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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... show more

Mathelirium

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15,826 次观看 • 9 个月前 •via X (Twitter)

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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. 🤯🤯🤩🤩🤗🫠

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

26,382 次观看 • 9 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

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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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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. 🤩🤗😍

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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. 🤩🤗🫠

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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!🫠❤️🫠

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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.

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