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Mathelirium

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Math, Physics & ML visuals - ex İTÜ & McMaster

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It's nuclear winter and some systems are still launching cruise missiles☠️💀 Mathematically, we solve the incompressible Navier-Stokes equations using an RK2 pressure-projection method, with Brinkman penalization enforcing no-slip flow on the missile.

It's nuclear winter and some systems are still launching cruise missiles☠️💀 Mathematically, we solve the incompressible Navier-Stokes equations using an RK2 pressure-projection method, with Brinkman penalization enforcing no-slip flow on the missile.

31,337 просмотров

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

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

25,928 просмотров

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

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 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).

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).

2,031,426 просмотров • 1 месяц назад

Two Black Holes spiral faster and faster as gravitational waves drain their orbital energy. The final orbit becomes a violent terminal chirp, the horizons merge, spacetime rings down, and one rapidly spinning black hole remains.

Two Black Holes spiral faster and faster as gravitational waves drain their orbital energy. The final orbit becomes a violent terminal chirp, the horizons merge, spacetime rings down, and one rapidly spinning black hole remains.

359,528 просмотров • 19 дней назад

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

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

670,491 просмотров • 3 месяцев назад

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

789,660 просмотров • 3 месяцев назад

A helium atom tearing itself apart. Two electrons circle the nucleus, swapping energy back and forth. Then one drops inward and kicks the other one out. Those rings rolling across the fabric are the escaping electron. Hydrogen can't do this because it only has one electron.

A helium atom tearing itself apart. Two electrons circle the nucleus, swapping energy back and forth. Then one drops inward and kicks the other one out. Those rings rolling across the fabric are the escaping electron. Hydrogen can't do this because it only has one electron.

39,212 просмотров • 6 дней назад

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.

175,921 просмотров • 1 месяц назад

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.

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.

356,130 просмотров • 2 месяцев назад

To think Mathematics did this is incredible! This is a simulation of air flowing through a donut ring, simulated with the lattice Boltzmann method.

To think Mathematics did this is incredible! This is a simulation of air flowing through a donut ring, simulated with the lattice Boltzmann method.

18,949 просмотров • 4 дней назад

A proton at the centre and a probability cloud of an electron with a Rydberg orbit. Just a creative visualization idea that came into my mind.

A proton at the centre and a probability cloud of an electron with a Rydberg orbit. Just a creative visualization idea that came into my mind.

58,312 просмотров • 15 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.

678,919 просмотров • 6 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.

1,052,168 просмотров • 1 год 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

771,360 просмотров • 10 месяцев назад

A spider, a web, and a handful of Mathematical constraints. Each step actually pulls on the silk, redistributes tension, and deforms the network. Geometry, mechanics, simple rules.

A spider, a web, and a handful of Mathematical constraints. Each step actually pulls on the silk, redistributes tension, and deforms the network. Geometry, mechanics, simple rules.

104,044 просмотров • 1 месяц назад

A simulation of Jupiter's magnetosphere, this time using the full equations of magnetohydrodynamics.

A simulation of Jupiter's magnetosphere, this time using the full equations of magnetohydrodynamics.

70,138 просмотров • 1 месяц назад

The best video on X right now, no contest! A Su-57 rolls left, rolls right, then rips a full Cobra maneuver. Simulated with the lattice Boltzmann method... a D3Q19 solver with a Smagorinsky turbulence model at Re = 1,000,000 (I learned after the render that its too low for a fighter jet... should've been between 50-100 mil). This took me 12 hrs😑 10 million tracer particles light up the wake by vorticity.

The best video on X right now, no contest! A Su-57 rolls left, rolls right, then rips a full Cobra maneuver. Simulated with the lattice Boltzmann method... a D3Q19 solver with a Smagorinsky turbulence model at Re = 1,000,000 (I learned after the render that its too low for a fighter jet... should've been between 50-100 mil). This took me 12 hrs😑 10 million tracer particles light up the wake by vorticity.

15,365 просмотров • 6 дней назад

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

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

579,450 просмотров • 11 месяцев назад

The Permanent Echo of Gravity A gravitational wave can pass through the universe and leave Spacetime permanently marked. The wave fades and the shaking stops, but freely falling objects do not return to exactly where they started. The universe keeps the displacement.

The Permanent Echo of Gravity A gravitational wave can pass through the universe and leave Spacetime permanently marked. The wave fades and the shaking stops, but freely falling objects do not return to exactly where they started. The universe keeps the displacement.

37,997 просмотров • 18 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

177,493 просмотров • 3 месяцев назад

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.

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.

97,374 просмотров • 1 месяц назад

The C-metric is an exact solution of Einstein’s vacuum field equations describing an accelerating black hole, driven by the tension of a cosmic string. This animation visualizes the C-metric-inspired geometry, including its black-hole and acceleration horizons.

The C-metric is an exact solution of Einstein’s vacuum field equations describing an accelerating black hole, driven by the tension of a cosmic string. This animation visualizes the C-metric-inspired geometry, including its black-hole and acceleration horizons.

24,112 просмотров • 12 дней назад