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Physics feels stable, predictable, and well-behaved largely because we learned it in 3D. That comfort hides a trap. In 1907, Paul Ehrenfest pointed out something unsettling. If you take the laws we treat as fundamental and transplant them into a different number of spatial dimensions, they often stop working...

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Mathelirium

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A Triple Spring-Mass Pendulum Just for The Fun of Physics We model the mass positions as r₁ = (x₁, y₁), r₂ = (x₂, y₂), r₃ = (x₃, y₃) with y positive downward in the simulation. The three spring segments are r₁ = r₁ − (0,0) r₁₂ = r₂ − r₁ r₂₃ = r₃ − r₂ Each spring applies a Hooke force along its own direction, F(r; k, L) = −k (|r| − L) (r / |r|) and each mass also feels gravity, Fgᵢ = (0, mᵢ g) So the equations of motion are m₁ r̈₁ = Fg₁ + F(r₁; k₁, L₁) − F(r₁₂; k₂, L₂) m₂ r̈₂ = Fg₂ + F(r₁₂; k₂, L₂) − F(r₂₃; k₃, L₃) m₃ r̈₃ = Fg₃ + F(r₂₃; k₃, L₃) We integrate these ODEs forward with RK4 using a small dt. Different scenes come from changing the mass ratios and from giving the third mass a small initial kick. The yellow trail is the path of the third mass. The inset phase portraits show the same motion in state space: (x₃, vₓ₃) and (−y₃, vᵧ₃) When those curves stay tight and loop cleanly, the motion is more regular. When they spread, fold, and smear, energy is being redistributed across the whole chain between gravity, spring stretch, and kinetic motion. #NonlinearDynamics #CoupledOscillators #SpringPendulum #ChaosInMotion #ComputationalPhysics #MathematicalVisualization
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Mathelirium

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💬

472,687 views • 1 year ago

Warmup to Statistical Mechanics What Exactly is a Hamiltonian A System? In ordinary Mechanics, you might begin with position and velocity. Hamiltonian Mechanics rewrites the same motion in a different language. Instead of position and velocity, it uses position and momentum. We write the position variables as q and the momentum variables as p. Then the full state of the system at one instant is (q, p) That pair is one point in phase space. Why do we do this? Because in these variables, the equations of motion take a remarkably clean form. Everything is generated by one single function, the Hamiltonian H(q, p) and in the simplest cases this Hamiltonian is just the total energy written in terms of position and momentum. So if you know H, you know the dynamics. You might wonder, but how can one function generate motion? The rule is dqᵢ/dt = ∂H/∂pᵢ dpᵢ/dt = −∂H/∂qᵢ These are Hamilton’s equations. Now read them slowly 😄 The rate of change of position comes from differentiating H with respect to momentum. The rate of change of momentum comes from differentiating H with respect to position, with a minus sign. This constitutes the whole engine. A simple example makes this less abstract: Take one particle of mass m moving in a potential V(q). Then the Hamiltonian is H(q, p) = p²/(2m) + V(q) The first term is kinetic energy. The second term is potential energy. Now apply Hamilton’s equations. First, dq/dt = ∂H/∂p = p/m So momentum tells you how position changes. Second, dp/dt = −∂H/∂q = −dV/dq Thus, momentum changes because of force. If you now combine these two equations, you recover ordinary Newtonian mechanics. Since p = m dq/dt, we get m d²q/dt² = −dV/dq So, Hamiltonian mechanics is not a different theory. It is the same mechanics, written in a form that exposes its geometric structure much more clearly. The animation The full 3D surface is the Hamiltonian itself, the energy landscape H(q, p). The floor underneath is phase space, marked by energy contours and the local flow field. The bright moving point is one actual state (q(t), p(t)) evolving under Hamilton’s equations. Its trail shows that the motion is not arbitrary. It is guided everywhere by the geometry of the same single function H. The render is doing more than illustrating a particle moving, it is showing how one function organizes the whole phase-space motion. The math breakdown: Start with one degree of freedom. The state is described by position q and momentum p. So the system lives in a two-dimensional phase space with coordinates (q, p) Now choose a Hamiltonian H(q, p) Think of H as the energy function. In many standard systems, H(q, p) = kinetic energy + potential energy For a particle of mass m in a potential V(q), this becomes H(q, p) = p²/(2m) + V(q) Hamilton’s equations say dq/dt = ∂H/∂p dp/dt = −∂H/∂q Now substitute this specific H. First compute the p derivative: ∂H/∂p = ∂/∂p (p²/(2m) + V(q)) = p/m So dq/dt = p/m Now compute the q derivative: ∂H/∂q = ∂/∂q (p²/(2m) + V(q)) = dV/dq So dp/dt = −dV/dq These two first-order equations completely determine the motion. Now, connect this back to Newton’s law. From dq/dt = p/m we get p = m dq/dt Differentiate both sides with respect to time: dp/dt = m d²q/dt² But Hamilton’s second equation gives dp/dt = −dV/dq So , together they imply m d²q/dt² = −dV/dq This is exactly Newton’s second law for motion in the potential V(q). Thus, Hamilton’s equations do not replace mechanic, they reorganize it. #HamiltonianMechanics #PhaseSpace #ClassicalMechanics #MathematicalPhysics #DifferentialEquations #Mathematics #Physics
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Mathelirium

50,370 views • 2 months ago

Quantum Mechanics Series Lecture 4 Lecture 1 established that ρ(x,t) = |ψ(x,t)|² behaves like a conserved probability density. Lecture 2 showed what drives that flow. We also saw that writing ψ = r exp(iθ) makes the probability current proportional to the phase gradient, making it clear that phase geometry literally steers the motion. Lecture 3 then showed that the centroid of that flow can move almost classically when the packet is tight and the external potential is smooth. However, that raises yet another question. If the centroid can look classical, why does the full wave still spread, bend, split, and interfere in ways no classical particle cloud would? This is because the wave is not driven only by the external potential. It is also driven by its own curvature. Write ψ(x,t) = r(x,t) exp(iθ(x,t)) with ρ = r². Then Schrödinger’s equation gives two coupled real equations. One is the continuity equation you already know. The other looks like a Hamilton-Jacobi equation, but with one extra term: Q = −(1/2m) ∇²r / r This is the so-called Quantum Potential. It depends entirely on how the amplitude bends across space. So, the wave is being shaped not only by V(x,t), but also by the geometry of its own envelope. In the animation, the upper surface is still |ψ| and its skin is still colored by arg(ψ). The glowing threads still trace the probability current. But now a second membrane hangs underneath. That lower membrane encodes the quantum potential Q itself. The porcelain bead marks the quantum centroid. The amber bead follows a classical centroid under the same external V. When those paths separate, the lower membrane tells you why. The difference is not magic but the extra term classical mechanics does not have. The math breakdown: Start from Schrödinger evolution in units with ħ = 1: i ∂ψ/∂t = [ −(1/2m) ∇² + V(x,t) ] ψ Write the state in polar form: ψ = r exp(iθ) Then ρ = |ψ|² = r² From the imaginary part, you recover probability conservation: ∂ρ/∂t + ∇·j = 0 with j = (1/m) Im(ψ* ∇ψ) = (ρ/m) ∇θ So the local velocity field is v = j / ρ = ∇θ / m Now take the real part of Schrödinger’s equation. That gives ∂θ/∂t + |∇θ|² / (2m) + V + Q = 0 where Q = −(1/2m) ∇²r / r This is the classical Hamilton-Jacobi equation with one extra term. That extra term is what makes quantum motion locally different from classical motion. Take a gradient of that phase equation and use v = ∇θ / m. Then the flow obeys an Euler-like equation: ∂v/∂t + (v·∇)v = −(1/m) ∇(V + Q) In other words, there are really two forces in the problem. One comes from the external potential V. The other comes from the wave’s own curvature through Q. That is why Ehrenfest is only approximate. The centroid can still satisfy d⟨x⟩/dt = ⟨p⟩/m d⟨p⟩/dt = −⟨∇V⟩ but the internal shape of the packet evolves under the combined influence of V and Q. When the packet stays broad and smooth, Q is gentle and the motion looks more classical. When the packet develops sharp curvature or interference structure, Q becomes strong and the classical picture breaks down. That is what this scene is designed to show live. #QuantumMechanics #Wavefunction #SchrodingerEquation #BornRule #ProbabilityCurrent #ContinuityEquation #Phase #EhrenfestTheorem #QuantumPotential #Madelung #HamiltonJacobi #MathematicalPhysics #Mathematics #Physics
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Quantum Mechanics Series Lecture 4 Lecture 1 established that ρ(x,t) = |ψ(x,t)|² behaves like a conserved probability density. Lecture 2 showed what drives that flow. We also saw that writing ψ = r exp(iθ) makes the probability current proportional to the phase gradient, making it clear that phase geometry literally steers the motion. Lecture 3 then showed that the centroid of that flow can move almost classically when the packet is tight and the external potential is smooth. However, that raises yet another question. If the centroid can look classical, why does the full wave still spread, bend, split, and interfere in ways no classical particle cloud would? This is because the wave is not driven only by the external potential. It is also driven by its own curvature. Write ψ(x,t) = r(x,t) exp(iθ(x,t)) with ρ = r². Then Schrödinger’s equation gives two coupled real equations. One is the continuity equation you already know. The other looks like a Hamilton-Jacobi equation, but with one extra term: Q = −(1/2m) ∇²r / r This is the so-called Quantum Potential. It depends entirely on how the amplitude bends across space. So, the wave is being shaped not only by V(x,t), but also by the geometry of its own envelope. In the animation, the upper surface is still |ψ| and its skin is still colored by arg(ψ). The glowing threads still trace the probability current. But now a second membrane hangs underneath. That lower membrane encodes the quantum potential Q itself. The porcelain bead marks the quantum centroid. The amber bead follows a classical centroid under the same external V. When those paths separate, the lower membrane tells you why. The difference is not magic but the extra term classical mechanics does not have. The math breakdown: Start from Schrödinger evolution in units with ħ = 1: i ∂ψ/∂t = [ −(1/2m) ∇² + V(x,t) ] ψ Write the state in polar form: ψ = r exp(iθ) Then ρ = |ψ|² = r² From the imaginary part, you recover probability conservation: ∂ρ/∂t + ∇·j = 0 with j = (1/m) Im(ψ* ∇ψ) = (ρ/m) ∇θ So the local velocity field is v = j / ρ = ∇θ / m Now take the real part of Schrödinger’s equation. That gives ∂θ/∂t + |∇θ|² / (2m) + V + Q = 0 where Q = −(1/2m) ∇²r / r This is the classical Hamilton-Jacobi equation with one extra term. That extra term is what makes quantum motion locally different from classical motion. Take a gradient of that phase equation and use v = ∇θ / m. Then the flow obeys an Euler-like equation: ∂v/∂t + (v·∇)v = −(1/m) ∇(V + Q) In other words, there are really two forces in the problem. One comes from the external potential V. The other comes from the wave’s own curvature through Q. That is why Ehrenfest is only approximate. The centroid can still satisfy d⟨x⟩/dt = ⟨p⟩/m d⟨p⟩/dt = −⟨∇V⟩ but the internal shape of the packet evolves under the combined influence of V and Q. When the packet stays broad and smooth, Q is gentle and the motion looks more classical. When the packet develops sharp curvature or interference structure, Q becomes strong and the classical picture breaks down. That is what this scene is designed to show live. #QuantumMechanics #Wavefunction #SchrodingerEquation #BornRule #ProbabilityCurrent #ContinuityEquation #Phase #EhrenfestTheorem #QuantumPotential #Madelung #HamiltonJacobi #MathematicalPhysics #Mathematics #Physics

Mathelirium

20,411 views • 2 months ago

Physics is often taught in dry equations but Walter Lewin turns it into an edge of your seat thriller. He proves that the laws of nature are not just suggestions they are absolute. By releasing the ball from exactly 2.5 times the radius of the loop (h = 5/2 R) he demonstrates the perfect balance between gravitational potential energy and the kinetic energy required to maintain centripetal force. Watch as the math comes to life and the ball defies gravity through pure conservation of mechanical energy. This is what happens when you stop memorizing formulas and start witnessing the universe in motion.
0:37

Physics is often taught in dry equations but Walter Lewin turns it into an edge of your seat thriller. He proves that the laws of nature are not just suggestions they are absolute. By releasing the ball from exactly 2.5 times the radius of the loop (h = 5/2 R) he demonstrates the perfect balance between gravitational potential energy and the kinetic energy required to maintain centripetal force. Watch as the math comes to life and the ball defies gravity through pure conservation of mechanical energy. This is what happens when you stop memorizing formulas and start witnessing the universe in motion.

Cosmos Archive

188,300 views • 2 months ago

Plasma isn't just raw energy—it's matter in motion, unbound and electrically alive. But when it self-organizes into a plasmoid—a coherent, toroidal form—it becomes something else entirely: a stable, structured source of energy. This isn’t science fiction. It’s cutting-edge physics… and we’re just beginning to understand its potential.
1:06

Plasma isn't just raw energy—it's matter in motion, unbound and electrically alive. But when it self-organizes into a plasmoid—a coherent, toroidal form—it becomes something else entirely: a stable, structured source of energy. This isn’t science fiction. It’s cutting-edge physics… and we’re just beginning to understand its potential.

Randall Carlson

22,712 views • 1 year ago

Elon Musk: 𝕏 is simply trying to adhere to the law. If the people in a country want the laws to be different, they should make the laws different. "What we're trying to do is simply adhere to the laws in a country. If something is illegal in the United States, or if it's illegal in Europe or Brazil or wherever it might be, then we will take it down, and we'll suspend the account because we're not there to make the laws. But if speech is not illegal, then what are we doing? Now we're injecting ourselves in as a censor. And where does it stop? And who decides? Where does that path lead? I think it leads to a bad place. So, if the people in a country want the laws to be different, they should make the laws different. But otherwise, we're going to obey the law in each jurisdiction. That's it. It's not more complicated than that. We're not trying to flout the law. We're trying to adhere to the law. And if the laws change, we will change. And if the laws don't change, we won't. We're just literally trying to adhere to the law." From: The All-In Podcast, September 9, 2024
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Elon Musk: 𝕏 is simply trying to adhere to the law. If the people in a country want the laws to be different, they should make the laws different. "What we're trying to do is simply adhere to the laws in a country. If something is illegal in the United States, or if it's illegal in Europe or Brazil or wherever it might be, then we will take it down, and we'll suspend the account because we're not there to make the laws. But if speech is not illegal, then what are we doing? Now we're injecting ourselves in as a censor. And where does it stop? And who decides? Where does that path lead? I think it leads to a bad place. So, if the people in a country want the laws to be different, they should make the laws different. But otherwise, we're going to obey the law in each jurisdiction. That's it. It's not more complicated than that. We're not trying to flout the law. We're trying to adhere to the law. And if the laws change, we will change. And if the laws don't change, we won't. We're just literally trying to adhere to the law." From: The All-In Podcast, September 9, 2024

ELON CLIPS

5,539,295 views • 1 year ago

Why String Theory Won't Die The string theory idea refuses to die because it's one of the few frameworks that forces a quantum object and spacetime geometry into the same picture. In the render, a closed string isn’t just a loop doing pretty vibrations. Its internal phase pattern is a quantum degree of freedom, shown as color. At the same time, its motion stirs ripples in the surrounding metric, shown as the fabric below. One object, two roles. A wave-like state living on the string, and a curvature-like response spreading through the background. You can literally watch quantum stuff and gravity stuff talk to each other in the same scene. #StringTheory #Physics #QuantumMechanics #GeneralRelativity #MathematicalPhysics #SciComm
1:00

Why String Theory Won't Die The string theory idea refuses to die because it's one of the few frameworks that forces a quantum object and spacetime geometry into the same picture. In the render, a closed string isn’t just a loop doing pretty vibrations. Its internal phase pattern is a quantum degree of freedom, shown as color. At the same time, its motion stirs ripples in the surrounding metric, shown as the fabric below. One object, two roles. A wave-like state living on the string, and a curvature-like response spreading through the background. You can literally watch quantum stuff and gravity stuff talk to each other in the same scene. #StringTheory #Physics #QuantumMechanics #GeneralRelativity #MathematicalPhysics #SciComm

Mathelirium

13,635 views • 4 months ago

Quantum mechanics has a reputation for being mystical mainly because people skip the rules and jump to interpretations. In this lecture series, we’re doing the opposite. We start from the rules, follow the algebra, and let the picture be the calculation. Classical Probability Theory combines alternatives by adding their probabilities. Quantum Theory combines them one step earlier…add complex amplitudes first, then square at the end. That swap in order is everything. Expand |a₁ + a₂|² and you don’t just get |a₁|² + |a₂|²…you get a cross-term, 2 Re(a₁ a₂*). Its sign is set by phase, so the same two contributions can reinforce or cancel. Interference is just the algebra of squaring a sum. In the 3D render, the surface height is proportional to |a(x)| (so peaks become bright bands after squaring), while the surface skin is colored by the local phase arg(a(x)). As the phase knob φ(t) is swept on path 2, the cross-term oscillates, and you literally watch the interference ridges slide across the screen. We model a detector screen with coordinates x in R² (think x = (x,y)). A quantum state assigns a complex amplitude a(x). The rule for outcomes is p(x) = |a(x)|² Now the key situation: two coherent alternatives contribute to the same outcome x. Let their amplitudes be a₁(x) and a₂(x). Quantum says a(x) = a₁(x) + a₂(x) So the probability density becomes p(x) = |a₁(x) + a₂(x)|² Expand it (this is the whole episode): p(x) = (a₁ + a₂)(a₁* + a₂*) = |a₁|² + |a₂|² + a₁ a₂* + a₁* a₂ = |a₁|² + |a₂|² + 2 Re(a₁ a₂*) That last term is the interference term. It can be positive or negative. To see phase explicitly, write each contribution in polar form: a₁(x) = r₁(x) exp(i θ₁(x)) a₂(x) = r₂(x) exp(i θ₂(x)) Then a₁ a₂* = r₁ r₂ exp(i(θ₁ − θ₂)) So the cross-term is 2 Re(a₁ a₂*) = 2 r₁ r₂ cos(θ₁(x) − θ₂(x)) That’s the fringe engine: p(x) = r₁² + r₂² + 2 r₁ r₂ cos(Δθ(x)) Now the phase knob we animate: Add a controllable phase shift φ to path 2: a₂(x) → a₂(x) exp(i φ) Then Δθ(x) → Δθ(x) − φ, so p(x; φ) = r₁² + r₂² + 2 r₁ r₂ cos(Δθ(x) − φ) As φ changes smoothly, the bright/dark pattern slides continuously. Same setup, same geometry, same magnitudes r₁,r₂, only phase changed. #QuantumMechanics #WaveInterference #ComplexAmplitudes #DoubleSlit #Physics #Mathematics
1:00

Quantum mechanics has a reputation for being mystical mainly because people skip the rules and jump to interpretations. In this lecture series, we’re doing the opposite. We start from the rules, follow the algebra, and let the picture be the calculation. Classical Probability Theory combines alternatives by adding their probabilities. Quantum Theory combines them one step earlier…add complex amplitudes first, then square at the end. That swap in order is everything. Expand |a₁ + a₂|² and you don’t just get |a₁|² + |a₂|²…you get a cross-term, 2 Re(a₁ a₂*). Its sign is set by phase, so the same two contributions can reinforce or cancel. Interference is just the algebra of squaring a sum. In the 3D render, the surface height is proportional to |a(x)| (so peaks become bright bands after squaring), while the surface skin is colored by the local phase arg(a(x)). As the phase knob φ(t) is swept on path 2, the cross-term oscillates, and you literally watch the interference ridges slide across the screen. We model a detector screen with coordinates x in R² (think x = (x,y)). A quantum state assigns a complex amplitude a(x). The rule for outcomes is p(x) = |a(x)|² Now the key situation: two coherent alternatives contribute to the same outcome x. Let their amplitudes be a₁(x) and a₂(x). Quantum says a(x) = a₁(x) + a₂(x) So the probability density becomes p(x) = |a₁(x) + a₂(x)|² Expand it (this is the whole episode): p(x) = (a₁ + a₂)(a₁* + a₂*) = |a₁|² + |a₂|² + a₁ a₂* + a₁* a₂ = |a₁|² + |a₂|² + 2 Re(a₁ a₂*) That last term is the interference term. It can be positive or negative. To see phase explicitly, write each contribution in polar form: a₁(x) = r₁(x) exp(i θ₁(x)) a₂(x) = r₂(x) exp(i θ₂(x)) Then a₁ a₂* = r₁ r₂ exp(i(θ₁ − θ₂)) So the cross-term is 2 Re(a₁ a₂*) = 2 r₁ r₂ cos(θ₁(x) − θ₂(x)) That’s the fringe engine: p(x) = r₁² + r₂² + 2 r₁ r₂ cos(Δθ(x)) Now the phase knob we animate: Add a controllable phase shift φ to path 2: a₂(x) → a₂(x) exp(i φ) Then Δθ(x) → Δθ(x) − φ, so p(x; φ) = r₁² + r₂² + 2 r₁ r₂ cos(Δθ(x) − φ) As φ changes smoothly, the bright/dark pattern slides continuously. Same setup, same geometry, same magnitudes r₁,r₂, only phase changed. #QuantumMechanics #WaveInterference #ComplexAmplitudes #DoubleSlit #Physics #Mathematics

Mathelirium

81,405 views • 6 months ago

Most people learned physics starting with particles. Tiny things. Moving through empty space. Forces pushing them around. Many engineers already know something different. Pixels don’t contain the image. Droplets don’t contain the pattern in a fountain display. Radio circuits don’t contain the music. In every case: individual events → timed in relation → repeating into stable patterns The structure isn’t in the components. It shows up through their coordination. Now follow that one step further. If patterns appear when events coordinate in phase, then what we call an object is a stabilized pattern of relations in a deeper field. That shift flips the usual physics story. Instead of starting with tiny parts and trying to glue the universe together, the demonstration in this video starts from global closure of the field itself. From that closure: • geometry is understood as the field closing on itself • forces and shapes arise as conjugate projections • quantum randomness is understood as information lost at measurement • particles show up as compressed views of coherent field structure Even “gravity” looks different. When geometry is understood as the signature of electromagnetic field closure, gravity is understood as large-scale nested electromagnetic interaction. Space itself is the field’s geometry responding to closure. And the same principle scales everywhere. Atoms close → molecules stabilize. Molecules close → cells stabilize. Each layer becomes the substrate for the next. So here’s the question worth sitting with: How does physics show up when we start from the field itself instead of the particles we measure inside it? 🔗 Link: Zenodo archive with The Grammar of Projection (formal articulation), it's conjugate, The Grammar of Persistence, nearly 500 pages of rigorous canonical derivations starting from global closure, and a couple supplemental documents. The description in the archive indexes the papers well. The recursion holds. 🌀
9:38

Most people learned physics starting with particles. Tiny things. Moving through empty space. Forces pushing them around. Many engineers already know something different. Pixels don’t contain the image. Droplets don’t contain the pattern in a fountain display. Radio circuits don’t contain the music. In every case: individual events → timed in relation → repeating into stable patterns The structure isn’t in the components. It shows up through their coordination. Now follow that one step further. If patterns appear when events coordinate in phase, then what we call an object is a stabilized pattern of relations in a deeper field. That shift flips the usual physics story. Instead of starting with tiny parts and trying to glue the universe together, the demonstration in this video starts from global closure of the field itself. From that closure: • geometry is understood as the field closing on itself • forces and shapes arise as conjugate projections • quantum randomness is understood as information lost at measurement • particles show up as compressed views of coherent field structure Even “gravity” looks different. When geometry is understood as the signature of electromagnetic field closure, gravity is understood as large-scale nested electromagnetic interaction. Space itself is the field’s geometry responding to closure. And the same principle scales everywhere. Atoms close → molecules stabilize. Molecules close → cells stabilize. Each layer becomes the substrate for the next. So here’s the question worth sitting with: How does physics show up when we start from the field itself instead of the particles we measure inside it? 🔗 Link: Zenodo archive with The Grammar of Projection (formal articulation), it's conjugate, The Grammar of Persistence, nearly 500 pages of rigorous canonical derivations starting from global closure, and a couple supplemental documents. The description in the archive indexes the papers well. The recursion holds. 🌀

R. Wade H. Marr

11,296 views • 3 months ago

Physicist Avi Loeb just told me that futuristic technology could enable humanity to travel “faster than the speed of light.” But we would have to move beyond the three dimensions we are familiar with. And he speculated that advanced alien civilizations could already have access to these dimensions. You need to hear his fascinating theory: “Think about living on the surface of a balloon.” “That is two dimensional.” “You might not be aware that there is a third dimension because you are just living on the surface of that balloon.” “If there is another being that is capable of taking advantage of the third dimension, then that being will cross the distance between two points on the surface of the balloon faster than you can imagine.” “Because the travel between the two points can go through the third dimension that connects the two points—not necessarily on the curved surface of the balloon.” “So there are, in principle, possibilities of navigating in more than the dimensions that we are familiar with.” “We are familiar with three spatial dimensions plus time.” “If there are more than three and there is a technological way of taking advantage of those … objects will appear and disappear in ways that we cannot understand.” “Einstein’s theory of relativity states that no material object can move faster than light.” “However, if there are extra dimensions, you might actually travel faster than light in the three dimensions, even though you’re traveling less than the speed of light in the extra dimensions.”
3:09

Physicist Avi Loeb just told me that futuristic technology could enable humanity to travel “faster than the speed of light.” But we would have to move beyond the three dimensions we are familiar with. And he speculated that advanced alien civilizations could already have access to these dimensions. You need to hear his fascinating theory: “Think about living on the surface of a balloon.” “That is two dimensional.” “You might not be aware that there is a third dimension because you are just living on the surface of that balloon.” “If there is another being that is capable of taking advantage of the third dimension, then that being will cross the distance between two points on the surface of the balloon faster than you can imagine.” “Because the travel between the two points can go through the third dimension that connects the two points—not necessarily on the curved surface of the balloon.” “So there are, in principle, possibilities of navigating in more than the dimensions that we are familiar with.” “We are familiar with three spatial dimensions plus time.” “If there are more than three and there is a technological way of taking advantage of those … objects will appear and disappear in ways that we cannot understand.” “Einstein’s theory of relativity states that no material object can move faster than light.” “However, if there are extra dimensions, you might actually travel faster than light in the three dimensions, even though you’re traveling less than the speed of light in the extra dimensions.”

Jan Jekielek

31,984 views • 3 months ago

#HungarianGP 🇭🇺 | Lewis Hamilton post practice: “Today was not a good day, for me. Big, big struggle with the balance of the car. A lot different to the previous years I’ve been here.” “We tried two different things, we try to rectify some of the balance problems we had in FP1. We changed the car in FP2, it’s just very ver inconsistent balance from corner to corner.” “I think it’s probably something to do with tyre temperatures maybe so we’re gonna try and figure that out for tomorrow.”
1:11

#HungarianGP 🇭🇺 | Lewis Hamilton post practice: “Today was not a good day, for me. Big, big struggle with the balance of the car. A lot different to the previous years I’ve been here.” “We tried two different things, we try to rectify some of the balance problems we had in FP1. We changed the car in FP2, it’s just very ver inconsistent balance from corner to corner.” “I think it’s probably something to do with tyre temperatures maybe so we’re gonna try and figure that out for tomorrow.”

deni

55,565 views • 11 months ago

Sometimes the thing that hurts you, isn’t the moment itself, it’s the fact that you keep reliving it. Not once. Not twice. But in patterns that look different outside and feel identical inside. You start wondering why it keeps happening. Why the same fears return. Why the same version of you shows up even when you swore you were done with him. Most people think life moves in a straight line. But the Vishnu Purana says something completely different: “Krita, Treta, Dvapara and Kali, these ages turn like a wheel.” A wheel. Not a timeline. A rotation that keeps circling back until something in you changes. Centuries later, Nietzsche said almost the same thing, his idea of “eternal recurrence.” If you had to live this day again and again, would you be proud of it, or terrified that you’re still choosing the same life? And even modern physics keeps stumbling toward the same shape, cyclic universe models, Big Bangs followed by cosmic “bounces,” universe after universe rising from the ashes of the last. Once you see this, your own life starts making a different kind of sense. You stop blaming fate and start noticing your habits. You stop praying for a new life while living the old one on autopilot. Because you’re not trapped in a cycle. You’re participating in one. So ask yourself, What keeps returning because you haven’t faced it yet? What pattern keeps knocking like a teacher you ignore? What moment repeats because you haven’t shown up differently? In a universe where time moves in circles, even the smallest choice isn’t small. It becomes a pattern. A future. A repetition, or a release. Maybe this is the moment where the wheel comes back around and you finally choose a different version of you. Maybe this is your cycle break.
2:55

Sometimes the thing that hurts you, isn’t the moment itself, it’s the fact that you keep reliving it. Not once. Not twice. But in patterns that look different outside and feel identical inside. You start wondering why it keeps happening. Why the same fears return. Why the same version of you shows up even when you swore you were done with him. Most people think life moves in a straight line. But the Vishnu Purana says something completely different: “Krita, Treta, Dvapara and Kali, these ages turn like a wheel.” A wheel. Not a timeline. A rotation that keeps circling back until something in you changes. Centuries later, Nietzsche said almost the same thing, his idea of “eternal recurrence.” If you had to live this day again and again, would you be proud of it, or terrified that you’re still choosing the same life? And even modern physics keeps stumbling toward the same shape, cyclic universe models, Big Bangs followed by cosmic “bounces,” universe after universe rising from the ashes of the last. Once you see this, your own life starts making a different kind of sense. You stop blaming fate and start noticing your habits. You stop praying for a new life while living the old one on autopilot. Because you’re not trapped in a cycle. You’re participating in one. So ask yourself, What keeps returning because you haven’t faced it yet? What pattern keeps knocking like a teacher you ignore? What moment repeats because you haven’t shown up differently? In a universe where time moves in circles, even the smallest choice isn’t small. It becomes a pattern. A future. A repetition, or a release. Maybe this is the moment where the wheel comes back around and you finally choose a different version of you. Maybe this is your cycle break.

Wisdom Walk

11,296 views • 6 months ago

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
1:00

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

149,346 views • 5 months ago

Reporter: Haven't the circumstances changed and because there is a dire emergency, shouldn't you say we should do something different? If you are in the middle of a ball game and that strategy changes, the coach calls a different play. Why not call a different play right now? Speaker Johnson: The ballgame analogy does not fit because we were not playing a game
1:31

Reporter: Haven't the circumstances changed and because there is a dire emergency, shouldn't you say we should do something different? If you are in the middle of a ball game and that strategy changes, the coach calls a different play. Why not call a different play right now? Speaker Johnson: The ballgame analogy does not fit because we were not playing a game

Acyn

157,687 views • 8 months ago

In 1998, Japanese Mathematician, Engineer and Machine Learning pioneer Shun-ichi Amari published "Natural Gradient Works Efficiently in Learning (722 Cites in Papers and 9 Cites in Patents)" and made a point that still feels fresh even today If your loss is L(θ), then the usual update θ̇ = −∇ L(θ) quietly assumes your parameter space is flat. Amari asked what happens when the model itself has geometry, described by G(θ), the Fisher information. Then the more natural direction becomes θ̇ = −G(θ)⁻¹∇ L(θ). This is the same loss, but with a different notion of distance. As the animation shows, ordinary Gradient Descent follows the raw slope and gets dragged around by distortion, while the natural gradient moves in a way that respects the geometry of the statistical model itself.
0:20

In 1998, Japanese Mathematician, Engineer and Machine Learning pioneer Shun-ichi Amari published "Natural Gradient Works Efficiently in Learning (722 Cites in Papers and 9 Cites in Patents)" and made a point that still feels fresh even today If your loss is L(θ), then the usual update θ̇ = −∇ L(θ) quietly assumes your parameter space is flat. Amari asked what happens when the model itself has geometry, described by G(θ), the Fisher information. Then the more natural direction becomes θ̇ = −G(θ)⁻¹∇ L(θ). This is the same loss, but with a different notion of distance. As the animation shows, ordinary Gradient Descent follows the raw slope and gets dragged around by distortion, while the natural gradient moves in a way that respects the geometry of the statistical model itself.

Mathelirium

45,750 views • 3 months ago

I believe that the interfaces of the future will disappear. Instead of buttons and menus, we will interact with objects. The process itself becomes the feedback. Building software is always a complex craft, but spatial computing is a different beast entirely. It demands a fusion of visual art, logic, and spatial UI where every interaction happens in the air around you. With OPERATOR, built for Apple Vision Pro, we explored this by turning music into a sculpture. The music track isn't a timeline anymore; it’s a 3D object you can walk around, touch, and modify from any angle.
0:21

I believe that the interfaces of the future will disappear. Instead of buttons and menus, we will interact with objects. The process itself becomes the feedback. Building software is always a complex craft, but spatial computing is a different beast entirely. It demands a fusion of visual art, logic, and spatial UI where every interaction happens in the air around you. With OPERATOR, built for Apple Vision Pro, we explored this by turning music into a sculpture. The music track isn't a timeline anymore; it’s a 3D object you can walk around, touch, and modify from any angle.

maxim zhestkov

102,634 views • 4 months ago

Jordan Ott on Stephen Curry's 35-point blast in Warriors win over Clippers. “Looked like the same Steph Curry. You guard him for however many minutes you think you’re guarding him and then all the sudden, the explosion happens." On playing small: "It's always the balance, but I think Deni (Avdija) got in the paint no matter who we had in the game. His ability to drive is pretty unique. Tomorrow is a whole new geometry. It's a different team the way they attack offensively. Similar, they want to shoot a bunch of 3s and how they attack is way different. They run a lot of off-ball screens to free up Curry and then they just react off of his gravity. Tomorrow will be a different challenge and in-game, you always have to be willing to adjust. You have a plan going into it. Got to be willing to adjust. Again, it's all hands on deck." #Suns #DubNation
1:53

Jordan Ott on Stephen Curry's 35-point blast in Warriors win over Clippers. “Looked like the same Steph Curry. You guard him for however many minutes you think you’re guarding him and then all the sudden, the explosion happens." On playing small: "It's always the balance, but I think Deni (Avdija) got in the paint no matter who we had in the game. His ability to drive is pretty unique. Tomorrow is a whole new geometry. It's a different team the way they attack offensively. Similar, they want to shoot a bunch of 3s and how they attack is way different. They run a lot of off-ball screens to free up Curry and then they just react off of his gravity. Tomorrow will be a different challenge and in-game, you always have to be willing to adjust. You have a plan going into it. Got to be willing to adjust. Again, it's all hands on deck." #Suns #DubNation

Duane Rankin

40,348 views • 2 months ago

🎙️ Moussa Diabate: "The team is different. We're moving different." "We understand that we're trying to accomplish something here, and we all know that it's very much still possible. And that is what we're trying to get to. We're gonna use that as a learning experience, and it definitely set the tone for sure." "The Hornets went through a lot, man. People tend to forget we had a lot of injuries last year, a lot of changes, the whole organization changing—just all those things coming in. And now us already making noise? It just shows that we're changing things around here. The culture is different, the players and the whole vision is being different."
1:09

🎙️ Moussa Diabate: "The team is different. We're moving different." "We understand that we're trying to accomplish something here, and we all know that it's very much still possible. And that is what we're trying to get to. We're gonna use that as a learning experience, and it definitely set the tone for sure." "The Hornets went through a lot, man. People tend to forget we had a lot of injuries last year, a lot of changes, the whole organization changing—just all those things coming in. And now us already making noise? It just shows that we're changing things around here. The culture is different, the players and the whole vision is being different."

r/CharlotteHornets

27,071 views • 4 months ago

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
0:56

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

Mathelirium

90,682 views • 7 days ago