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The Torus Becomes the Canvas A Jacobi Theta Function lives naturally on a complex torus. Instead of drawing it on a flat plane, we wrapped the mathematics onto the torus. The surface is driven by θ₁(z|τ), with its moving zeros, phase winding, and logarithmic derivative shaping the colour, seams,... show more

Mathelirium

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22,197 次观看 • 2 个月前 •via X (Twitter)

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I took a Jacobi Theta divisor, Θ(z,t) = ∏ⱼ θ₁(z - cⱼ(t) | τ(t)), and let it breathe across a torus, with the small spheres marking its moving zeros... the precise points where Θ(z,t) vanishes.

I took a Jacobi Theta divisor, Θ(z,t) = ∏ⱼ θ₁(z - cⱼ(t) | τ(t)), and let it breathe across a torus, with the small spheres marking its moving zeros... the precise points where Θ(z,t) vanishes.

Mathelirium

42,499 次观看 • 20 天前

Lanterns on the Logarithmic Sea Zeros and poles of a complex rational function R(z,t) drift across a 3D landscape carved from the logarithmic potential U(z,t)=log|R(z,t)|. Every ridge, valley, and glowing thread comes from the moving R(z,t), while the particles follow the tangent field of its level curves. The golden lantern ovals mark the singularities directly. So, the whole scene becomes a living map of where the function vanishes, where it blows up, and how the surrounding geometry bends in response.

Lanterns on the Logarithmic Sea Zeros and poles of a complex rational function R(z,t) drift across a 3D landscape carved from the logarithmic potential U(z,t)=log|R(z,t)|. Every ridge, valley, and glowing thread comes from the moving R(z,t), while the particles follow the tangent field of its level curves. The golden lantern ovals mark the singularities directly. So, the whole scene becomes a living map of where the function vanishes, where it blows up, and how the surrounding geometry bends in response.

Mathelirium

14,941 次观看 • 1 个月前

The Branch-Point Loom Some equations do not draw curves but instead draw weave sheets. This scene comes from the algebraic curve y² = Δ(z,t), where Δ(z,t) = Πⱼ(z - βⱼ(t)). The moving points βⱼ are branch points. These are special places where two sheets of the curve touch and swap identities. In the animation, those branch points appear as glowing mineral spindles moving across a dark surface. The landscape is shaped by -log|Δ(z,t)|. Therefore, the branch points rise into sharp seams while the surrounding surface folds around them. The bright threads follow the horizontal trajectories of the quadratic differential Δ(z,t)dz². #Mathematics #ComplexAnalysis #AlgebraicGeometry #MathArt #MathematicalArt #Geometry #PhysicsVisualization #STEM #Art

The Branch-Point Loom Some equations do not draw curves but instead draw weave sheets. This scene comes from the algebraic curve y² = Δ(z,t), where Δ(z,t) = Πⱼ(z - βⱼ(t)). The moving points βⱼ are branch points. These are special places where two sheets of the curve touch and swap identities. In the animation, those branch points appear as glowing mineral spindles moving across a dark surface. The landscape is shaped by -log|Δ(z,t)|. Therefore, the branch points rise into sharp seams while the surrounding surface folds around them. The bright threads follow the horizontal trajectories of the quadratic differential Δ(z,t)dz². #Mathematics #ComplexAnalysis #AlgebraicGeometry #MathArt #MathematicalArt #Geometry #PhysicsVisualization #STEM #Art

Mathelirium

15,287 次观看 • 1 个月前

This is a living map of a complex rational field: cool regions are root-dominant (|R| 1), where R is the complex rational function. Phase stripes come from arg S (the logarithmic derivative of our complex field R) and particles surf the Newton flow. 🤩🤗😍

This is a living map of a complex rational field: cool regions are root-dominant (|R| 1), where R is the complex rational function. Phase stripes come from arg S (the logarithmic derivative of our complex field R) and particles surf the Newton flow. 🤩🤗😍

Mathelirium

29,237 次观看 • 10 个月前

What you're watching here is a torus cycling through its eigenfunctions...the surface's pure standing-wave modes. An eigenfunction is a mode that, when the surface Laplacian operator acts on it, reproduces itself scaled by a constant...that constant fixes the natural frequency. Each mode is indexed by two counts: how many oscillations wrap the small tube and how many wrap the big ring. Dark rings are nodes (no motion) and the bright bands between them are antinodes. As the counts rise, nodes crowd together, ridges sharpen, and the torus takes on a pine-cone look. #Eigenfunctions #SpectralGeometry #MathArt #Math

What you're watching here is a torus cycling through its eigenfunctions...the surface's pure standing-wave modes. An eigenfunction is a mode that, when the surface Laplacian operator acts on it, reproduces itself scaled by a constant...that constant fixes the natural frequency. Each mode is indexed by two counts: how many oscillations wrap the small tube and how many wrap the big ring. Dark rings are nodes (no motion) and the bright bands between them are antinodes. As the counts rise, nodes crowd together, ridges sharpen, and the torus takes on a pine-cone look. #Eigenfunctions #SpectralGeometry #MathArt #Math

Mathelirium

54,411 次观看 • 9 个月前

STAR TRAIL TIME LAPSE AT THE EQUATOR The stars reveal the unseen forever swirling etheric vortex of the torus field, which encompasses our stationary earth plane. On the right, stars rising up = EAST. On the left, stars flowing down into horizon = WEST. The middle point between = NORTH. Edges where they wrap around = SOUTH "If the universe is a torus field—then the universe is a sound wave because the torus is formed by a vibration at its core. That vibration makes a sound. And that wave illuminates the ether all around us." ~ Nicole Murphy ✨🙌🏾💫

STAR TRAIL TIME LAPSE AT THE EQUATOR The stars reveal the unseen forever swirling etheric vortex of the torus field, which encompasses our stationary earth plane. On the right, stars rising up = EAST. On the left, stars flowing down into horizon = WEST. The middle point between = NORTH. Edges where they wrap around = SOUTH "If the universe is a torus field—then the universe is a sound wave because the torus is formed by a vibration at its core. That vibration makes a sound. And that wave illuminates the ether all around us." ~ Nicole Murphy ✨🙌🏾💫

🧬Maxpein🧬

13,701 次观看 • 25 天前

Pseudospectral Mirage of Matrices We usually judge a matrix by its eigenvalues, but sometimes the eigenvalues are only the bait. The real monster is hiding in the pseudospectrum. Most of the time, a matrix is shown as a few points in the complex plane called eigenvalues. But for some matrices, especially non-normal matrices, those few points do not tell the whole story. In this scene, the complex plane becomes a moving landscape. The bright anchors are the true eigenvalues. Around them, huge pseudospectral continents grow, split, and collapse as the matrix changes. The moving threads show where the matrix is most sensitive. In those regions, even a tiny change can create a huge response. This is the hidden instability of a matrix made visible. Algebra becomes a pressure map and the pseudospectrum becomes terrain. #Mathematics #LinearAlgebra #SpectralTheory #NumericalAnalysis #MatrixTheory #MathArt #MathematicalArt #STEM #Art

Pseudospectral Mirage of Matrices We usually judge a matrix by its eigenvalues, but sometimes the eigenvalues are only the bait. The real monster is hiding in the pseudospectrum. Most of the time, a matrix is shown as a few points in the complex plane called eigenvalues. But for some matrices, especially non-normal matrices, those few points do not tell the whole story. In this scene, the complex plane becomes a moving landscape. The bright anchors are the true eigenvalues. Around them, huge pseudospectral continents grow, split, and collapse as the matrix changes. The moving threads show where the matrix is most sensitive. In those regions, even a tiny change can create a huge response. This is the hidden instability of a matrix made visible. Algebra becomes a pressure map and the pseudospectrum becomes terrain. #Mathematics #LinearAlgebra #SpectralTheory #NumericalAnalysis #MatrixTheory #MathArt #MathematicalArt #STEM #Art

Mathelirium

29,171 次观看 • 1 个月前

Stokes Phenomenon - When an Invisible Term Switches On In Asymptotic Mathematics, an expression can hide terms that are almost invisible, then suddenly make them appear when you cross a special direction. Here, we visualize the phase function Φ(z; s) = z³/3 - sz, where both z and s are complex numbers. The saddle points are found from Φ′(z; s) = z² - s = 0, and thus the two saddles are z₊ = √s, z₋ = -√s. Near each saddle, the asymptotic contribution looks like exp(Φ(z₊; s)/h) or exp(Φ(z₋; s)/h), where h is a small parameter. The Stokes condition is given by Im(Φ(z₊; s) - Φ(z₋; s)) = 0. This is the direction where the two exponential phases align. Across that line, one saddle contribution switches on or off. The glowing curves are steepest-descent paths. Along them, Im Φ = constant, while Re Φ decreases. This scene is the hidden switchboard of Asymptotic Analysis... two saddles, invisible exponentials, and a glowing Stokes line where a term suddenly becomes real. #Mathematics #Asymptotics #ComplexAnalysis #StokesPhenomenon #MathematicalPhysics #AppliedMathematics #ScientificVisualization #MathArt #STEM

Stokes Phenomenon - When an Invisible Term Switches On In Asymptotic Mathematics, an expression can hide terms that are almost invisible, then suddenly make them appear when you cross a special direction. Here, we visualize the phase function Φ(z; s) = z³/3 - sz, where both z and s are complex numbers. The saddle points are found from Φ′(z; s) = z² - s = 0, and thus the two saddles are z₊ = √s, z₋ = -√s. Near each saddle, the asymptotic contribution looks like exp(Φ(z₊; s)/h) or exp(Φ(z₋; s)/h), where h is a small parameter. The Stokes condition is given by Im(Φ(z₊; s) - Φ(z₋; s)) = 0. This is the direction where the two exponential phases align. Across that line, one saddle contribution switches on or off. The glowing curves are steepest-descent paths. Along them, Im Φ = constant, while Re Φ decreases. This scene is the hidden switchboard of Asymptotic Analysis... two saddles, invisible exponentials, and a glowing Stokes line where a term suddenly becomes real. #Mathematics #Asymptotics #ComplexAnalysis #StokesPhenomenon #MathematicalPhysics #AppliedMathematics #ScientificVisualization #MathArt #STEM

Mathelirium

20,929 次观看 • 1 个月前

Phase Space Is the Real Stage of Dynamics. So, Stop Watching the Object and Watch the State. Ordinary Space tells you where something is. Phase Space tells you what state it is in. Position alone is not enough because you also need the variable that tells you where the system is trying to go. In the animation, the left panel shows the particle moving in Ordinary Space, while the right panel shows the same system as a moving point in Phase Space. For one degree of freedom, the state is (x, p) and as time evolves, that state traces a trajectory t ↦ (x(t), p(t)) Instead of thinking of a differential equation as just a formula, Phase Space lets you see it as a flow on the space of states. For this lecture we use ẋ = p ṗ = x − x³ with Hamiltonian H(x,p) = 0.5 p² + 0.25 x⁴ − 0.5 x² #PhaseSpace #DynamicalSystems #NonlinearDynamics #HamiltonianMechanics #Mathematics

Phase Space Is the Real Stage of Dynamics. So, Stop Watching the Object and Watch the State. Ordinary Space tells you where something is. Phase Space tells you what state it is in. Position alone is not enough because you also need the variable that tells you where the system is trying to go. In the animation, the left panel shows the particle moving in Ordinary Space, while the right panel shows the same system as a moving point in Phase Space. For one degree of freedom, the state is (x, p) and as time evolves, that state traces a trajectory t ↦ (x(t), p(t)) Instead of thinking of a differential equation as just a formula, Phase Space lets you see it as a flow on the space of states. For this lecture we use ẋ = p ṗ = x − x³ with Hamiltonian H(x,p) = 0.5 p² + 0.25 x⁴ − 0.5 x² #PhaseSpace #DynamicalSystems #NonlinearDynamics #HamiltonianMechanics #Mathematics

Mathelirium

30,340 次观看 • 4 个月前

$The plane we live in is all about syncretism — Everything is connected, from light to language, from electromagnetic waves to sacred geometry, from your torus field to the movement of the sun and stars. They told you science and spirituality were separate. But in truth… it's all one system. ⚡ The Earth is electromagnetic 💧 You are water and vibration 🌈 Light is frequency and color 🧬 DNA is a fractal antenna 🌪️ The Torus is the key to it all These aren’t random sketches… They’re codes of the universe — hidden in plain sight.$

The plane we live in is all about syncretism — Everything is connected, from light to language, from electromagnetic waves to sacred geometry, from your torus field to the movement of the sun and stars. They told you science and spirituality were separate. But in truth… it's all one system. ⚡ The Earth is electromagnetic 💧 You are water and vibration 🌈 Light is frequency and color 🧬 DNA is a fractal antenna 🌪️ The Torus is the key to it all These aren’t random sketches… They’re codes of the universe — hidden in plain sight.

vegastar

30,055 次观看 • 1 年前

String Theory Lecture 2 In Conformal Gauge, the String Becomes a Wave Equation Episode 1 showed the geometric jump point particle -> worldline string -> worldsheet Episode 2 is the dynamical jump. The Nambu-Goto action measures the area of the worldsheet, S = −T ∫ dτ dσ √[−det hₐᵦ] but the square-root determinant is awkward to work with. So we usually rewrite the same classical theory in Polyakov form, S = −(T/2) ∫ dτ dσ √[−γ] γᵃᵇ ∂ₐXᵘ ∂ᵦXᵤ Here Xᵘ(τ,σ) tells us where each point of the string’s worldsheet sits in spacetime, and γₐᵦ is the metric we put on the worldsheet. The power of this form is that we can choose a convenient gauge. In conformal gauge, the equations of motion simplify to (∂²/∂τ² − ∂²/∂σ²) Xᵘ = 0 So the string’s spacetime coordinates behave like waves living on the worldsheet. The τ-derivative measures how the string changes in worldsheet time. The σ-derivative measures how it bends along its own length. For a closed string, σ is periodic Xᵘ(τ, σ + 2π) = Xᵘ(τ,σ) and the wave equation splits into two traveling pieces, Xᵘ(τ,σ) = Fᵘ(τ + σ) + Gᵘ(τ − σ) One family moves one way around the string and the other moves the opposite way. These are the left-moving and right-moving modes. In the render, the bright loop is the string at the present moment. The glowing cylinder behind it is the worldsheet it has swept out. The cyan curves trace one traveling family, and the gold curves trace the other. They are the visual version of τ + σ and τ − σ. Therefore, the theory has an internal wave equation, and its normal modes are the raw material for the string spectrum. #StringTheory #TheoreticalPhysics #ConformalGauge #Worldsheet #WaveEquation #Physics #Mathematics #MathematicalPhysics #QuantumGravity #ScienceVisuals

String Theory Lecture 2 In Conformal Gauge, the String Becomes a Wave Equation Episode 1 showed the geometric jump point particle -> worldline string -> worldsheet Episode 2 is the dynamical jump. The Nambu-Goto action measures the area of the worldsheet, S = −T ∫ dτ dσ √[−det hₐᵦ] but the square-root determinant is awkward to work with. So we usually rewrite the same classical theory in Polyakov form, S = −(T/2) ∫ dτ dσ √[−γ] γᵃᵇ ∂ₐXᵘ ∂ᵦXᵤ Here Xᵘ(τ,σ) tells us where each point of the string’s worldsheet sits in spacetime, and γₐᵦ is the metric we put on the worldsheet. The power of this form is that we can choose a convenient gauge. In conformal gauge, the equations of motion simplify to (∂²/∂τ² − ∂²/∂σ²) Xᵘ = 0 So the string’s spacetime coordinates behave like waves living on the worldsheet. The τ-derivative measures how the string changes in worldsheet time. The σ-derivative measures how it bends along its own length. For a closed string, σ is periodic Xᵘ(τ, σ + 2π) = Xᵘ(τ,σ) and the wave equation splits into two traveling pieces, Xᵘ(τ,σ) = Fᵘ(τ + σ) + Gᵘ(τ − σ) One family moves one way around the string and the other moves the opposite way. These are the left-moving and right-moving modes. In the render, the bright loop is the string at the present moment. The glowing cylinder behind it is the worldsheet it has swept out. The cyan curves trace one traveling family, and the gold curves trace the other. They are the visual version of τ + σ and τ − σ. Therefore, the theory has an internal wave equation, and its normal modes are the raw material for the string spectrum. #StringTheory #TheoreticalPhysics #ConformalGauge #Worldsheet #WaveEquation #Physics #Mathematics #MathematicalPhysics #QuantumGravity #ScienceVisuals

Mathelirium

15,607 次观看 • 3 个月前

String Theory Lecture 1 A String Does Not Move Like a Point A point particle traces a line through spacetime. A string traces a surface. This is the first geometric shift in String Theory. Particle mechanics asks where one object is at time t, so its history is a curve. String Theory asks where every point of an extended object is at worldsheet time τ, so we need another coordinate telling us where we are along the string. For a point particle x(t) So, for one input of time we get a position in Spacetime. For a string Xᵘ(τ,σ) Here τ plays the role of time on the worldsheet, while σ labels position along the string. Freeze τ and vary σ, and you see the string at one instant. Let τ move, and that curve sweeps out a two-dimensional surface... the worldsheet. The same comparison appears in the action. For a relativistic point particle, the geometric action measures worldline length S = −m ∫ ds If we parameterize the path by t, the action has one integral, one parameter, and one tangent vector dxᵘ/dt For a string, the same idea grows by one dimension. The action measures area, not length. In Nambu-Goto form, S = −T ∫ dτ dσ √[−det hₐᵦ] Here T is the string tension. It plays a role similar to mass, but for an extended object. It weights the area of a surface rather than the length of a line. The particle action has ∫ dt because the history is one-dimensional. The string action has ∫ dτ dσ because the history is two-dimensional. We are no longer summing along a path, we are summing over a surface. The geometry changes for the same reason. For the particle, one derivative is enough dxᵘ/dt For the string, the geometry is built from two derivatives: ∂τXᵘ and ∂σXᵘ The first tells you how the string changes as worldsheet time flows. The second tells you how the embedding changes as you move along the string. Together they define the induced worldsheet metric hₐᵦ = ∂ₐXᵘ ∂ᵦXᵤ In plain terms, hₐᵦ measures tangent lengths and tangent angles on the worldsheet. From it, the area element is dA = dτ dσ √[−det hₐᵦ] This, the Nambu-Goto action is the direct analogue of the point-particle length action. The point particle extremizes length and the string extremizes area. For calculations, people usually switch to the Polyakov action: S = −(T/2) ∫ dτ dσ √[−γ] γᵃᵇ ∂ₐXᵘ ∂ᵦXᵤ This describes the same classical string dynamics, but the algebra is cleaner. After choosing conformal gauge, varying with respect to Xᵘ gives (∂²/∂τ² − ∂²/∂σ²) Xᵘ = 0 This is the first real dynamical payoff... a two-dimensional wave equation on the worldsheet. For a point particle, the equation of motion tells you how one position evolves along one path. For a string, it tells you how an entire curve evolves, with waves traveling along it. The term ∂²Xᵘ/∂τ² measures acceleration in worldsheet time, while ∂²Xᵘ/∂σ² measures curvature along the string. The time evolution is balanced by how the string bends along its own length. This is why strings have oscillation modes. A point particle has one trajectory. A string has many possible vibration patterns, each one a normal mode of the worldsheet wave equation. For a closed string, σ wraps around the loop Xᵘ(τ, σ + 2π) = Xᵘ(τ, σ) For an open string, one standard free-end condition is ∂σXᵘ = 0 at the endpoints. Solving the wave equation gives waves moving in opposite directions along the string Xᵘ(τ,σ) = Fᵘ(τ + σ) + Gᵘ(τ − σ) A function of τ + σ moves one way. A function of τ − σ moves the other. Therefore, a particle has a worldline, its action measures length, and its geometry uses one tangent. The string has a worldsheet, its action measures area, and its geometry uses two tangent directions. #StringTheory #TheoreticalPhysics #MathematicalPhysics #Physics #Spacetime

String Theory Lecture 1 A String Does Not Move Like a Point A point particle traces a line through spacetime. A string traces a surface. This is the first geometric shift in String Theory. Particle mechanics asks where one object is at time t, so its history is a curve. String Theory asks where every point of an extended object is at worldsheet time τ, so we need another coordinate telling us where we are along the string. For a point particle x(t) So, for one input of time we get a position in Spacetime. For a string Xᵘ(τ,σ) Here τ plays the role of time on the worldsheet, while σ labels position along the string. Freeze τ and vary σ, and you see the string at one instant. Let τ move, and that curve sweeps out a two-dimensional surface... the worldsheet. The same comparison appears in the action. For a relativistic point particle, the geometric action measures worldline length S = −m ∫ ds If we parameterize the path by t, the action has one integral, one parameter, and one tangent vector dxᵘ/dt For a string, the same idea grows by one dimension. The action measures area, not length. In Nambu-Goto form, S = −T ∫ dτ dσ √[−det hₐᵦ] Here T is the string tension. It plays a role similar to mass, but for an extended object. It weights the area of a surface rather than the length of a line. The particle action has ∫ dt because the history is one-dimensional. The string action has ∫ dτ dσ because the history is two-dimensional. We are no longer summing along a path, we are summing over a surface. The geometry changes for the same reason. For the particle, one derivative is enough dxᵘ/dt For the string, the geometry is built from two derivatives: ∂τXᵘ and ∂σXᵘ The first tells you how the string changes as worldsheet time flows. The second tells you how the embedding changes as you move along the string. Together they define the induced worldsheet metric hₐᵦ = ∂ₐXᵘ ∂ᵦXᵤ In plain terms, hₐᵦ measures tangent lengths and tangent angles on the worldsheet. From it, the area element is dA = dτ dσ √[−det hₐᵦ] This, the Nambu-Goto action is the direct analogue of the point-particle length action. The point particle extremizes length and the string extremizes area. For calculations, people usually switch to the Polyakov action: S = −(T/2) ∫ dτ dσ √[−γ] γᵃᵇ ∂ₐXᵘ ∂ᵦXᵤ This describes the same classical string dynamics, but the algebra is cleaner. After choosing conformal gauge, varying with respect to Xᵘ gives (∂²/∂τ² − ∂²/∂σ²) Xᵘ = 0 This is the first real dynamical payoff... a two-dimensional wave equation on the worldsheet. For a point particle, the equation of motion tells you how one position evolves along one path. For a string, it tells you how an entire curve evolves, with waves traveling along it. The term ∂²Xᵘ/∂τ² measures acceleration in worldsheet time, while ∂²Xᵘ/∂σ² measures curvature along the string. The time evolution is balanced by how the string bends along its own length. This is why strings have oscillation modes. A point particle has one trajectory. A string has many possible vibration patterns, each one a normal mode of the worldsheet wave equation. For a closed string, σ wraps around the loop Xᵘ(τ, σ + 2π) = Xᵘ(τ, σ) For an open string, one standard free-end condition is ∂σXᵘ = 0 at the endpoints. Solving the wave equation gives waves moving in opposite directions along the string Xᵘ(τ,σ) = Fᵘ(τ + σ) + Gᵘ(τ − σ) A function of τ + σ moves one way. A function of τ − σ moves the other. Therefore, a particle has a worldline, its action measures length, and its geometry uses one tangent. The string has a worldsheet, its action measures area, and its geometry uses two tangent directions. #StringTheory #TheoreticalPhysics #MathematicalPhysics #Physics #Spacetime

Mathelirium

31,560 次观看 • 3 个月前

A Hamiltonian system is a way of describing motion where position and momentum evolve together as one coupled system. The plot shows an energy landscape in phase space, with the motion of the system traced directly on top of it and projected onto the underlying phase portrait. #HamiltonianSystems #PhaseSpace #PhysicsSimulation #DynamicalSystems #MathematicalPhysics

A Hamiltonian system is a way of describing motion where position and momentum evolve together as one coupled system. The plot shows an energy landscape in phase space, with the motion of the system traced directly on top of it and projected onto the underlying phase portrait. #HamiltonianSystems #PhaseSpace #PhysicsSimulation #DynamicalSystems #MathematicalPhysics

Mathelirium

35,799 次观看 • 2 个月前

Look at the projection on real numbers of your beloved curve taking a ride on its complex projective torus Anon 🤠 The colors identifies the different parts of the moving loop What a mesmeric journey ! It goes inside out in the middle 😵‍💫 3D version for more details soon™

Look at the projection on real numbers of your beloved curve taking a ride on its complex projective torus Anon 🤠 The colors identifies the different parts of the moving loop What a mesmeric journey ! It goes inside out in the middle 😵‍💫 3D version for more details soon™

Pantamis

14,067 次观看 • 3 年前

You're watching a Quadratic-Differential Teichmüller flow unfold. We drive the scene with a time-varying quadratic differential built from moving zeros and poles. Its log-magnitude sets the color, and its phase defines the horizontal foliation (the direction is half the phase). Then a Teichmüller stretch biases things along vs. against that foliation...so the surface deforms in a geodesic-like way. 🤩🤗🫠

You're watching a Quadratic-Differential Teichmüller flow unfold. We drive the scene with a time-varying quadratic differential built from moving zeros and poles. Its log-magnitude sets the color, and its phase defines the horizontal foliation (the direction is half the phase). Then a Teichmüller stretch biases things along vs. against that foliation...so the surface deforms in a geodesic-like way. 🤩🤗🫠

Mathelirium

11,938 次观看 • 9 个月前

Here's the rotating movement of the stars on our flat Earth. It's an electromagnetic torus field with a plane of inertia in the middle. That's how the universe works! Believing that you live on a spinning globe traveling in space and chasing the sun, is their biggest mockery of your intelligence. I believe that one day this truth will be known and clear to all of humanity.

Here's the rotating movement of the stars on our flat Earth. It's an electromagnetic torus field with a plane of inertia in the middle. That's how the universe works! Believing that you live on a spinning globe traveling in space and chasing the sun, is their biggest mockery of your intelligence. I believe that one day this truth will be known and clear to all of humanity.

vegastar

300,095 次观看 • 2 年前

Have you ever wondered how Starlink can keep talking to satellites moving across the sky? It uses a Phased-Array Antenna. It has many small antenna elements on a flat surface, and each one is given a tiny phase delay so the radio wave adds up in one chosen direction. That lets the beam steer electronically and stay locked onto a fast-moving satellite. In this animation, the beam pattern was generated using computational electromagnetics, by computing how the array’s directional pattern changes as those phases are updated.

Have you ever wondered how Starlink can keep talking to satellites moving across the sky? It uses a Phased-Array Antenna. It has many small antenna elements on a flat surface, and each one is given a tiny phase delay so the radio wave adds up in one chosen direction. That lets the beam steer electronically and stay locked onto a fast-moving satellite. In this animation, the beam pattern was generated using computational electromagnetics, by computing how the array’s directional pattern changes as those phases are updated.

Mathelirium

42,442 次观看 • 4 个月前

2D → 3D → 4D → Beyond? Is GEOMETRY the language of ENERGY? 2D, Pi (π) and Tau (τ) map the CIRCLE... division (π) and completion (τ). Duality and Wholeness. 3D, the circle becomes the SPHERE... the Yin-Yang begins to move, spiral, and flow. The TORUS (TAUrus) forms: a self-sustaining field of energy. 4D, we reach the KLEIN BOTTLE and Möbius Strip... strange, looping topologies where inside and outside become ONE. Unity beyond opposites. And beyond? In what we might call 5D, sacred geometry emerges... the Flower of Life, STAR tetrahedrons - patterns encoding the structure of LIGHT, consciousness, and space-time itself. 🌀 Beginning as a simple CIRCLE... becoming the architecture of the universe?

2D → 3D → 4D → Beyond? Is GEOMETRY the language of ENERGY? 2D, Pi (π) and Tau (τ) map the CIRCLE... division (π) and completion (τ). Duality and Wholeness. 3D, the circle becomes the SPHERE... the Yin-Yang begins to move, spiral, and flow. The TORUS (TAUrus) forms: a self-sustaining field of energy. 4D, we reach the KLEIN BOTTLE and Möbius Strip... strange, looping topologies where inside and outside become ONE. Unity beyond opposites. And beyond? In what we might call 5D, sacred geometry emerges... the Flower of Life, STAR tetrahedrons - patterns encoding the structure of LIGHT, consciousness, and space-time itself. 🌀 Beginning as a simple CIRCLE... becoming the architecture of the universe?

WhatdoIknow

90,633 次观看 • 1 年前

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

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

81,501 次观看 • 7 个月前

The Field Learns to Sew Itself This animation uses a moving quadratic differential q(z,t)dz², where zeros and double poles steer thousands of particles along the field’s horizontal trajectories, turning the complex plane into a living fabric of mathematically guided threads.

The Field Learns to Sew Itself This animation uses a moving quadratic differential q(z,t)dz², where zeros and double poles steer thousands of particles along the field’s horizontal trajectories, turning the complex plane into a living fabric of mathematically guided threads.

Mathelirium

10,952 次观看 • 2 个月前