a parameter path. Do that 50,000 times, say, and you no longer have a polynomial. You have an ensemble. For each realization, the goal is the same Find its roots. Instead of attacking p(z)=0 directly with something like fsolve, you rewrite the polynomial as a matrix problem. Specifically, you build its Companion Matrix. For a monic polynomial zⁿ + bₙ₋₁zⁿ⁻¹ + … + b₀ the companion matrix is an n×n matrix whose last column contains the coefficients (with minus signs), and whose subdiagonal is filled with ones. Everything else is zero. It’s sparse, structured and deterministic. The theorem that makes this worth doing is simple and beautiful The eigenvalues of the companion matrix are exactly the roots of the polynomial. So each polynomial sample becomes a matrix. Each matrix becomes an eigenvalue problem. And eigenvalues are something numerical linear algebra is very, very good at computing.

Mathelirium

15,566 次观看 • 5 个月前

Boost A Mass To Light Speed, And Gravity Becomes A Shockwave. The Aichelburg-Sexl Ultraboost is an exact solution of General Relativity obtained by boosting the Schwarzschild gravitational field toward light speed while keeping the energy finite.

Boost A Mass To Light Speed, And Gravity Becomes A Shockwave. The Aichelburg-Sexl Ultraboost is an exact solution of General Relativity obtained by boosting the Schwarzschild gravitational field toward light speed while keeping the energy finite.

Mathelirium

12,414 次观看 • 3 个月前

When the Lorenz Attractor Becomes a Cosmos Many of us have seen plots of the famous Lorenz system that famous butterfly-shaped chaos: ẋ = σ(y - x) ẏ = x(ρ - z) - y ż = xy - βz But here, we went one layer deeper. At every point (x, y, z), we computed the local Jacobian matrix, J(x, y, z) = [ -σ σ 0 ] [ ρ - z -1 - x ] [ y x - β ] Then we found its eigenvalues λ and plotted them in the complex plane. Re(λ) tells us whether nearby motion is locally growing or decaying. Im(λ) tells us whether that local motion is rotating. That gives us the spectral flow of the Lorenz system and not the attractor itself, but the hidden local linear geometry driving its instability. When I saw that flow, it looked strangely cosmic with two mirrored swarms curling like particles trapped in a mathematical whirlpool. So I mapped that spectral flow onto our living fabric, not as Einstein Spacetime this time, but as a creative geometric stage for chaos itself. #LorenzSystem #ChaosTheory #DynamicalSystems #Eigenvalues #SpectralFlow #ComplexPlane #NonlinearDynamics #Mathematics #Physics

When the Lorenz Attractor Becomes a Cosmos Many of us have seen plots of the famous Lorenz system that famous butterfly-shaped chaos: ẋ = σ(y - x) ẏ = x(ρ - z) - y ż = xy - βz But here, we went one layer deeper. At every point (x, y, z), we computed the local Jacobian matrix, J(x, y, z) = [ -σ σ 0 ] [ ρ - z -1 - x ] [ y x - β ] Then we found its eigenvalues λ and plotted them in the complex plane. Re(λ) tells us whether nearby motion is locally growing or decaying. Im(λ) tells us whether that local motion is rotating. That gives us the spectral flow of the Lorenz system and not the attractor itself, but the hidden local linear geometry driving its instability. When I saw that flow, it looked strangely cosmic with two mirrored swarms curling like particles trapped in a mathematical whirlpool. So I mapped that spectral flow onto our living fabric, not as Einstein Spacetime this time, but as a creative geometric stage for chaos itself. #LorenzSystem #ChaosTheory #DynamicalSystems #Eigenvalues #SpectralFlow #ComplexPlane #NonlinearDynamics #Mathematics #Physics

Mathelirium

22,460 次观看 • 1 个月前

Quantum Probability Surface Goes Negative A Schrödinger-Cat state is built from two separated coherent wavepackets, |ψ⟩=N(|α⟩+|−α⟩), and its Wigner function reveals the part of Quantum Mechanics that ordinary probability cannot show. The two bright lobes behave like classical possibilities, while the interference between them carves negative valleys into phase space, where W(x,p)<0. In this animation, the surface height is the signed Wigner function. So, gold rises above zero, cyan sinks below it, and the blades between them expose the nonclassical structure of the wavefunction. #QuantumMechanics #WignerFunction #SchrodingerCat #Physics #QuantumArt #ScienceVisualization

Quantum Probability Surface Goes Negative A Schrödinger-Cat state is built from two separated coherent wavepackets, |ψ⟩=N(|α⟩+|−α⟩), and its Wigner function reveals the part of Quantum Mechanics that ordinary probability cannot show. The two bright lobes behave like classical possibilities, while the interference between them carves negative valleys into phase space, where W(x,p)<0. In this animation, the surface height is the signed Wigner function. So, gold rises above zero, cyan sinks below it, and the blades between them expose the nonclassical structure of the wavefunction. #QuantumMechanics #WignerFunction #SchrodingerCat #Physics #QuantumArt #ScienceVisualization

Mathelirium

12,603 次观看 • 2 个月前

Gravity makes things orbit...but what happens when the mass distribution itself is rotating and dragging phase space into resonances and chaos? In this scene we follow tens of thousands of stars in a toy barred galaxy: a flat disc with a central mass and a rotating bar of extra density that is not rigid, but wobbling, swaying, and breathing in strength over time. The stars start on almost circular orbits in a calm ring, but as the bar turns and its pattern speed and strength modulate, they feel a time-dependent gravitational shove that nudges some onto resonant tracks. 🤩😍 #Mathelirium #Astrophysics #GalacticDynamics #BarredGalaxy #PhaseSpace #HamiltonianChaos #OrbitalResonance

Gravity makes things orbit...but what happens when the mass distribution itself is rotating and dragging phase space into resonances and chaos? In this scene we follow tens of thousands of stars in a toy barred galaxy: a flat disc with a central mass and a rotating bar of extra density that is not rigid, but wobbling, swaying, and breathing in strength over time. The stars start on almost circular orbits in a calm ring, but as the bar turns and its pattern speed and strength modulate, they feel a time-dependent gravitational shove that nudges some onto resonant tracks. 🤩😍 #Mathelirium #Astrophysics #GalacticDynamics #BarredGalaxy #PhaseSpace #HamiltonianChaos #OrbitalResonance

Mathelirium

152,143 次观看 • 8 个月前