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d’Alembert’s Paradox: The 1752 Paradox That Stood in the Way of Real Aerodynamics In 1752, d’Alembert found a result that still feels wrong at first sight: An ideal fluid can flow around a body and produce no drag at all. In the first animation, viscosity is set to zero. The fluid bends around the object, accelerates, slows down, and stitches itself back together downstream. You get no wake, no loss and no scar in the flow. The equations have removed the very thing that would let the body leave a trace. That is Euler flow: ρ(∂u/∂t + (u · ∇)u) = −∇p ∇ · u = 0 For steady irrotational flow, u = ∇φ and ∇²φ = 0. The pressure field stays perfectly symmetric front to back, so when you integrate pressure over the body, the drag cancels: D = 0 The second animation changes one thing: viscosity is allowed back in. ρ(∂u/∂t + (u · ∇)u) = −∇p + μ∇²u ∇ · u = 0 Now, the surface can grip the fluid. A boundary layer forms. It can separate, vorticity rolls off the body and wake appears. Drag is now built by the flow. Therefore, making viscosity tiny is not the same as deleting it. ν → 0 still has a boundary layer. ν = 0 has no memory.

d’Alembert’s Paradox: The 1752 Paradox That Stood in the Way of Real Aerodynamics In 1752, d’Alembert found a result that still feels wrong at first sight: An ideal fluid can flow around a body and produce no drag at all. In the first animation, viscosity is set to zero. The fluid bends around the object, accelerates, slows down, and stitches itself back together downstream. You get no wake, no loss and no scar in the flow. The equations have removed the very thing that would let the body leave a trace. That is Euler flow: ρ(∂u/∂t + (u · ∇)u) = −∇p ∇ · u = 0 For steady irrotational flow, u = ∇φ and ∇²φ = 0. The pressure field stays perfectly symmetric front to back, so when you integrate pressure over the body, the drag cancels: D = 0 The second animation changes one thing: viscosity is allowed back in. ρ(∂u/∂t + (u · ∇)u) = −∇p + μ∇²u ∇ · u = 0 Now, the surface can grip the fluid. A boundary layer forms. It can separate, vorticity rolls off the body and wake appears. Drag is now built by the flow. Therefore, making viscosity tiny is not the same as deleting it. ν → 0 still has a boundary layer. ν = 0 has no memory.

43,784 просмотров

What Happens When Prime Numbers Enter The Complex Plane? Prime numbers live on the number line. Gaussian primes live on the complex lattice. Instead of looking only at numbers like 2, 3, 5, 7, we look at points z = a + bi, where a and b are integers. The key quantity is the norm: N(a+bi) = a² + b². This norm decides which lattice points behave like primes. For points away from the axes, a+bi is a Gaussian prime when a² + b² is an ordinary prime. On the axes, the rule changes. A point is prime only when the nonzero coordinate is an ordinary prime of the form p ≡ 3 mod 4. And the prime 2 is special: 2 = (1+i)(1−i). That is why the animation grows as a structured prime landscape. Off-axis primes appear in symmetric families. Axis primes survive only in the 4k+3 class. The norm-2 points glow as the ramified prime. The result is a square lattice where prime numbers stop looking like isolated dots and start behaving like geometry. #PrimeNumbers #NumberTheory #GaussianIntegers #ComplexPlane #Mathematics #Mathelirium

What Happens When Prime Numbers Enter The Complex Plane? Prime numbers live on the number line. Gaussian primes live on the complex lattice. Instead of looking only at numbers like 2, 3, 5, 7, we look at points z = a + bi, where a and b are integers. The key quantity is the norm: N(a+bi) = a² + b². This norm decides which lattice points behave like primes. For points away from the axes, a+bi is a Gaussian prime when a² + b² is an ordinary prime. On the axes, the rule changes. A point is prime only when the nonzero coordinate is an ordinary prime of the form p ≡ 3 mod 4. And the prime 2 is special: 2 = (1+i)(1−i). That is why the animation grows as a structured prime landscape. Off-axis primes appear in symmetric families. Axis primes survive only in the 4k+3 class. The norm-2 points glow as the ramified prime. The result is a square lattice where prime numbers stop looking like isolated dots and start behaving like geometry. #PrimeNumbers #NumberTheory #GaussianIntegers #ComplexPlane #Mathematics #Mathelirium

10,339 просмотров

d’Alembert’s Paradox: ν → 0 Is Not ν = 0 In 1752, Jean-le-Rond d’Alembert proved a result that still trips people up: An inviscid, incompressible, steady flow exerts zero drag on a body. No wake. No resistance. The equations let the fluid slip past as if the object weren’t there. The first animation shows exactly that world. The flow is ideal Euler flow. Streamlines bend around the body, accelerate, slow down, then recombine perfectly downstream. Nothing is left behind. The motion you see comes from pathlines moving through a steady velocity field, not from any evolving structure in the flow itself. The setup is the ideal fluid model: Euler (ν = 0) ρ(∂u/∂t + (u·∇)u) = −∇p ∇·u = 0 Assume steady flow and zero viscosity and the picture locks in. If the flow is also irrotational, ∇×u = 0, you can write u = ∇φ and the problem collapses to potential flow: ∇²φ = 0 u = ∇φ The force on the body comes entirely from pressure: F = −∮ p n dS D = F·eₓ Under these assumptions the pressure field is perfectly front–back symmetric, so the integral gives D = 0 That’s the paradox. Not a small correction. Zero. Now look at the second animation. This is the same geometry and the same inflow, but with viscosity turned on, even if it’s only a small amount. Navier–Stokes (ν > 0) ρ(∂u/∂t + (u·∇)u) = −∇p + μ∇²u ∇·u = 0 That extra term changes everything. A thin boundary layer forms near the surface. Separation becomes possible. Vorticity is generated and shed. A wake appears. Drag is no longer optional. The contrast is the point. Letting viscosity go to zero is not the same thing as setting it to zero. The inviscid limit deletes the mechanism that breaks time-reversal symmetry and allows energy dissipation. Once that mechanism is gone, wakes can’t exist, and drag vanishes by construction. #FluidDynamics #NavierStokes #EulerEquations #DAlembertParadox #BoundaryLayer #Physics

d’Alembert’s Paradox: ν → 0 Is Not ν = 0 In 1752, Jean-le-Rond d’Alembert proved a result that still trips people up: An inviscid, incompressible, steady flow exerts zero drag on a body. No wake. No resistance. The equations let the fluid slip past as if the object weren’t there. The first animation shows exactly that world. The flow is ideal Euler flow. Streamlines bend around the body, accelerate, slow down, then recombine perfectly downstream. Nothing is left behind. The motion you see comes from pathlines moving through a steady velocity field, not from any evolving structure in the flow itself. The setup is the ideal fluid model: Euler (ν = 0) ρ(∂u/∂t + (u·∇)u) = −∇p ∇·u = 0 Assume steady flow and zero viscosity and the picture locks in. If the flow is also irrotational, ∇×u = 0, you can write u = ∇φ and the problem collapses to potential flow: ∇²φ = 0 u = ∇φ The force on the body comes entirely from pressure: F = −∮ p n dS D = F·eₓ Under these assumptions the pressure field is perfectly front–back symmetric, so the integral gives D = 0 That’s the paradox. Not a small correction. Zero. Now look at the second animation. This is the same geometry and the same inflow, but with viscosity turned on, even if it’s only a small amount. Navier–Stokes (ν > 0) ρ(∂u/∂t + (u·∇)u) = −∇p + μ∇²u ∇·u = 0 That extra term changes everything. A thin boundary layer forms near the surface. Separation becomes possible. Vorticity is generated and shed. A wake appears. Drag is no longer optional. The contrast is the point. Letting viscosity go to zero is not the same thing as setting it to zero. The inviscid limit deletes the mechanism that breaks time-reversal symmetry and allows energy dissipation. Once that mechanism is gone, wakes can’t exist, and drag vanishes by construction. #FluidDynamics #NavierStokes #EulerEquations #DAlembertParadox #BoundaryLayer #Physics

27,053 просмотров

The Breather surface is the surface of Gaussian curvature -1 that corresponds to the solution of the Sine-Gordon equation

The Breather surface is the surface of Gaussian curvature -1 that corresponds to the solution of the Sine-Gordon equation

16,570 просмотров

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A Black Hole Can Be Dragged Through Spacetime A Black Hole does not have to sit still. Einstein’s equations allow a stranger possibility... a Black Hole that accelerates. This is the C-metric, an exact solution of General Relativity describing an accelerating Black Hole. The Black Hole accelerates because the geometry of Spacetime says it must. The C-metric is one of those exact solutions that reminds us how wild General Relativity really is. Gravity is not just attraction, and Spacetime is not just a stage. Sometimes the stage itself contains the force. #Astrophysics #GeneralRelativity #Einstein #CMetric #BlackHole #CosmicStrings #Spacetime #Cosmology #Mathematics #Physics

A Black Hole Can Be Dragged Through Spacetime A Black Hole does not have to sit still. Einstein’s equations allow a stranger possibility... a Black Hole that accelerates. This is the C-metric, an exact solution of General Relativity describing an accelerating Black Hole. The Black Hole accelerates because the geometry of Spacetime says it must. The C-metric is one of those exact solutions that reminds us how wild General Relativity really is. Gravity is not just attraction, and Spacetime is not just a stage. Sometimes the stage itself contains the force. #Astrophysics #GeneralRelativity #Einstein #CMetric #BlackHole #CosmicStrings #Spacetime #Cosmology #Mathematics #Physics

788,559 просмотров • 28 дней назад

Near a Black Hole, gravity stretches time so profoundly that seconds for one observer can become centuries for another. So the question isn’t what time is it? The question is: When are we?

Near a Black Hole, gravity stretches time so profoundly that seconds for one observer can become centuries for another. So the question isn’t what time is it? The question is: When are we?

93,014 просмотров • 5 дней назад

A Neural Sheet Discovers the Shape of Data A self-organizing map (SOM) starts as a flat sheet of neurons with no knowledge of the data around it. For every input sample, it finds the best-matching neuron, c = argminᵢ ‖x - wᵢ‖, and updates that neuron together with its neighbours, wᵢ ← wᵢ + η(t)hᶜᵢ(t)(x - wᵢ). As the neighbourhood radius shrinks during training, the sheet gradually bends and folds onto the hidden geometry of the dataset while preserving its neighbourhood structure. Developed by Finnish computer scientist Teuvo Kohonen, the Self-Organizing Map remains one of the most elegant examples of competitive learning, where order emerges from thousands of simple local updates. #MachineLearning #ArtificialIntelligence #SelfOrganizingMap #NeuralNetworks #DataVisualization #Mathematics #ComputerScience #Finland #TeuvoKohonen

A Neural Sheet Discovers the Shape of Data A self-organizing map (SOM) starts as a flat sheet of neurons with no knowledge of the data around it. For every input sample, it finds the best-matching neuron, c = argminᵢ ‖x - wᵢ‖, and updates that neuron together with its neighbours, wᵢ ← wᵢ + η(t)hᶜᵢ(t)(x - wᵢ). As the neighbourhood radius shrinks during training, the sheet gradually bends and folds onto the hidden geometry of the dataset while preserving its neighbourhood structure. Developed by Finnish computer scientist Teuvo Kohonen, the Self-Organizing Map remains one of the most elegant examples of competitive learning, where order emerges from thousands of simple local updates. #MachineLearning #ArtificialIntelligence #SelfOrganizingMap #NeuralNetworks #DataVisualization #Mathematics #ComputerScience #Finland #TeuvoKohonen

159,296 просмотров • 9 дней назад

String Theory needed a place for open strings to end. So, D-Branes were born. D-Branes are hidden surfaces in higher dimensions where open strings anchor and give rise to nature’s forces, while closed strings roam the surrounding space carrying gravity.

String Theory needed a place for open strings to end. So, D-Branes were born. D-Branes are hidden surfaces in higher dimensions where open strings anchor and give rise to nature’s forces, while closed strings roam the surrounding space carrying gravity.

45,384 просмотров • 6 дней назад

It is often said that the lift on a wing is generated because the flow moving over the top surface has a longer distance to travel and therefore needs to go faster. This common explanation is actually wrong.

It is often said that the lift on a wing is generated because the flow moving over the top surface has a longer distance to travel and therefore needs to go faster. This common explanation is actually wrong.

678,733 просмотров • 3 месяцев назад

A Pulsar (the tiny, ultra-dense core left behind after a massive star explodes) is one of the most accurate clocks in the Universe. But when its radio pulses pass near a Black Hole, gravity delays their arrival. In other words, the pulse never changed and the clock stayed perfect but gravity changed the timing.

A Pulsar (the tiny, ultra-dense core left behind after a massive star explodes) is one of the most accurate clocks in the Universe. But when its radio pulses pass near a Black Hole, gravity delays their arrival. In other words, the pulse never changed and the clock stayed perfect but gravity changed the timing.

26,555 просмотров • 4 дней назад

String Theory makes an ambitious proposal: Quantum excitations and Spacetime geometry may arise from the same extended objects.

String Theory makes an ambitious proposal: Quantum excitations and Spacetime geometry may arise from the same extended objects.

31,910 просмотров • 7 дней назад

The Chinese Physicist Who Turned Force Into Geometry In 1954, Chinese physicist Chen-Ning Yang, together with Robert Mills, introduced a new idea that changed modern physics: Maybe a force is not just something that pushes or pulls. Maybe a force can come from geometry. In Yang-Mills Theory, every point in space carries a hidden internal direction. As you move through space, that direction can twist. If you go around a closed loop and the direction does not return exactly the same, the field has curvature. That curvature is what we see as force. In this animation, the glowing surface shows where the gauge field is strongest. The changing colours show how the hidden internal direction varies across space. The moving rectangular loops are Wilson loops. They travel through the field and measure how much the internal direction changes after completing a closed path. The key equation behind the scene is Fₓᵧ = ∂ₓAᵧ - ∂ᵧAₓ + Aₓ × Aᵧ. The final term, Aₓ × Aᵧ, is the special part. It means the field can interact with itself. That self-interaction is one reason Yang-Mills theory became one of the foundations of particle physics. #China #ChinesePhysics #YangMills #GaugeTheory #QuantumFieldTheory #NonAbelianGeometry #WilsonLoops #ModernPhysics #STEM

The Chinese Physicist Who Turned Force Into Geometry In 1954, Chinese physicist Chen-Ning Yang, together with Robert Mills, introduced a new idea that changed modern physics: Maybe a force is not just something that pushes or pulls. Maybe a force can come from geometry. In Yang-Mills Theory, every point in space carries a hidden internal direction. As you move through space, that direction can twist. If you go around a closed loop and the direction does not return exactly the same, the field has curvature. That curvature is what we see as force. In this animation, the glowing surface shows where the gauge field is strongest. The changing colours show how the hidden internal direction varies across space. The moving rectangular loops are Wilson loops. They travel through the field and measure how much the internal direction changes after completing a closed path. The key equation behind the scene is Fₓᵧ = ∂ₓAᵧ - ∂ᵧAₓ + Aₓ × Aᵧ. The final term, Aₓ × Aᵧ, is the special part. It means the field can interact with itself. That self-interaction is one reason Yang-Mills theory became one of the foundations of particle physics. #China #ChinesePhysics #YangMills #GaugeTheory #QuantumFieldTheory #NonAbelianGeometry #WilsonLoops #ModernPhysics #STEM

81,786 просмотров • 22 дней назад

In 1996, James Sethian showed something almost unfair...you can find shortest routes through a messy world by letting a wave expand once...no trial paths, no search beams, just one growing front. Here’s how: we solve for an arrival-time field T(x,y) so that T literally means how long the wave needs to reach this point. The rule is ||∇ T|| = 1/F, where the medium is fast (F large) the front sprints, where it’s slow it trudges, and obstacles are speed ≈ 0, so the front wraps around them because that’s the only way forward. Then comes the satisfying part: once T exists, a path doesn’t need to search at all...drop a bead anywhere and let it follow ẋ ∝ -∇ T, it slides downhill on the time landscape and traces a globally fastest route back to the source. This “wave = optimal control” viewpoint is exactly what Tsitsiklis (1995) made precise from the Hamilton-Jacobi side...compute the value/arrival-time function and the optimal trajectories fall out from it. #FastMarching #EikonalEquation

In 1996, James Sethian showed something almost unfair...you can find shortest routes through a messy world by letting a wave expand once...no trial paths, no search beams, just one growing front. Here’s how: we solve for an arrival-time field T(x,y) so that T literally means how long the wave needs to reach this point. The rule is ||∇ T|| = 1/F, where the medium is fast (F large) the front sprints, where it’s slow it trudges, and obstacles are speed ≈ 0, so the front wraps around them because that’s the only way forward. Then comes the satisfying part: once T exists, a path doesn’t need to search at all...drop a bead anywhere and let it follow ẋ ∝ -∇ T, it slides downhill on the time landscape and traces a globally fastest route back to the source. This “wave = optimal control” viewpoint is exactly what Tsitsiklis (1995) made precise from the Hamilton-Jacobi side...compute the value/arrival-time function and the optimal trajectories fall out from it. #FastMarching #EikonalEquation

766,230 просмотров • 7 месяцев назад

The Non-Traversable Bridge Einstein Found Inside Gravity Einstein’s equations not only describe gravity as a force but as the shape of Spacetime as well. In 1935, Einstein and Rosen found that the Schwarzschild solution could be written in a way that reveals something astonishing: Two separate exterior regions of Spacetime joined by a bridge-like throat Rᵤᵥ = 0 This became known as the Einstein-Rosen Bridge. It is often called a Wormhole, but the classical Einstein-Rosen bridge is not a science-fiction tunnel. In the standard Schwarzschild geometry, it is non-traversable. The throat does not remain open long enough for ordinary travel from one region to the other. Its real power is mathematical. It shows that General Relativity is not merely about objects moving through space. The theory allows Spacetime to have global structure such as different regions, horizons, extensions, and hidden connections that are not visible from one local patch alone. The Einstein-Rosen bridge is one of the earliest glimpses of that deeper idea. Gravity is geometry. And sometimes, geometry has two worlds joined by a throat. #Astrophysics #GeneralRelativity #Einstein #BlackHole #Spacetime #Cosmology #Gravity #Wormhole #Mathematics #Physics

The Non-Traversable Bridge Einstein Found Inside Gravity Einstein’s equations not only describe gravity as a force but as the shape of Spacetime as well. In 1935, Einstein and Rosen found that the Schwarzschild solution could be written in a way that reveals something astonishing: Two separate exterior regions of Spacetime joined by a bridge-like throat Rᵤᵥ = 0 This became known as the Einstein-Rosen Bridge. It is often called a Wormhole, but the classical Einstein-Rosen bridge is not a science-fiction tunnel. In the standard Schwarzschild geometry, it is non-traversable. The throat does not remain open long enough for ordinary travel from one region to the other. Its real power is mathematical. It shows that General Relativity is not merely about objects moving through space. The theory allows Spacetime to have global structure such as different regions, horizons, extensions, and hidden connections that are not visible from one local patch alone. The Einstein-Rosen bridge is one of the earliest glimpses of that deeper idea. Gravity is geometry. And sometimes, geometry has two worlds joined by a throat. #Astrophysics #GeneralRelativity #Einstein #BlackHole #Spacetime #Cosmology #Gravity #Wormhole #Mathematics #Physics

91,291 просмотров • 26 дней назад

Black Holes Don’t Simply Suck Things In. They Sort Motion by Geometry. A particle falling toward a black hole is not automatically doomed. Its path depends on how it approaches, the distance, the angle, and the momentum it carries. Some paths cross the horizon. Some skim the edge and whirl around before falling or escaping. Others get sharply deflected and fly back out into space.

Black Holes Don’t Simply Suck Things In. They Sort Motion by Geometry. A particle falling toward a black hole is not automatically doomed. Its path depends on how it approaches, the distance, the angle, and the momentum it carries. Some paths cross the horizon. Some skim the edge and whirl around before falling or escaping. Others get sharply deflected and fly back out into space.

275,186 просмотров • 2 месяцев назад

Meet AETHRA-E...a drone without rotors or wings. Slot microjets embedded in the hull vector a continuous thrust sheet to lift, steer, and stabilize. Edge-blown monocoque: the airframe itself becomes the nozzle. Just shaped thrust. No props. No wings.

Meet AETHRA-E...a drone without rotors or wings. Slot microjets embedded in the hull vector a continuous thrust sheet to lift, steer, and stabilize. Edge-blown monocoque: the airframe itself becomes the nozzle. Just shaped thrust. No props. No wings.

1,044,555 просмотров • 11 месяцев назад

Memory Chip Replacement 🫠🫠🫠🫠🫠🫠

Memory Chip Replacement 🫠🫠🫠🫠🫠🫠

308,054 просмотров • 4 месяцев назад

A realm of my own construct... engineered from math.

A realm of my own construct... engineered from math.

579,450 просмотров • 8 месяцев назад

A White Hole is a Black Hole Running Backward A Black Hole is famous because it lets things fall in, but never lets them come back out. A White Hole is the time-reversed version of that idea. In the full Schwarzschild solution of General Relativity, the Black-Hole region and the White-Hole region appear as two different causal parts of the same exact geometry. Outside the horizon, the metric is the usual Schwarzschild metric ds² = -(1 - 2M/r)dt² + (1 - 2M/r)⁻¹dr² + r²dΩ². The horizon sits at rₛ = 2M, using G = c = 1. For an ordinary Black Hope, future-directed paths can cross this surface inward. Once they do, they cannot return to the outside universe. For a White Hole, the arrow is reversed and particles and light can come out from the horizon, but outside observers cannot enter it. No one has observed a physical White Hole. It may be an ideal solution rather than an object that nature actually forms. But as Mathematics, it is one of the strangest lessons of General Relativity: The equations allow a horizon that throws the universe outward. #WhiteHoles #GeneralRelativity #Schwarzschild #Spacetime #BlackHoles #Cosmology #Physics #Mathematics #Astrophysics #Astronomy

A White Hole is a Black Hole Running Backward A Black Hole is famous because it lets things fall in, but never lets them come back out. A White Hole is the time-reversed version of that idea. In the full Schwarzschild solution of General Relativity, the Black-Hole region and the White-Hole region appear as two different causal parts of the same exact geometry. Outside the horizon, the metric is the usual Schwarzschild metric ds² = -(1 - 2M/r)dt² + (1 - 2M/r)⁻¹dr² + r²dΩ². The horizon sits at rₛ = 2M, using G = c = 1. For an ordinary Black Hope, future-directed paths can cross this surface inward. Once they do, they cannot return to the outside universe. For a White Hole, the arrow is reversed and particles and light can come out from the horizon, but outside observers cannot enter it. No one has observed a physical White Hole. It may be an ideal solution rather than an object that nature actually forms. But as Mathematics, it is one of the strangest lessons of General Relativity: The equations allow a horizon that throws the universe outward. #WhiteHoles #GeneralRelativity #Schwarzschild #Spacetime #BlackHoles #Cosmology #Physics #Mathematics #Astrophysics #Astronomy

44,045 просмотров • 17 дней назад

Aharonov-Bohm Flux Braider A Quantum wave splits around a hidden magnetic flux tube, then recombines with a phase difference that cannot be read from the local magnetic field alone. The core is blocked, the field is trapped near the centre, yet the vector potential A = α(-y, x)/(x² + y² + a²) still changes the phase picked up by paths winding around it. The animation follows i∂ₜψ = 1/2(-i∇ - A)²ψ + Vψ inside an annular interferometer. The density ρ = |ψ|² shows where the wave lives, while the smooth threads trace the probability current j = Im(ψ*∇ψ) - Aρ. Those braided paths are the visual signature of the Aharonov-Bohm effect where the wave feels the hidden flux through phase, not through a classical force. #QuantumMechanics #AharonovBohm #SchrodingerEquation #MathematicalPhysics #QuantumPhase #Wavefunction #PhysicsAnimation #ScientificVisualization #MathematicalArt

Aharonov-Bohm Flux Braider A Quantum wave splits around a hidden magnetic flux tube, then recombines with a phase difference that cannot be read from the local magnetic field alone. The core is blocked, the field is trapped near the centre, yet the vector potential A = α(-y, x)/(x² + y² + a²) still changes the phase picked up by paths winding around it. The animation follows i∂ₜψ = 1/2(-i∇ - A)²ψ + Vψ inside an annular interferometer. The density ρ = |ψ|² shows where the wave lives, while the smooth threads trace the probability current j = Im(ψ*∇ψ) - Aρ. Those braided paths are the visual signature of the Aharonov-Bohm effect where the wave feels the hidden flux through phase, not through a classical force. #QuantumMechanics #AharonovBohm #SchrodingerEquation #MathematicalPhysics #QuantumPhase #Wavefunction #PhysicsAnimation #ScientificVisualization #MathematicalArt

25,867 просмотров • 10 дней назад

Black Hole Can Ring Like a Bell A Black Hole can vibrate after it is disturbed. Instead of settling instantly, its horizon releases gravitational waves in fading tones called quasinormal modes. This is commonly known as Ringdown. Each tone carries information about the black hole’s final mass and spin, which is why physicists call this Black-Hole Spectroscopy. In a way, when Spacetime strikes a Black Hole, the Black Hole answers with its own gravitational sound. #BlackHole #GeneralRelativity #GravitationalWaves #Spacetime #Einstein #Astrophysics #Physics #Mathematics #Cosmology

Black Hole Can Ring Like a Bell A Black Hole can vibrate after it is disturbed. Instead of settling instantly, its horizon releases gravitational waves in fading tones called quasinormal modes. This is commonly known as Ringdown. Each tone carries information about the black hole’s final mass and spin, which is why physicists call this Black-Hole Spectroscopy. In a way, when Spacetime strikes a Black Hole, the Black Hole answers with its own gravitational sound. #BlackHole #GeneralRelativity #GravitationalWaves #Spacetime #Einstein #Astrophysics #Physics #Mathematics #Cosmology

30,456 просмотров • 13 дней назад

What Happens When a Black Hole Does More Than Fall? In General Relativity, a rotating Black Hole is described by the Kerr Metric. Its spin does something strange to spacetime... it drags the geometry around with it. The horizon sits at r₊ = M + √(M² - a²) but outside it there is another surface, the Ergosphere, r_E(θ) = M + √(M² - a²cos²θ) Inside that region, Spacetime is being swept around so strongly that remaining still is no longer possible. Every particle, every light thread, every local observer is forced to rotate with the geometry. For this scene, the C-metric acceleration remains alive in the background and the Black Hole still moves through the fabric, pulled along an axial defect. But the main event is Kerr frame dragging. The fabric twists. The Ergosphere breathes. Light paths shear into spirals. Particles that try to cross the region get carried around before falling inward. #GeneralRelativity #BlackHole #KerrMetric #FrameDragging #Ergosphere #Spacetime #EinsteinEquations #Astrophysics #Physics

What Happens When a Black Hole Does More Than Fall? In General Relativity, a rotating Black Hole is described by the Kerr Metric. Its spin does something strange to spacetime... it drags the geometry around with it. The horizon sits at r₊ = M + √(M² - a²) but outside it there is another surface, the Ergosphere, r_E(θ) = M + √(M² - a²cos²θ) Inside that region, Spacetime is being swept around so strongly that remaining still is no longer possible. Every particle, every light thread, every local observer is forced to rotate with the geometry. For this scene, the C-metric acceleration remains alive in the background and the Black Hole still moves through the fabric, pulled along an axial defect. But the main event is Kerr frame dragging. The fabric twists. The Ergosphere breathes. Light paths shear into spirals. Particles that try to cross the region get carried around before falling inward. #GeneralRelativity #BlackHole #KerrMetric #FrameDragging #Ergosphere #Spacetime #EinsteinEquations #Astrophysics #Physics

26,484 просмотров • 12 дней назад

A Lens That Takes Derivatives US Patent Basis: US8610839B2 - Optical Processing System for Computing Derivatives. In a 4f Optical Processor, the first lens takes the incoming field and forms its Fourier spectrum. At that middle plane, a tiny optical mask multiplies the spectrum by iξ. This is the derivative operator written in Fourier language u(x) -> U(ξ) -> iξU(ξ) -> ∂u/∂x Then the second lens brings the field back to the real space. What comes out is no longer just a focused beam. It is the spatial derivative of the input field, computed by light as it propagates. So, this is the serious promise of optical computing. A physical optical train can perform operations that usually live inside numerical code: differentiation, filtering, convolution, edge detection, correlation, and many other linear transforms.

A Lens That Takes Derivatives US Patent Basis: US8610839B2 - Optical Processing System for Computing Derivatives. In a 4f Optical Processor, the first lens takes the incoming field and forms its Fourier spectrum. At that middle plane, a tiny optical mask multiplies the spectrum by iξ. This is the derivative operator written in Fourier language u(x) -> U(ξ) -> iξU(ξ) -> ∂u/∂x Then the second lens brings the field back to the real space. What comes out is no longer just a focused beam. It is the spatial derivative of the input field, computed by light as it propagates. So, this is the serious promise of optical computing. A physical optical train can perform operations that usually live inside numerical code: differentiation, filtering, convolution, edge detection, correlation, and many other linear transforms.

146,719 просмотров • 2 месяцев назад

A neural network can begin as a flat sheet and learn the shape of hidden data A self-organizing map turns learning into geometry. Each data point pulls one winning neuron toward it, but nearby neurons move too, and so the whole lattice bends without losing its neighborhood structure. The strange part is that the network is not given the roll shape. It discovers the shape through competition and local cooperation. Paper: Self-Organized Formation of Topologically Correct Feature Maps Authors: Teuvo Kohonen Year: 1982

A neural network can begin as a flat sheet and learn the shape of hidden data A self-organizing map turns learning into geometry. Each data point pulls one winning neuron toward it, but nearby neurons move too, and so the whole lattice bends without losing its neighborhood structure. The strange part is that the network is not given the roll shape. It discovers the shape through competition and local cooperation. Paper: Self-Organized Formation of Topologically Correct Feature Maps Authors: Teuvo Kohonen Year: 1982

129,378 просмотров • 2 месяцев назад