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Mathelirium

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This Navier-Stokes simulation shows a vertical-axis wind turbine spinning

This Navier-Stokes simulation shows a vertical-axis wind turbine spinning

24,077 views

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 views

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 views

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 views

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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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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).

Mathelirium

2,027,777 views • 12 days ago

Mathematics always surprises us with how naturally it can model things that feel alive and real. Here, an abstract graph becomes a living Neural Network, carrying signals through branching connections into a central hypergraph.
0:57
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Mathematics always surprises us with how naturally it can model things that feel alive and real. Here, an abstract graph becomes a living Neural Network, carrying signals through branching connections into a central hypergraph.

Mathelirium

26,734 views • 2 days ago

Two spinning Black Holes spiral through a warped Spacetime fabric, dragging it into twisting wells before merging into a single ringing remnant.
0:40
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Two spinning Black Holes spiral through a warped Spacetime fabric, dragging it into twisting wells before merging into a single ringing remnant.

Mathelirium

640,125 views • 1 month ago

Have you seen anything more beautiful?!! 🤩😍🙌🏽 A magnetohydrodynamics (MHD) simulation of solar-wknd compression, magnetic reconnection, plasmoid release, particle acceleration, and auroral activity around Earth.
0:55
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Have you seen anything more beautiful?!! 🤩😍🙌🏽 A magnetohydrodynamics (MHD) simulation of solar-wknd compression, magnetic reconnection, plasmoid release, particle acceleration, and auroral activity around Earth.

Mathelirium

354,225 views • 1 month ago

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
0:59
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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

Mathelirium

789,561 views • 2 months ago

Did you know that a charged particle can actually outrun light? Not light in a vacuum of course. That limit still stands. But inside a dielectric, light travels more slowly. If an electron moves through that material faster than the local phase velocity, something incredible happens... the EM field can no longer rearrange itself smoothly around the particle. A coherent shock front forms. This is known as Cherenkov radiation, named after Russian physicist Pavel Cherenkov who experimentally observed the effect in the 1930s.
0:30
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Did you know that a charged particle can actually outrun light? Not light in a vacuum of course. That limit still stands. But inside a dielectric, light travels more slowly. If an electron moves through that material faster than the local phase velocity, something incredible happens... the EM field can no longer rearrange itself smoothly around the particle. A coherent shock front forms. This is known as Cherenkov radiation, named after Russian physicist Pavel Cherenkov who experimentally observed the effect in the 1930s.

Mathelirium

96,550 views • 11 days ago

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.
0:59
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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

24,101 views • 5 days ago

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.
0:52
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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.

Mathelirium

678,827 views • 5 months ago

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.
0:40
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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.

Mathelirium

35,400 views • 10 days ago

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
0:59
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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

Mathelirium

176,645 views • 1 month ago

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
0:31
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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

Mathelirium

766,230 views • 8 months ago

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.
0:48
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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.

Mathelirium

1,052,168 views • 1 year ago

This is a simulation of a Hydrogen electron described by a superposition of several highly excited Rydberg states. The superposition forms a localized quantum wavepacket that moves around the atom, spreads, splits, and later reforms as the different energy components return to phase.
0:59
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This is a simulation of a Hydrogen electron described by a superposition of several highly excited Rydberg states. The superposition forms a localized quantum wavepacket that moves around the atom, spreads, splits, and later reforms as the different energy components return to phase.

Mathelirium

41,000 views • 13 days ago

Quantum Entanglement is difficult to picture because the correlations belong to the quantum state as a whole, and not to little signals travelling between particles. Here, I represent a many-body quantum state as a changing graph and hypergraph. Pairwise correlations form the luminous links
0:55
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Quantum Entanglement is difficult to picture because the correlations belong to the quantum state as a whole, and not to little signals travelling between particles. Here, I represent a many-body quantum state as a changing graph and hypergraph. Pairwise correlations form the luminous links

Mathelirium

37,668 views • 13 days ago

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.
0:41
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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.

Mathelirium

33,210 views • 12 days ago

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.
1:00
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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.

Mathelirium

275,282 views • 4 months ago

This is Quantum Chaos inside a Bunimovich stadium. The motion is governed by the Schrödinger equation. The whirlpools in the flow are Quantum phase vortices.
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This is Quantum Chaos inside a Bunimovich stadium. The motion is governed by the Schrödinger equation. The whirlpools in the flow are Quantum phase vortices.

Mathelirium

33,450 views • 14 days ago

A realm of my own construct... engineered from math.
3:02
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A realm of my own construct... engineered from math.

Mathelirium

579,450 views • 9 months ago

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.
0:30
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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.

Mathelirium

103,032 views • 1 month ago

Memory Chip Replacement 🫠🫠🫠🫠🫠🫠
0:28
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Memory Chip Replacement 🫠🫠🫠🫠🫠🫠

Mathelirium

308,098 views • 5 months ago