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Animation requested by our channel Subscriber @TheShiftReportX The Million-Pendulum Chaos FabricA double pendulum is usually shown as one swinging system with a trail behind it.Here, every pixel is a double pendulum.The horizontal direction is the first starting angleθ₁(0)The vertical direction is the second starting angleθ₂(0)Thus, the square is a dense map of initial conditions(θ₁, θ₂) ∈ S¹ × S¹Each pixel evolves under the full nonlinear double-pendulum system, with state y = (θ₁, ω₁, θ₂, ω₂)T

Animation requested by our channel Subscriber @TheShiftReportX Th…

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Ever Heard of a Metric in Differential Geometry? How About a Geodesic?These two ideas are central to how Mathematics and Physics describe Spacetime.A metric is how a surface measures distance and angle. A geodesic is the path that counts as "straight" after that measuring rule is chosen.Once the metric changes from point to point, straightness is no longer something you judge by eye.On a curved surface, the shortest paths are geodesics. They are chosen by the metric.Take a surface written asr(x,

Ever Heard of a Metric in Differential Geometry? How About a Geodesic…

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Demis Hassabis just told a room full of academics that they’re running out of time.Not the engineers. Not the technologists.The economists. The philosophers. The people who are supposed to understand what a civilization actually is.Hassabis: “It’s very urgent that we really think about the second-order consequences.”He wasn’t talking about the technology.He was talking about everything that comes after it.Hassabis: “I’m always a little bit astounded when I talk to economists about what’s happeni

Demis Hassabis just told a room full of academics that they’re running …

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The Branch-Point LoomSome 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 surr

The Branch-Point LoomSome equations do not draw curves but …

4.6K views1 week ago

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Stokes Phenomenon - When an Invisible Term Switches OnIn 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 functionΦ(z; s) = z³/3 - sz,where both z and s are complex numbers.The saddle points are found fromΦ′(z; s) = z² - s = 0,and thus the two saddles arez₊ = √s, z₋ = -√s.Near each saddle, the asymptotic contribution looks likeexp(Φ(z₊; s)/h) or exp(Φ(z₋; s)/h),where h is

Stokes Phenomenon - When an Invisible Term Switches OnIn Asy…

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The Exact Einstein Solution Where Light Twists GeometryA Gyraton is an exact solution of Einstein’s equations describing a beam of null radiation carrying angular momentum. Think of light with spin... it travels at the speed of light, but it also drags the geometry around its path.This is General Relativity in one of its strangest forms telling us that when light carries enough energy and angular momentum, it becomes part of the gravitational field shaping Spacetime.#GeneralRelativity #EinsteinE

The Exact Einstein Solution Where Light Twists GeometryA Gyraton i…

2.4K views1 week ago

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Faculty of Mathematics | Undergraduate Programs | University of Waterloo

Faculty of Mathematics | Undergraduate Programs | Univer…

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