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This is an animation of a Photonic Molecule. Light becomes trapped inside a tiny ring and then coupled into another without the two ever touching. The scene is evolved from Maxwell's equations using Finite Difference Time Domain (FDTD) on a dense Yee grid.

Mathelirium

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19,395 просмотров • 3 дней назад •via X (Twitter)

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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.

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A sharp 90-degree turn is a brutal request for an Electromagnetic wave. Yet inside a Photonic Crystal, something remarkable can happen. The repeating dielectric structure creates ranges of frequency for which propagation through the bulk crystal is strongly suppressed. Remove a line of that crystal and suddenly the defect becomes an allowed route the EM field.

A sharp 90-degree turn is a brutal request for an Electromagnetic wave. Yet inside a Photonic Crystal, something remarkable can happen. The repeating dielectric structure creates ranges of frequency for which propagation through the bulk crystal is strongly suppressed. Remove a line of that crystal and suddenly the defect becomes an allowed route the EM field.

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What you're seeing here is a simulation of a solar-wind pressure interacting with Earth's magnetosphere. Computations are based on the equations of resistive magnetohydrodynamics (MHD). In the model, the plasma is treated as an electrically conducting fluid whose motion is coupled to the evolution of the magnetic field.

What you're seeing here is a simulation of a solar-wind pressure interacting with Earth's magnetosphere. Computations are based on the equations of resistive magnetohydrodynamics (MHD). In the model, the plasma is treated as an electrically conducting fluid whose motion is coupled to the evolution of the magnetic field.

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Finite Element Methods use fine meshes only where greater detail is needed, allowing the focusing behavior of a convex lens to be simulated accurately without the cost of refining the entire domain.

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Mathelirium

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$Built an FDTD Maxwell solver from scratch and rendered 45 seconds of it. What you’re watching: A plane EM wave hits a PEC barrier with two slits 90 cells apart. λ = 26 cells. The slits diffract. The beams interfere. Fringes form. Then a biconvex dielectric lens (ε = 3.8) refracts the transmitted field and partially focuses it. Young’s double-slit experiment. Fresnel diffraction. Snell’s law refraction. All emergent. None of it coded in it falls out of four update equations on a 480×480 Yee grid. This is what Maxwell’s equations actually look like when you let them run.$

Built an FDTD Maxwell solver from scratch and rendered 45 seconds of it. What you’re watching: A plane EM wave hits a PEC barrier with two slits 90 cells apart. λ = 26 cells. The slits diffract. The beams interfere. Fringes form. Then a biconvex dielectric lens (ε = 3.8) refracts the transmitted field and partially focuses it. Young’s double-slit experiment. Fresnel diffraction. Snell’s law refraction. All emergent. None of it coded in it falls out of four update equations on a 480×480 Yee grid. This is what Maxwell’s equations actually look like when you let them run.

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A Luneburg lens bends an Electromagnetic wave without a curved boundary. Instead its dielectric constant changes continuously from the centre to the edge. This forces different parts of the wavelength to travel at different speeds. Here, we solve the full 2D TM EM wave equation on an unstructured triangular mesh using the Finite Element Method (FEM).

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$When gravitational potential rotates, it can lock onto an orbit’s phase and start rearranging motion. Order can turn into visible structure, and structure can slide into chaos, without any collisions or extra bodies. This is a toy barred galaxy...a softened central potential plus a rotating bar-shaped overdensity. The bar is not a metronome. Its angle sways and wobbles, and its strength breathes over time, so the force field keeps changing. I seed about 45k test stars on near-circular orbits in a ring and evolve them by using Newton’s 2nd law step by step (velocity Verlet scheme)...the potential is time-dependent. As the bar turns, resonances trap some stars into long-lived orbital families. Others get stretched into spiral-like streams as the ring is sheared. A fraction drift into chaotic layers, where tiny timing differences blow up because the resonance boundaries are moving. What you see in the trails is phase space being rewritten...stable islands, separating boundaries, broken tori, and mixing. #Astrophysics #GalacticDynamics #BarredGalaxy #HamiltonianChaos #PhaseSpace #OrbitalResonance$

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Mathelirium

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An intense pulse of laser light enters a plasma and what follows is almost unbelievable. The laser pushes plasma electron aside, carving out a charged cavity. As the plasma rushes back, it creates an electric wake that can trap and accelerate another bunch of electrons. This is known as Laser Wakefield Acceleration (LWFA) and is being explored for future accelerator Physics. For this animation, we used a reduced particle-in-cell (PIC) method, coupling particle motion to the self-consistent plasma field on a computational mesh.

An intense pulse of laser light enters a plasma and what follows is almost unbelievable. The laser pushes plasma electron aside, carving out a charged cavity. As the plasma rushes back, it creates an electric wake that can trap and accelerate another bunch of electrons. This is known as Laser Wakefield Acceleration (LWFA) and is being explored for future accelerator Physics. For this animation, we used a reduced particle-in-cell (PIC) method, coupling particle motion to the self-consistent plasma field on a computational mesh.

Mathelirium

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The Elusive Concept of Time. Your instincts treat time like a background meter the whole universe shares. Relativity does not take time away, it forces you to earn it operationally. Events are points. Motion is a curve through them. A clock is not a metaphor, it is a worldline with a number attached to it. Two observers can disagree on which distant events are simultaneous, and nothing contradictory happens, because causal structure is still pinned down by light cones and invariants. Even in weak gravity the rule shows up. A clock deeper in a gravitational potential ticks more slowly relative to one far away. Near a compact object the difference becomes hard to ignore. Add rotation and spacetime itself picks up a twist. That twist is frame dragging, not a new force, just geometry telling you that time and angle are coupled in a rotating spacetime. In the animation You are watching a geometry lesson disguised as a black hole scene. The fabric is a visualization of the field shaping clock-rates and the paths light can take. The ripples are driven by local proper time, so their phase visibly slows as you approach the horizon. The accretion disk is lensed through Kerr ray tracing, and its brightness is pushed by redshift and beaming so the approaching side can flare while the receding side dims. Beacon points at different radii pulse at different rates, so you can see time dilation without any labels. The bead ring is a redshift tracer, with intensity scaled by a g³ proxy so deeper emission arrives weaker and shifted. The math breakdown Start with what a clock actually measures. Proper time τ is the accumulated time along an observer’s worldline. In special relativity, the invariant interval is ds² = c² dt² − dx² − dy² − dz² Along a timelike path, dτ = (1/c) √(ds²) = √( dt² − (1/c²)(dx²+dy²+dz²) ) If the observer moves with speed v, so dx²+dy²+dy²+dz² = v² dt², then dτ = dt √(1 − v²/c²) That is time dilation as geometry. The moving clock accumulates less τ between the same pair of events. Now add gravity. General relativity replaces the flat interval with a metric gᵤᵥ that depends on position: ds² = gᵤᵥ dxᵘ dxᵛ For a stationary clock in Schwarzschild geometry (mass M), the time component is g_tt = −(1 − 2GM/(rc²)) If the clock sits at fixed r (no spatial motion), ds² = g_tt c² dt², so dτ = dt √(1 − 2GM/(rc²)) Closer to the mass means a smaller factor, so the clock ticks more slowly relative to a clock far away. That is the rule used to drive the fabric phase in the animation. Now connect time to light. A gravitational field shifts photon frequency. Between an emitter at rₑ and an observer at rₒ, f_obs / f_emit = √( (1 − 2GM/(rₑ c²)) / (1 − 2GM/(rₒ c²)) ) For a far-away observer rₒ → ∞, f_obs / f_emit = √(1 − 2GM/(rₑ c²)) Deeper emission arrives redshifted. Lower frequency. Lower energy per photon. In the render, the disk intensity uses a Kerr-derived redshift factor g (clipped for stability). The bead ring uses a simple radiative proxy I_obs ∝ g³ I_emit to make that effect visible. Finally, why rotation looks like a twist. A rotating black hole is Kerr geometry. The key structural change is a nonzero g_tφ term, which couples time to angle. That coupling is frame dragging in equations. Near the hole, being stationary is not the same notion everywhere, because the local inertial frames are being pulled around the spin axis. So the moral stays clean. Time is not a universal substance flowing everywhere at one rate. It is what clocks accumulate along worldlines. Light cones constrain what can influence what. Invariants are what everyone agrees on. The rest is operational detail that only feels universal because our daily corner of the universe is slow and mild. #GeneralRelativity #Gravity #FrameDragging #BlackHoles #Spacetime

The Elusive Concept of Time. Your instincts treat time like a background meter the whole universe shares. Relativity does not take time away, it forces you to earn it operationally. Events are points. Motion is a curve through them. A clock is not a metaphor, it is a worldline with a number attached to it. Two observers can disagree on which distant events are simultaneous, and nothing contradictory happens, because causal structure is still pinned down by light cones and invariants. Even in weak gravity the rule shows up. A clock deeper in a gravitational potential ticks more slowly relative to one far away. Near a compact object the difference becomes hard to ignore. Add rotation and spacetime itself picks up a twist. That twist is frame dragging, not a new force, just geometry telling you that time and angle are coupled in a rotating spacetime. In the animation You are watching a geometry lesson disguised as a black hole scene. The fabric is a visualization of the field shaping clock-rates and the paths light can take. The ripples are driven by local proper time, so their phase visibly slows as you approach the horizon. The accretion disk is lensed through Kerr ray tracing, and its brightness is pushed by redshift and beaming so the approaching side can flare while the receding side dims. Beacon points at different radii pulse at different rates, so you can see time dilation without any labels. The bead ring is a redshift tracer, with intensity scaled by a g³ proxy so deeper emission arrives weaker and shifted. The math breakdown Start with what a clock actually measures. Proper time τ is the accumulated time along an observer’s worldline. In special relativity, the invariant interval is ds² = c² dt² − dx² − dy² − dz² Along a timelike path, dτ = (1/c) √(ds²) = √( dt² − (1/c²)(dx²+dy²+dz²) ) If the observer moves with speed v, so dx²+dy²+dy²+dz² = v² dt², then dτ = dt √(1 − v²/c²) That is time dilation as geometry. The moving clock accumulates less τ between the same pair of events. Now add gravity. General relativity replaces the flat interval with a metric gᵤᵥ that depends on position: ds² = gᵤᵥ dxᵘ dxᵛ For a stationary clock in Schwarzschild geometry (mass M), the time component is g_tt = −(1 − 2GM/(rc²)) If the clock sits at fixed r (no spatial motion), ds² = g_tt c² dt², so dτ = dt √(1 − 2GM/(rc²)) Closer to the mass means a smaller factor, so the clock ticks more slowly relative to a clock far away. That is the rule used to drive the fabric phase in the animation. Now connect time to light. A gravitational field shifts photon frequency. Between an emitter at rₑ and an observer at rₒ, f_obs / f_emit = √( (1 − 2GM/(rₑ c²)) / (1 − 2GM/(rₒ c²)) ) For a far-away observer rₒ → ∞, f_obs / f_emit = √(1 − 2GM/(rₑ c²)) Deeper emission arrives redshifted. Lower frequency. Lower energy per photon. In the render, the disk intensity uses a Kerr-derived redshift factor g (clipped for stability). The bead ring uses a simple radiative proxy I_obs ∝ g³ I_emit to make that effect visible. Finally, why rotation looks like a twist. A rotating black hole is Kerr geometry. The key structural change is a nonzero g_tφ term, which couples time to angle. That coupling is frame dragging in equations. Near the hole, being stationary is not the same notion everywhere, because the local inertial frames are being pulled around the spin axis. So the moral stays clean. Time is not a universal substance flowing everywhere at one rate. It is what clocks accumulate along worldlines. Light cones constrain what can influence what. Invariants are what everyone agrees on. The rest is operational detail that only feels universal because our daily corner of the universe is slow and mild. #GeneralRelativity #Gravity #FrameDragging #BlackHoles #Spacetime

Mathelirium

149,835 просмотров • 7 месяцев назад

A Million-Dollar Storm Mathematics Still Cannot Tame. Turbulence matters because it is the default behavior of real fluids once motion becomes strong enough, e.g. air over wings, blood through vessels, weather in the atmosphere, water in oceans, fuel in engines, and flow in pipes and turbines. We can write down the Navier-Stokes equations for all of this, but in 3D one of the deepest open questions is still whether smooth solutions always stay smooth, or can break down in finite time. That is crucial because it asks whether the basic equations of fluid motion remain mathematically under control forever, or can form a true singularity where the theory itself breaks. Solve that and you got yourself a million dollars

A Million-Dollar Storm Mathematics Still Cannot Tame. Turbulence matters because it is the default behavior of real fluids once motion becomes strong enough, e.g. air over wings, blood through vessels, weather in the atmosphere, water in oceans, fuel in engines, and flow in pipes and turbines. We can write down the Navier-Stokes equations for all of this, but in 3D one of the deepest open questions is still whether smooth solutions always stay smooth, or can break down in finite time. That is crucial because it asks whether the basic equations of fluid motion remain mathematically under control forever, or can form a true singularity where the theory itself breaks. Solve that and you got yourself a million dollars

Mathelirium

30,056 просмотров • 4 месяцев назад

We know gravity pulls things together in space… but have you ever watched it work in phase space? In this scene we’re looking at a thin probability sheet in (x, v) position-velocity space evolving under its own self-gravity. In the collisionless limit, Liouville’s theorem says the sheet can stretch, fold, and swirl, but it can’t tear and it can’t change area. Any collapse in x is paid for by stretching in v. Numerically this is a 1D cold-dark-matter toy universe: a sheet of particles evolved under the Vlasov-Poisson system. We solve Poisson on a periodic line to get the self-consistent gravitational field, then integrate Hamilton’s equations so the sheet folds into a multi-layer spiral. The glowing filaments are the places the sheet keeps revisiting dense, sharp caustics carved into phase space. Gravity here isn’t stuff falling down. It’s area-preserving origami in (x, v). #Physics #VlasovPoisson #PhaseSpace #Liouville #DarkMatter #HamiltonianFlow #ProbabilityFlow #Mathematics

We know gravity pulls things together in space… but have you ever watched it work in phase space? In this scene we’re looking at a thin probability sheet in (x, v) position-velocity space evolving under its own self-gravity. In the collisionless limit, Liouville’s theorem says the sheet can stretch, fold, and swirl, but it can’t tear and it can’t change area. Any collapse in x is paid for by stretching in v. Numerically this is a 1D cold-dark-matter toy universe: a sheet of particles evolved under the Vlasov-Poisson system. We solve Poisson on a periodic line to get the self-consistent gravitational field, then integrate Hamilton’s equations so the sheet folds into a multi-layer spiral. The glowing filaments are the places the sheet keeps revisiting dense, sharp caustics carved into phase space. Gravity here isn’t stuff falling down. It’s area-preserving origami in (x, v). #Physics #VlasovPoisson #PhaseSpace #Liouville #DarkMatter #HamiltonianFlow #ProbabilityFlow #Mathematics

Mathelirium

16,437 просмотров • 7 месяцев назад

We often hear that machine learning models "learn patterns in data". But what does that actually look like in geometry? If you dropped a little elastic mesh into a cloud of points and let it learn, how would it fold itself to match the shape of the data? In this scene we watch a self-organizing map...a simple unsupervised neural model...learn the shape of a two-dimensional dataset arranged in a spiral arm. On top of this, we lay down a square grid of neurons whose weights live in the same plane. At the start, this grid is just a flat net floating across the cloud...it knows nothing about the structure underneath. Learning is a repeated game: pick a random data point, find the neuron whose weight is closest, and then nudge that neuron and its neighbours toward the point. Do this again and again, while slowly shrinking how far the neighbourhood influence spreads. #MachineLearning #ManifoldLearning #UnsupervisedLearning #NeuralMaps #GeometricML

We often hear that machine learning models "learn patterns in data". But what does that actually look like in geometry? If you dropped a little elastic mesh into a cloud of points and let it learn, how would it fold itself to match the shape of the data? In this scene we watch a self-organizing map...a simple unsupervised neural model...learn the shape of a two-dimensional dataset arranged in a spiral arm. On top of this, we lay down a square grid of neurons whose weights live in the same plane. At the start, this grid is just a flat net floating across the cloud...it knows nothing about the structure underneath. Learning is a repeated game: pick a random data point, find the neuron whose weight is closest, and then nudge that neuron and its neighbours toward the point. Do this again and again, while slowly shrinking how far the neighbourhood influence spreads. #MachineLearning #ManifoldLearning #UnsupervisedLearning #NeuralMaps #GeometricML

Mathelirium

139,757 просмотров • 9 месяцев назад

Have You Ever Heard of Molecular Communication Systems? Instead of using radio waves or light, molecular communications send information carefully through released chemical particles. A transmitter emits molecules into a medium, the channel reshapes that signal through diffusion, flow, and randomness. The receiver reads the message from the molecules that arrive and bind. It is communication theory at microscopic scale, where physics, probability and biology all become part of the channel. Some applications you may know: Targeted drug delivery inside the body, microscopic health monitoring with nanosensors, and lab-on-a-chip.

Have You Ever Heard of Molecular Communication Systems? Instead of using radio waves or light, molecular communications send information carefully through released chemical particles. A transmitter emits molecules into a medium, the channel reshapes that signal through diffusion, flow, and randomness. The receiver reads the message from the molecules that arrive and bind. It is communication theory at microscopic scale, where physics, probability and biology all become part of the channel. Some applications you may know: Targeted drug delivery inside the body, microscopic health monitoring with nanosensors, and lab-on-a-chip.

Mathelirium

19,452 просмотров • 5 месяцев назад

This is the process of making a ring from magnetic sand using an iPhone.

This is the process of making a ring from magnetic sand using an iPhone.

Arrayan Taysir

2,916,168 просмотров • 10 месяцев назад

We know gravity pulls things together in space...but have you ever seen what it does in phase space? In this scene we're not watching a blob of matter in x–space. We're watching a thin probability sheet in (x, v).... position–velocity space... being pulled by its own gravity. In the collisionless limit, Liouville’s theorem says this sheet can stretch, fold, and swirl, but never tear or change area. Collapse in x is "paid for" by stretching in v. Numerically, this is a 1D cold-dark-matter toy universe: a sheet of particles evolved under the Vlasov-Poisson system. We solve Poisson on a periodic line, get the self-consistent gravitational field, then integrate Hamilton’s equations so the sheet folds into a multi-layer spiral. The glowing filaments are where the sheet keeps revisiting, carving out dense caustics in phase space. It's gravity not as "stuff falling down," but as area-preserving origami in phase space. #Mathelirium #VlasovPoisson #PhaseSpace #Liouville #DarkMatter #HamiltonianFlow #ProbabilityFlow

We know gravity pulls things together in space...but have you ever seen what it does in phase space? In this scene we're not watching a blob of matter in x–space. We're watching a thin probability sheet in (x, v).... position–velocity space... being pulled by its own gravity. In the collisionless limit, Liouville’s theorem says this sheet can stretch, fold, and swirl, but never tear or change area. Collapse in x is "paid for" by stretching in v. Numerically, this is a 1D cold-dark-matter toy universe: a sheet of particles evolved under the Vlasov-Poisson system. We solve Poisson on a periodic line, get the self-consistent gravitational field, then integrate Hamilton’s equations so the sheet folds into a multi-layer spiral. The glowing filaments are where the sheet keeps revisiting, carving out dense caustics in phase space. It's gravity not as "stuff falling down," but as area-preserving origami in phase space. #Mathelirium #VlasovPoisson #PhaseSpace #Liouville #DarkMatter #HamiltonianFlow #ProbabilityFlow

Mathelirium

47,651 просмотров • 9 месяцев назад

There truly is an underlying order in every chaotic system In this scene, we have 2500 identical triple pendulums on a grid with almost the same initial conditions.

There truly is an underlying order in every chaotic system In this scene, we have 2500 identical triple pendulums on a grid with almost the same initial conditions.

Mathelirium

21,363 просмотров • 5 месяцев назад