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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

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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.

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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.

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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

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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

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What you're watching here is a torus cycling through its eigenfunctions...the surface's pure standing-wave modes. An eigenfunction is a mode that, when the surface Laplacian operator acts on it, reproduces itself scaled by a constant...that constant fixes the natural frequency. Each mode is indexed by two counts: how many oscillations wrap the small tube and how many wrap the big ring. Dark rings are nodes (no motion) and the bright bands between them are antinodes. As the counts rise, nodes crowd together, ridges sharpen, and the torus takes on a pine-cone look. #Eigenfunctions #SpectralGeometry #MathArt #Math

What you're watching here is a torus cycling through its eigenfunctions...the surface's pure standing-wave modes. An eigenfunction is a mode that, when the surface Laplacian operator acts on it, reproduces itself scaled by a constant...that constant fixes the natural frequency. Each mode is indexed by two counts: how many oscillations wrap the small tube and how many wrap the big ring. Dark rings are nodes (no motion) and the bright bands between them are antinodes. As the counts rise, nodes crowd together, ridges sharpen, and the torus takes on a pine-cone look. #Eigenfunctions #SpectralGeometry #MathArt #Math

Mathelirium

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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

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This is Lithium, visualized as a quantum wavepacket. The bright structures are 3D isosurfaces of the outer electron’s probability density, while colour represents the complex phase. Unlike Hydrogen, Lithium’s valence electron moves in a screened potential created by the nucleus and the closed 1s² core. The animation solves that effective Schrödinger problem and follows a coherent mixture of s- and p-type Rydberg states as they beat, breathe, shift and reform around the core.

This is Lithium, visualized as a quantum wavepacket. The bright structures are 3D isosurfaces of the outer electron’s probability density, while colour represents the complex phase. Unlike Hydrogen, Lithium’s valence electron moves in a screened potential created by the nucleus and the closed 1s² core. The animation solves that effective Schrödinger problem and follows a coherent mixture of s- and p-type Rydberg states as they beat, breathe, shift and reform around the core.

Mathelirium

175,921 Aufrufe • vor 1 Monat

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

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$This scene asks a stranger question : What happens if the material has negative refractive index? As the electron moves through the metamaterial slab, the emitted radiation no longer follows the familiar Cherenkov geometry (see below). The energy flow forms a backward-directed cone, extending opposite to the electron's motion. Method used: Finite Element Method with Poynting style energy-flow tracers.$

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A coherent wave passes through two finite slits and the downstream field is rebuilt by a Huygens sum. The filaments trace the probability current j = Im(ψ*∇ψ). The interference bands on detector grow from individual impacts sampled from |ψ|².

A coherent wave passes through two finite slits and the downstream field is rebuilt by a Huygens sum. The filaments trace the probability current j = Im(ψ*∇ψ). The interference bands on detector grow from individual impacts sampled from |ψ|².

Mathelirium

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Normalize Working in Phase Space Ordinary space shows where the particle is. Phase space shows its complete state, position and momentum together, revealing the hidden geometry of its motion.

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To preserve the Declaration of Independence, the original document is kept in the dark in bulletproof cases that are guarded around the clock.

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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 Aufrufe • vor 4 Monaten

We evolve a complex quantum field ψ(x,y,t) with a Schrödinger-Poisson flow: i ψ_t = -(1/2)Δψ + (V_conf(x,y) + G Φ(x,y,t)) ψ, with ρ=|ψ|² and ΔΦ = ρ - ⟨ρ⟩. That bracket (V_conf + GΦ) is just the potential energy landscape. V_conf is the trap you choose by hand (a smooth bowl that keeps the wave from drifting away). Φ is the field generated by the wave’s own density...compute ρ, solve the Poisson equation, feed Φ back into the evolution. G is the knob that sets how strongly the wave feels its own Φ. Visually Hue is phase φ=arg(ψ), brightness is density ρ^γ, the banded helicoids are phase ribbons cos(mφ+θ(t)), the bright ridges are boosted by |∇ρ|, and the tiny white pearls mark vortices where the phase winding satisfies ∮∇φ·dl = 2πk. #QuantumMechanics #SchrodingerPoisson #ComputationalPhysics #PhysicsVisualization

We evolve a complex quantum field ψ(x,y,t) with a Schrödinger-Poisson flow: i ψ_t = -(1/2)Δψ + (V_conf(x,y) + G Φ(x,y,t)) ψ, with ρ=|ψ|² and ΔΦ = ρ - ⟨ρ⟩. That bracket (V_conf + GΦ) is just the potential energy landscape. V_conf is the trap you choose by hand (a smooth bowl that keeps the wave from drifting away). Φ is the field generated by the wave’s own density...compute ρ, solve the Poisson equation, feed Φ back into the evolution. G is the knob that sets how strongly the wave feels its own Φ. Visually Hue is phase φ=arg(ψ), brightness is density ρ^γ, the banded helicoids are phase ribbons cos(mφ+θ(t)), the bright ridges are boosted by |∇ρ|, and the tiny white pearls mark vortices where the phase winding satisfies ∮∇φ·dl = 2πk. #QuantumMechanics #SchrodingerPoisson #ComputationalPhysics #PhysicsVisualization

Mathelirium

18,700 Aufrufe • vor 9 Monaten

I love turning basic Math concepts into art. This is a Riemann sphere driven by a moving rational function R(z) = ∏(z-aⱼ) / ∏(z-bₖ) The glowing roots and poles orbit across S² while log|R| and Φ = arg R continuously reshape the surface. The small particles ride the phase gradient ∇Φ.

I love turning basic Math concepts into art. This is a Riemann sphere driven by a moving rational function R(z) = ∏(z-aⱼ) / ∏(z-bₖ) The glowing roots and poles orbit across S² while log|R| and Φ = arg R continuously reshape the surface. The small particles ride the phase gradient ∇Φ.

Mathelirium

24,841 Aufrufe • vor 1 Monat

The Branch-Point Loom Some equations do not draw curves but instead draw weave sheets. This scene comes from the algebraic curve y² = Δ(z,t), where Δ(z,t) = Πⱼ(z - βⱼ(t)). The moving points βⱼ are branch points. These are special places where two sheets of the curve touch and swap identities. In the animation, those branch points appear as glowing mineral spindles moving across a dark surface. The landscape is shaped by -log|Δ(z,t)|. Therefore, the branch points rise into sharp seams while the surrounding surface folds around them. The bright threads follow the horizontal trajectories of the quadratic differential Δ(z,t)dz². #Mathematics #ComplexAnalysis #AlgebraicGeometry #MathArt #MathematicalArt #Geometry #PhysicsVisualization #STEM #Art

The Branch-Point Loom Some equations do not draw curves but instead draw weave sheets. This scene comes from the algebraic curve y² = Δ(z,t), where Δ(z,t) = Πⱼ(z - βⱼ(t)). The moving points βⱼ are branch points. These are special places where two sheets of the curve touch and swap identities. In the animation, those branch points appear as glowing mineral spindles moving across a dark surface. The landscape is shaped by -log|Δ(z,t)|. Therefore, the branch points rise into sharp seams while the surrounding surface folds around them. The bright threads follow the horizontal trajectories of the quadratic differential Δ(z,t)dz². #Mathematics #ComplexAnalysis #AlgebraicGeometry #MathArt #MathematicalArt #Geometry #PhysicsVisualization #STEM #Art

Mathelirium

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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

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