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The Hopf fibration maps a 4D hypersphere onto a regular sphere using circles that never touch but link through each other. It's the same structure that describes qubit states on the Bloch sphere - the geometry every quantum computer runs on. Most people think four dimensions are impossible to... show more

Eron

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14,881 次观看 • 21 天前 •via X (Twitter)

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Heinz Hopf discovered this in 1931. Roger Penrose called it "an element of the architecture of our world." Eric Weinstein brought it up on Joe Rogan - and the silence in the room said everything. The Hopf fibration maps a 4D hypersphere onto a regular sphere using circles that never intersect but each links through every other exactly once. It shows up in at least 8 areas of physics - including the Bloch sphere geometry that every qubit in a quantum computer lives on. 6 minutes. Bookmark & watch today. The most important shape nobody talks about.

Heinz Hopf discovered this in 1931. Roger Penrose called it "an element of the architecture of our world." Eric Weinstein brought it up on Joe Rogan - and the silence in the room said everything. The Hopf fibration maps a 4D hypersphere onto a regular sphere using circles that never intersect but each links through every other exactly once. It shows up in at least 8 areas of physics - including the Bloch sphere geometry that every qubit in a quantum computer lives on. 6 minutes. Bookmark & watch today. The most important shape nobody talks about.

Eron

120,593 次观看 • 20 天前

Hamilton spent 10 years trying to build a 3D number system. On October 16, 1843, walking across a bridge in Dublin, he realized he needed 4 dimensions instead - and carved the equation into the stone on the spot. Quaternions are what every robotics system, 3D game engine, and LLM positional encoding falls back on when rotation math gets hard. This video builds the full 4D geometric intuition using stereographic projection - the same trick that maps a sphere onto a flat screen. 30 minutes. Bookmark & watch today. The Alice in Wonderland connection alone is worth it.

Hamilton spent 10 years trying to build a 3D number system. On October 16, 1843, walking across a bridge in Dublin, he realized he needed 4 dimensions instead - and carved the equation into the stone on the spot. Quaternions are what every robotics system, 3D game engine, and LLM positional encoding falls back on when rotation math gets hard. This video builds the full 4D geometric intuition using stereographic projection - the same trick that maps a sphere onto a flat screen. 30 minutes. Bookmark & watch today. The Alice in Wonderland connection alone is worth it.

Eron

158,073 次观看 • 1 个月前

Every quantum state in the universe is a vector - not in 3D space, but in a complex space with potentially infinite dimensions where the sum of squared coefficients always equals exactly 1. That constraint is the same math LLMs use to normalize probability distributions over tokens. A wavefunction collapsing to a measurement and a softmax choosing the next word are the same operation - projecting an infinite-dimensional vector onto a basis. Claude and GPT do quantum math on every forward pass. 14 minutes. Bookmark & watch today. Quantum mechanics and LLMs speak the same language.

Every quantum state in the universe is a vector - not in 3D space, but in a complex space with potentially infinite dimensions where the sum of squared coefficients always equals exactly 1. That constraint is the same math LLMs use to normalize probability distributions over tokens. A wavefunction collapsing to a measurement and a softmax choosing the next word are the same operation - projecting an infinite-dimensional vector onto a basis. Claude and GPT do quantum math on every forward pass. 14 minutes. Bookmark & watch today. Quantum mechanics and LLMs speak the same language.

Eron

18,166 次观看 • 23 天前

Geometry 🤝 Topology What you're seeing here is called Hopf Fibration. In pure-math terms, S³ (the unit sphere in 4D) can be decomposed into circles called fibers. The Hopf map sends each of those circles to a single point on S². So you don’t fill S³ with random loops, you fill it with a perfectly organized family of circles, one circle sitting above each point of S². For the animation, we sample thousands of fibers on S³, apply a smooth SO(4) rotation (a real rotation in 4D, not a 3D cheat), then stereographically project into R³ so we can actually draw it. Pure mathematics is beautiful alz #HopfFibration #Topology #Geometry #StereographicProjection #PureMathematics

Geometry 🤝 Topology What you're seeing here is called Hopf Fibration. In pure-math terms, S³ (the unit sphere in 4D) can be decomposed into circles called fibers. The Hopf map sends each of those circles to a single point on S². So you don’t fill S³ with random loops, you fill it with a perfectly organized family of circles, one circle sitting above each point of S². For the animation, we sample thousands of fibers on S³, apply a smooth SO(4) rotation (a real rotation in 4D, not a 3D cheat), then stereographically project into R³ so we can actually draw it. Pure mathematics is beautiful alz #HopfFibration #Topology #Geometry #StereographicProjection #PureMathematics

Mathelirium

15,775 次观看 • 7 个月前

Without the Pauli Exclusion Principle, every electron in every atom would collapse to the lowest energy level. No chemistry. No molecules. No life. No hardware to run Claude on. It's not a fundamental law — it's a consequence of wavefunctions being anti-symmetric. Two identical fermions in the same quantum state cancel each other out completely. Every GPU chip, every transistor, every bit of silicon that runs GPT exists because of this one sign flip. 19 minutes. Bookmark & watch today. The universe runs on antisymmetry.

Without the Pauli Exclusion Principle, every electron in every atom would collapse to the lowest energy level. No chemistry. No molecules. No life. No hardware to run Claude on. It's not a fundamental law — it's a consequence of wavefunctions being anti-symmetric. Two identical fermions in the same quantum state cancel each other out completely. Every GPU chip, every transistor, every bit of silicon that runs GPT exists because of this one sign flip. 19 minutes. Bookmark & watch today. The universe runs on antisymmetry.

Eron

21,199 次观看 • 27 天前

Using the BLUEPRINT of reality? Classical computers compute with BITS: 0 or 1 - ONE state at a time. Quantum computers use QUBITS: 0 and 1 or both "simultaneously" through superposition. These states EVOLVE continuously on something like a Bloch sphere - the same MATHEMATICS that describes spin, atoms and electron orbitals. The GEOMETRY feels familiar. Wheels within wheels. Computation, now operating in the rule-set of REALITY itself. Engineering in the same language of nature. THIS is a profound shift. So what follows? 📽 credits: uncover.quantum

Using the BLUEPRINT of reality? Classical computers compute with BITS: 0 or 1 - ONE state at a time. Quantum computers use QUBITS: 0 and 1 or both "simultaneously" through superposition. These states EVOLVE continuously on something like a Bloch sphere - the same MATHEMATICS that describes spin, atoms and electron orbitals. The GEOMETRY feels familiar. Wheels within wheels. Computation, now operating in the rule-set of REALITY itself. Engineering in the same language of nature. THIS is a profound shift. So what follows? 📽 credits: uncover.quantum

WhatdoIknow

21,205 次观看 • 6 个月前

Fourier wrapped a signal around a circle in 1822. Every Spotify equalizer, every noise filter in your AirPods, and every frequency layer inside ChatGPT runs on what he found. The trick is simple. Spin the signal at different speeds. When the spin matches a hidden frequency, the center of mass shifts. That shift is the answer. Claude and GPT use this same idea to separate relevant tokens from noise. 20 minutes. Bookmark & watch today. The math is older than the telephone.

Fourier wrapped a signal around a circle in 1822. Every Spotify equalizer, every noise filter in your AirPods, and every frequency layer inside ChatGPT runs on what he found. The trick is simple. Spin the signal at different speeds. When the spin matches a hidden frequency, the center of mass shifts. That shift is the answer. Claude and GPT use this same idea to separate relevant tokens from noise. 20 minutes. Bookmark & watch today. The math is older than the telephone.

Eron

57,431 次观看 • 1 个月前

🌀🕳️🪐 Why is the wormhole in Interstellar a sphere instead of a flat circle? ⭕️🕳️🪐* Because a real wormhole wouldn't look like a hole in space. It would look like a sphere. In movies and cartoons, wormholes are often shown as flat portals - like a doorway. But in real physics, space isn't 2D. It's three-dimensional. So if a wormhole connects two distant points in space-time, its entrance would extend in all directions. That's why in Interstellar, the wormhole appears as a floating sphere. No matter which direction you approach it from... You're still entering the same tunnel through space-time. The filmmakers actually worked with physicist Kip Thorne to make the depiction as scientifically accurate as possible. So that strange sphere in the movie? It might be the most realistic wormhole ever shown on screen. Sometimes science fiction is closer to real science than we think.*

🌀🕳️🪐 Why is the wormhole in Interstellar a sphere instead of a flat circle? ⭕️🕳️🪐* Because a real wormhole wouldn't look like a hole in space. It would look like a sphere. In movies and cartoons, wormholes are often shown as flat portals - like a doorway. But in real physics, space isn't 2D. It's three-dimensional. So if a wormhole connects two distant points in space-time, its entrance would extend in all directions. That's why in Interstellar, the wormhole appears as a floating sphere. No matter which direction you approach it from... You're still entering the same tunnel through space-time. The filmmakers actually worked with physicist Kip Thorne to make the depiction as scientifically accurate as possible. So that strange sphere in the movie? It might be the most realistic wormhole ever shown on screen. Sometimes science fiction is closer to real science than we think.*

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Complex numbers aren't imaginary. They're the only way to describe a wave that has both amplitude and phase at the same time - and every LLM processes language as exactly that kind of wave through its attention layers. The same math that describes quantum wave interference is what makes Fourier transforms work, which is what lets GPT process tokens in parallel instead of one at a time. Descartes named them "imaginary" before anyone understood what they were. 20 minutes. Bookmark & watch today. The confusion clears up faster than any textbook manages.

Complex numbers aren't imaginary. They're the only way to describe a wave that has both amplitude and phase at the same time - and every LLM processes language as exactly that kind of wave through its attention layers. The same math that describes quantum wave interference is what makes Fourier transforms work, which is what lets GPT process tokens in parallel instead of one at a time. Descartes named them "imaginary" before anyone understood what they were. 20 minutes. Bookmark & watch today. The confusion clears up faster than any textbook manages.

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Is the Universe Actually a Hologram? “Is everything we see in 3D space, 4D spacetime, actually projected from a 2D surface like a hologram? I would say, in some sense, yeah, by definition. But how physical is it? If you can describe things the same way, you might as well use whatever definition is useful. I don’t always take it too literally in the sense of, like, we are in the celestial sphere. It’s more like, can I convert everything to variables on that celestial sphere, and does that help me organize scattering computations?”

Is the Universe Actually a Hologram? “Is everything we see in 3D space, 4D spacetime, actually projected from a 2D surface like a hologram? I would say, in some sense, yeah, by definition. But how physical is it? If you can describe things the same way, you might as well use whatever definition is useful. I don’t always take it too literally in the sense of, like, we are in the celestial sphere. It’s more like, can I convert everything to variables on that celestial sphere, and does that help me organize scattering computations?”

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Since Newton, every law of physics has been written as a differential equation. The same math that describes a pendulum swinging is what makes AI training unstable when learning rates are too large. Differential equations are hard not because the concepts are complex, but because most of them simply cannot be solved exactly. This video shows why - and how phase space diagrams let you understand them anyway without solving anything. 25 minutes. Bookmark & watch today. The chaos theory connection at the end alone is worth it.

Since Newton, every law of physics has been written as a differential equation. The same math that describes a pendulum swinging is what makes AI training unstable when learning rates are too large. Differential equations are hard not because the concepts are complex, but because most of them simply cannot be solved exactly. This video shows why - and how phase space diagrams let you understand them anyway without solving anything. 25 minutes. Bookmark & watch today. The chaos theory connection at the end alone is worth it.

Eron

26,000 次观看 • 1 个月前

🛩️ This is so cool: A Redditor living under SFO's takeoff path built a ceiling projection that maps every plane flying over their house in real time, using ADS-B, the open radio signal aircraft broadcast on 1090 MHz. Same feed as FlightRadar24, picked up with a cheap SDR dongle and beamed onto the ceiling.

🛩️ This is so cool: A Redditor living under SFO's takeoff path built a ceiling projection that maps every plane flying over their house in real time, using ADS-B, the open radio signal aircraft broadcast on 1090 MHz. Same feed as FlightRadar24, picked up with a cheap SDR dongle and beamed onto the ceiling.

International Cyber Digest

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Shawn Ryan Show Earth isn’t a perfect sphere...its equatorial diameter is 26 miles larger than its polar diameter. That 13-mile “bulge” on each side works like a stabilizing flywheel, helping keep the planet’s axis steady. Skeptics call the repeating numerical patterns coincidence, but the consistency is hard to ignore. Over and over, the geometry of the Earth echoes the same proportional template, hinting at something deeper woven into the structure of the planet itself.

Shawn Ryan Show Earth isn’t a perfect sphere...its equatorial diameter is 26 miles larger than its polar diameter. That 13-mile “bulge” on each side works like a stabilizing flywheel, helping keep the planet’s axis steady. Skeptics call the repeating numerical patterns coincidence, but the consistency is hard to ignore. Over and over, the geometry of the Earth echoes the same proportional template, hinting at something deeper woven into the structure of the planet itself.

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10,467 次观看 • 9 个月前

We usually never seen a Hero and the background dancers wear the exact same costume. In "God Mode," Suriya and background dancers wear the same color costume; Suriya is given a red shirt in only one shot. Most of the time, we see costumes and lighting that are designed to highlight the hero, but it isn't like that here. When hundreds or even thousands of people are on screen, but your eyes automatically lock onto just one person—that is screen presence. Even in such a wide shot with so many people in the frame, he still stands out!! Aura God for a reason Suriya Sivakumar ♥️

We usually never seen a Hero and the background dancers wear the exact same costume. In "God Mode," Suriya and background dancers wear the same color costume; Suriya is given a red shirt in only one shot. Most of the time, we see costumes and lighting that are designed to highlight the hero, but it isn't like that here. When hundreds or even thousands of people are on screen, but your eyes automatically lock onto just one person—that is screen presence. Even in such a wide shot with so many people in the frame, he still stands out!! Aura God for a reason Suriya Sivakumar ♥️

Rocky Bhai

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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

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

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Four ways to watch a ball fall. Newton: a force pushes it. Lagrange: out of every possible path, it picks the most efficient one. Hamilton: position and momentum chase each other forever. Quantum: it takes every path at once, you just see the ones that didn’t cancel out. Life requires you to navigate all of these trajectories at the same time. Consciousness is moving toward its purpose faster than the physical universe allows.

Four ways to watch a ball fall. Newton: a force pushes it. Lagrange: out of every possible path, it picks the most efficient one. Hamilton: position and momentum chase each other forever. Quantum: it takes every path at once, you just see the ones that didn’t cancel out. Life requires you to navigate all of these trajectories at the same time. Consciousness is moving toward its purpose faster than the physical universe allows.

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APPLE BUILT A $10,000 COMPUTER THAT RUNS WHAT USED TO REQUIRE AN ENTIRE SERVER ROOM 06:40 he shows the number that breaks everything - one Mac Studio is more powerful than four Dell or AMD machines combined and uses half the energy DeepSeek R1 doesn't run on even two competitors - you need four - one Mac Studio does it alone runs quieter than a fan in the room and uses less energy than a kettle in the kitchen four Mac Studios on a desk connected through a regular cable run a 1 trillion parameter model - what used to require an entire data center anyone can put this at home and Jeff Geerling proved it - run the same models that big companies use without cloud, without subscriptions, without data leaving the room

APPLE BUILT A $10,000 COMPUTER THAT RUNS WHAT USED TO REQUIRE AN ENTIRE SERVER ROOM 06:40 he shows the number that breaks everything - one Mac Studio is more powerful than four Dell or AMD machines combined and uses half the energy DeepSeek R1 doesn't run on even two competitors - you need four - one Mac Studio does it alone runs quieter than a fan in the room and uses less energy than a kettle in the kitchen four Mac Studios on a desk connected through a regular cable run a 1 trillion parameter model - what used to require an entire data center anyone can put this at home and Jeff Geerling proved it - run the same models that big companies use without cloud, without subscriptions, without data leaving the room

Noisy

26,305 次观看 • 2 个月前

Rick Rubin describes how the same creative aesthetic is present in everything he does: “You're at Shangri-La and you walk through the space and you see it doesn't feel like other places you've been. It has the same aesthetic as the records I make, or the things that I work on, or things that I buy.” “It’s the same aesthetic. It’s all fitting into a [creative] worldview.” ——— And then Rick and I tied this to the same creative personality type you see in every field: David: “I've had my first podcast for a decade. And it started out just me reading biographies of people that build businesses. But over time I realized it’s the same personality type: The artist, the entrepreneur, the filmmaker, the musician. I've done podcasts on all of these kind of people and I don't see any distinction between them.” Rick: “It's the same. It’s the same. It's all the creative spirit.” “Most of what I do is not really about music. I happen to work in music but it's not about the music. Does that make sense?”

Rick Rubin describes how the same creative aesthetic is present in everything he does: “You're at Shangri-La and you walk through the space and you see it doesn't feel like other places you've been. It has the same aesthetic as the records I make, or the things that I work on, or things that I buy.” “It’s the same aesthetic. It’s all fitting into a [creative] worldview.” ——— And then Rick and I tied this to the same creative personality type you see in every field: David: “I've had my first podcast for a decade. And it started out just me reading biographies of people that build businesses. But over time I realized it’s the same personality type: The artist, the entrepreneur, the filmmaker, the musician. I've done podcasts on all of these kind of people and I don't see any distinction between them.” Rick: “It's the same. It’s the same. It's all the creative spirit.” “Most of what I do is not really about music. I happen to work in music but it's not about the music. Does that make sense?”

David Senra

15,872 次观看 • 3 个月前

The one step that most people miss when doing a parasite cleanse: the binder. Detox protocols are good at one thing: releasing what your body's been holding onto. But release isn't the same as removal. Once unwanted compounds are mobilized, they need somewhere to go and without the right support, they can simply recirculate. 🪱 I’ve been on the search for a good binder and I finally found one that supports your gut lining. Visit the link in my bio. Code TRUTHSEEKER saves you 10% on every order.

The one step that most people miss when doing a parasite cleanse: the binder. Detox protocols are good at one thing: releasing what your body's been holding onto. But release isn't the same as removal. Once unwanted compounds are mobilized, they need somewhere to go and without the right support, they can simply recirculate. 🪱 I’ve been on the search for a good binder and I finally found one that supports your gut lining. Visit the link in my bio. Code TRUTHSEEKER saves you 10% on every order.

TruthSeeker01011

18,835 次观看 • 22 天前

Asmongold explains why the Jew hate obsession on the right is a subversion designed to keep conservatives fighting each other “If every single Jew in the world disappeared tomorrow, all of these problems would stay the same. That’s what I think. This fixation around blaming everything on them is just stupid.” “When you say that kind of stuff, you find yourself on the same side of every single organization and group that wants to destroy the United States and Western civilization. Every single group thinks the exact same thing. That’s crazy.” “I think it’s a subversion that’s been created to poison the right wing and the conservative movement, to keep it fighting amongst itself rather than recognizing where the real enemies are. You’ve been poison-pilled.”

Asmongold explains why the Jew hate obsession on the right is a subversion designed to keep conservatives fighting each other “If every single Jew in the world disappeared tomorrow, all of these problems would stay the same. That’s what I think. This fixation around blaming everything on them is just stupid.” “When you say that kind of stuff, you find yourself on the same side of every single organization and group that wants to destroy the United States and Western civilization. Every single group thinks the exact same thing. That’s crazy.” “I think it’s a subversion that’s been created to poison the right wing and the conservative movement, to keep it fighting amongst itself rather than recognizing where the real enemies are. You’ve been poison-pilled.”

Asmongold Updates

114,848 次观看 • 2 个月前