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A vector field assigns a vector to every point in space, giving each location both a direction and a magnitude. You can imagine placing a tiny arrow at every point on a map: the arrow points in the direction something would move, while its length represents how strongly or...

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mathswithmuza

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Stokes' Theorem is a classic result from vector calculus. It tells us that the line integral of a vector field over a loop is equal to the surface integral of the curl of the vector field over some enclosed surface.
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Stokes' Theorem is a classic result from vector calculus. It tells us that the line integral of a vector field over a loop is equal to the surface integral of the curl of the vector field over some enclosed surface.

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One of the odd things about learning mathematics is how long you can manipulate an equation without having much of a picture of it in your head. You learn the notation, follow the derivation, get the answer, and move on. Then somebody shows you the same idea visually and you realize there was a geometric intuition hiding underneath the symbols the whole time. A Fourier sum becomes rotating vectors added head-to-tail. Attention becomes a weighted combination you can watch being formed. SVD becomes a sequence of transformations you can follow one step at a time. Even the equation behind a diffusion process feels different when you can see a point following a changing vector field. None of this replaces the mathematics. The equations are still where the precision lives. But once you have a picture in your head, the notation has something to attach itself to. I think this is why some mathematical ideas suddenly become much easier years after we first learned them. We finally stop seeing the equation as a line of symbols to manipulate and start seeing the process it describes. I wish more mathematics had been taught that way.
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One of the odd things about learning mathematics is how long you can manipulate an equation without having much of a picture of it in your head. You learn the notation, follow the derivation, get the answer, and move on. Then somebody shows you the same idea visually and you realize there was a geometric intuition hiding underneath the symbols the whole time. A Fourier sum becomes rotating vectors added head-to-tail. Attention becomes a weighted combination you can watch being formed. SVD becomes a sequence of transformations you can follow one step at a time. Even the equation behind a diffusion process feels different when you can see a point following a changing vector field. None of this replaces the mathematics. The equations are still where the precision lives. But once you have a picture in your head, the notation has something to attach itself to. I think this is why some mathematical ideas suddenly become much easier years after we first learned them. We finally stop seeing the equation as a line of symbols to manipulate and start seeing the process it describes. I wish more mathematics had been taught that way.

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Electromagnetic radiation: oscillating electric and magnetic fields propagate outward from a source at the speed of light. The fields E and B remain mutually perpendicular and orthogonal to the radial unit vector r̂. Intensity I of the radiation decreases with the inverse square of the distance, I ∝ r⁻².
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Electromagnetic radiation: oscillating electric and magnetic fields propagate outward from a source at the speed of light. The fields E and B remain mutually perpendicular and orthogonal to the radial unit vector r̂. Intensity I of the radiation decreases with the inverse square of the distance, I ∝ r⁻².

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The Lévy C curve is a fascinating fractal created by repeatedly replacing every straight line segment with two shorter segments that meet at a right angle. With each iteration, the curve becomes more intricate, folding over itself in a pattern that is surprisingly orderly despite its complexity. Even though the construction follows a very simple rule, the resulting shape has an infinitely detailed boundary and displays self-similarity, meaning that small sections resemble the whole curve when viewed at different scales. One of the most striking features of the Lévy C curve is how a simple line gradually transforms into what looks like a crumpled ribbon while still remaining a continuous path. It has applications in mathematics, computer graphics, and procedural art, demonstrating how complex structures can emerge from repeated, straightforward operations. The curve is a beautiful example of how recursion and iteration can produce patterns that are both mathematically rich and visually captivating. #math #art #fractal #fractalart #visual
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The Lévy C curve is a fascinating fractal created by repeatedly replacing every straight line segment with two shorter segments that meet at a right angle. With each iteration, the curve becomes more intricate, folding over itself in a pattern that is surprisingly orderly despite its complexity. Even though the construction follows a very simple rule, the resulting shape has an infinitely detailed boundary and displays self-similarity, meaning that small sections resemble the whole curve when viewed at different scales. One of the most striking features of the Lévy C curve is how a simple line gradually transforms into what looks like a crumpled ribbon while still remaining a continuous path. It has applications in mathematics, computer graphics, and procedural art, demonstrating how complex structures can emerge from repeated, straightforward operations. The curve is a beautiful example of how recursion and iteration can produce patterns that are both mathematically rich and visually captivating. #math #art #fractal #fractalart #visual

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The Helmholtz decomposition is one of the fundamental results of vector calculus. It says any well-behaved vector field can be split into two parts, one capturing sources and sinks through divergence, and one capturing rotation through curl.
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The Helmholtz decomposition is one of the fundamental results of vector calculus. It says any well-behaved vector field can be split into two parts, one capturing sources and sinks through divergence, and one capturing rotation through curl.

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A guy from China is reshaping glowing particles into a jellyfish, a statue, a dragon, and a samurai just by moving his hands in front of a camera No controller, no mouse, no keyboard Just a camera reading his hand position and a particle system recalculating thousands of points in real time, morphing one shape straight into the next Clench both fists at the end, and everything collapses into nothing This is running live at 60fps, not pre-rendered The entire scene sits inside a wireframe box tracking every motion as raw geometry reshapes into whatever his hands tell it to become
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A guy from China is reshaping glowing particles into a jellyfish, a statue, a dragon, and a samurai just by moving his hands in front of a camera No controller, no mouse, no keyboard Just a camera reading his hand position and a particle system recalculating thousands of points in real time, morphing one shape straight into the next Clench both fists at the end, and everything collapses into nothing This is running live at 60fps, not pre-rendered The entire scene sits inside a wireframe box tracking every motion as raw geometry reshapes into whatever his hands tell it to become

pulmencr

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When a spacecraft leaves Earth, it doesn’t just fire its engines and head straight to its destination. In many missions, especially those going beyond low Earth orbit, there’s a more subtle and elegant strategy at play, one that uses gravity itself as part of the navigation system. This is often called a gravity assist, or a slingshot maneuver. But in the case of missions like #Artemis II, what’s being used is a closely related idea known as a free-return trajectory. At first glance, it might sound simple: the spacecraft goes to the Moon, loops around it, and comes back. But the physics behind it is anything but simple. Instead of relying on continuous propulsion, the spacecraft follows a carefully calculated path through the gravitational field of the Earth–Moon system. It is launched with just the right speed and direction so that, as it approaches the Moon, the Moon’s gravity bends its trajectory. The spacecraft is effectively flung around the Moon, redirected onto a path that naturally brings it back toward Earth. No major engine burn is needed for the return. Small trajectory corrections may still be required, but gravity does the heavy lifting. That’s the key. This kind of trajectory is not just efficient, it’s also safe. If something goes wrong with the spacecraft’s engines or onboard systems, gravity itself ensures the return. It’s an inherent backup plan, built into the trajectory from the very beginning. The same fundamental idea appears in gravity assists used across the Solar System. When a spacecraft flies past a planet, it can gain or lose speed by exchanging momentum with that planet. From the spacecraft’s point of view, it’s as if it has been accelerated without using fuel. In reality, it has borrowed a tiny amount of orbital energy from the planet itself. That’s how missions like Voyager reached the outer planets, and how probes continue to explore regions far beyond what their onboard fuel alone would allow. But there’s an important distinction. An interplanetary gravity assist is typically used to change speed and direction, often increasing the spacecraft’s energy. A free-return trajectory, like the one used in Artemis II, is designed for something more specific: a path that naturally loops back to Earth without requiring additional propulsion. It’s less about gaining energy, and more about shaping a trajectory that guarantees a return. To understand why this works, it helps to stop thinking in straight lines. In space, motion follows curves defined by gravity. The spacecraft is constantly falling, first toward Earth, then toward the Moon, and then back toward Earth again. What looks like a loop is really a continuous free fall through a changing gravitational landscape. This way of navigating space reveals something deeper. We tend to think of engines as the drivers of motion, but once a spacecraft is on its way, gravity does most of the work. The art of spaceflight is not just about thrust. It’s about knowing when not to use it. #GoodLuck #Artemis NASA Artemis
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When a spacecraft leaves Earth, it doesn’t just fire its engines and head straight to its destination. In many missions, especially those going beyond low Earth orbit, there’s a more subtle and elegant strategy at play, one that uses gravity itself as part of the navigation system. This is often called a gravity assist, or a slingshot maneuver. But in the case of missions like #Artemis II, what’s being used is a closely related idea known as a free-return trajectory. At first glance, it might sound simple: the spacecraft goes to the Moon, loops around it, and comes back. But the physics behind it is anything but simple. Instead of relying on continuous propulsion, the spacecraft follows a carefully calculated path through the gravitational field of the Earth–Moon system. It is launched with just the right speed and direction so that, as it approaches the Moon, the Moon’s gravity bends its trajectory. The spacecraft is effectively flung around the Moon, redirected onto a path that naturally brings it back toward Earth. No major engine burn is needed for the return. Small trajectory corrections may still be required, but gravity does the heavy lifting. That’s the key. This kind of trajectory is not just efficient, it’s also safe. If something goes wrong with the spacecraft’s engines or onboard systems, gravity itself ensures the return. It’s an inherent backup plan, built into the trajectory from the very beginning. The same fundamental idea appears in gravity assists used across the Solar System. When a spacecraft flies past a planet, it can gain or lose speed by exchanging momentum with that planet. From the spacecraft’s point of view, it’s as if it has been accelerated without using fuel. In reality, it has borrowed a tiny amount of orbital energy from the planet itself. That’s how missions like Voyager reached the outer planets, and how probes continue to explore regions far beyond what their onboard fuel alone would allow. But there’s an important distinction. An interplanetary gravity assist is typically used to change speed and direction, often increasing the spacecraft’s energy. A free-return trajectory, like the one used in Artemis II, is designed for something more specific: a path that naturally loops back to Earth without requiring additional propulsion. It’s less about gaining energy, and more about shaping a trajectory that guarantees a return. To understand why this works, it helps to stop thinking in straight lines. In space, motion follows curves defined by gravity. The spacecraft is constantly falling, first toward Earth, then toward the Moon, and then back toward Earth again. What looks like a loop is really a continuous free fall through a changing gravitational landscape. This way of navigating space reveals something deeper. We tend to think of engines as the drivers of motion, but once a spacecraft is on its way, gravity does most of the work. The art of spaceflight is not just about thrust. It’s about knowing when not to use it. #GoodLuck #Artemis NASA Artemis

Erika 

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🚨 PHYSICISTS JUST SPLIT A SINGLE PHOTON AND IT TURNED INTO AN IMPROBABLE SWARM OF PARTICLES. In a striking experiment, researchers have shown that a photon can be split apart in such a way that it produces a large number of particles, creating what they describe as a “mixture from zero to infinity.” Instead of the usual clean splitting into two photons (as seen in spontaneous parametric down-conversion), this process generated a complex, broad swarm of particles. The result challenges conventional intuition about how photons behave when pushed into extreme nonlinear regimes. Why this matters: • It demonstrates a rare and complex form of photon splitting that was previously very difficult to observe cleanly • Such processes could help simulate high-energy particle physics in table-top experiments • It opens new possibilities for generating exotic quantum states of light • It provides deeper insight into nonlinear quantum electrodynamics (QED) in strong fields The deeper implication: Photons are usually thought of as indivisible quanta of light. But under the right extreme conditions, a single photon can effectively “break apart” into many particles. This isn’t just a curiosity it touches on fundamental questions about the nature of light and matter, and could eventually lead to new tools for quantum technologies and for studying physics that normally requires particle accelerators. We’re seeing light behave in ways that blur the line between a single quantum and a many-particle system. How do you think being able to controllably split photons into swarms of particles could impact quantum optics or fundamental physics research? Follow for more frontier quantum physics and breakthroughs in light-matter interaction.
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🚨 PHYSICISTS JUST SPLIT A SINGLE PHOTON AND IT TURNED INTO AN IMPROBABLE SWARM OF PARTICLES. In a striking experiment, researchers have shown that a photon can be split apart in such a way that it produces a large number of particles, creating what they describe as a “mixture from zero to infinity.” Instead of the usual clean splitting into two photons (as seen in spontaneous parametric down-conversion), this process generated a complex, broad swarm of particles. The result challenges conventional intuition about how photons behave when pushed into extreme nonlinear regimes. Why this matters: • It demonstrates a rare and complex form of photon splitting that was previously very difficult to observe cleanly • Such processes could help simulate high-energy particle physics in table-top experiments • It opens new possibilities for generating exotic quantum states of light • It provides deeper insight into nonlinear quantum electrodynamics (QED) in strong fields The deeper implication: Photons are usually thought of as indivisible quanta of light. But under the right extreme conditions, a single photon can effectively “break apart” into many particles. This isn’t just a curiosity it touches on fundamental questions about the nature of light and matter, and could eventually lead to new tools for quantum technologies and for studying physics that normally requires particle accelerators. We’re seeing light behave in ways that blur the line between a single quantum and a many-particle system. How do you think being able to controllably split photons into swarms of particles could impact quantum optics or fundamental physics research? Follow for more frontier quantum physics and breakthroughs in light-matter interaction.

TheNewPhysics

25,874 görüntüleme • 4 ay önce

Vector Database by Hand ✍️ Vector databases are revolutionizing how we search and analyze complex data. They have become the backbone of Retrieval Augmented Generation (#RAG). How do vector databases work? [1] Given ↳ A dataset of three sentences, each has 3 words (or tokens) ↳ In practice, a dataset may contain millions or billions of sentences. The max number of tokens may be tens of thousands (e.g., 32,768 mistral-7b). Process "how are you" [2] 🟨 Word Embeddings ↳ For each word, look up corresponding word embedding vector from a table of 22 vectors, where 22 is the vocabulary size. ↳ In practice, the vocabulary size can be tens of thousands. The word embedding dimensions are in the thousands (e.g., 1024, 4096) [3] 🟩 Encoding ↳ Feed the sequence of word embeddings to an encoder to obtain a sequence of feature vectors, one per word. ↳ Here, the encoder is a simple one layer perceptron (linear layer + ReLU) ↳ In practice, the encoder is a transformer or one of its many variants. [4] 🟩 Mean Pooling ↳ Merge the sequence of feature vectors into a single vector using "mean pooling" which is to average across the columns. ↳ The result is a single vector. We often call it "text embeddings" or "sentence embeddings." ↳ Other pooling techniques are possible, such as CLS. But mean pooling is the most common. [5] 🟦 Indexing ↳ Reduce the dimensions of the text embedding vector by a projection matrix. The reduction rate is 50% (4->2). ↳ In practice, the values in this projection matrix is much more random. ↳ The purpose is similar to that of hashing, which is to obtain a short representation to allow faster comparison and retrieval. ↳ The resulting dimension-reduced index vector is saved in the vector storage. [6] Process "who are you" ↳ Repeat [2]-[5] [7] Process "who am I" ↳ Repeat [2]-[5] Now we have indexed our dataset in the vector database. [8] 🟥 Query: "am I you" ↳ Repeat [2]-[5] ↳ The result is a 2-d query vector. [9] 🟥 Dot Products ↳ Take dot product between the query vector and database vectors. They are all 2-d. ↳ The purpose is to use dot product to estimate similarity. ↳ By transposing the query vector, this step becomes a matrix multiplication. [10] 🟥 Nearest Neighbor ↳ Find the largest dot product by linear scan. ↳ The sentence with the highest dot product is "who am I" ↳ In practice, because scanning billions of vectors is slow, we use an Approximate Nearest Neighbor (ANN) algorithm like the Hierarchical Navigable Small Worlds (HNSW).
0:17

Vector Database by Hand ✍️ Vector databases are revolutionizing how we search and analyze complex data. They have become the backbone of Retrieval Augmented Generation (#RAG). How do vector databases work? [1] Given ↳ A dataset of three sentences, each has 3 words (or tokens) ↳ In practice, a dataset may contain millions or billions of sentences. The max number of tokens may be tens of thousands (e.g., 32,768 mistral-7b). Process "how are you" [2] 🟨 Word Embeddings ↳ For each word, look up corresponding word embedding vector from a table of 22 vectors, where 22 is the vocabulary size. ↳ In practice, the vocabulary size can be tens of thousands. The word embedding dimensions are in the thousands (e.g., 1024, 4096) [3] 🟩 Encoding ↳ Feed the sequence of word embeddings to an encoder to obtain a sequence of feature vectors, one per word. ↳ Here, the encoder is a simple one layer perceptron (linear layer + ReLU) ↳ In practice, the encoder is a transformer or one of its many variants. [4] 🟩 Mean Pooling ↳ Merge the sequence of feature vectors into a single vector using "mean pooling" which is to average across the columns. ↳ The result is a single vector. We often call it "text embeddings" or "sentence embeddings." ↳ Other pooling techniques are possible, such as CLS. But mean pooling is the most common. [5] 🟦 Indexing ↳ Reduce the dimensions of the text embedding vector by a projection matrix. The reduction rate is 50% (4->2). ↳ In practice, the values in this projection matrix is much more random. ↳ The purpose is similar to that of hashing, which is to obtain a short representation to allow faster comparison and retrieval. ↳ The resulting dimension-reduced index vector is saved in the vector storage. [6] Process "who are you" ↳ Repeat [2]-[5] [7] Process "who am I" ↳ Repeat [2]-[5] Now we have indexed our dataset in the vector database. [8] 🟥 Query: "am I you" ↳ Repeat [2]-[5] ↳ The result is a 2-d query vector. [9] 🟥 Dot Products ↳ Take dot product between the query vector and database vectors. They are all 2-d. ↳ The purpose is to use dot product to estimate similarity. ↳ By transposing the query vector, this step becomes a matrix multiplication. [10] 🟥 Nearest Neighbor ↳ Find the largest dot product by linear scan. ↳ The sentence with the highest dot product is "who am I" ↳ In practice, because scanning billions of vectors is slow, we use an Approximate Nearest Neighbor (ANN) algorithm like the Hierarchical Navigable Small Worlds (HNSW).

Tom Yeh

192,216 görüntüleme • 2 yıl önce

🚨 BREAKING What if mass… isn’t a fixed property? Scientists just found evidence that mass can change. Not in theory. Inside real matter. They detected a rare state called an η′-mesic nucleus where a particle gets trapped inside a nucleus… and its mass shifts. Let that sink in. Same particle. Same physics. Different environment… different mass. That means: Mass isn’t just “what something is.” It’s how it interacts with the vacuum around it. And the vacuum? It’s not empty. It’s a hidden structure that gives particles their properties. Mass may come from the vacuum The vacuum changes in dense matter So mass itself can change This is huge. Because it flips the idea that mass is fundamental. Instead: mass might be emergent. In my view: Mass isn’t a property… it’s a response to time/field structure. Change the structure… you change the mass. So the real question is: Are particles “things”… or just stable patterns in a deeper field? Follow me physics isn’t discovering matter… it’s discovering what creates it.
0:10

🚨 BREAKING What if mass… isn’t a fixed property? Scientists just found evidence that mass can change. Not in theory. Inside real matter. They detected a rare state called an η′-mesic nucleus where a particle gets trapped inside a nucleus… and its mass shifts. Let that sink in. Same particle. Same physics. Different environment… different mass. That means: Mass isn’t just “what something is.” It’s how it interacts with the vacuum around it. And the vacuum? It’s not empty. It’s a hidden structure that gives particles their properties. Mass may come from the vacuum The vacuum changes in dense matter So mass itself can change This is huge. Because it flips the idea that mass is fundamental. Instead: mass might be emergent. In my view: Mass isn’t a property… it’s a response to time/field structure. Change the structure… you change the mass. So the real question is: Are particles “things”… or just stable patterns in a deeper field? Follow me physics isn’t discovering matter… it’s discovering what creates it.

TheNewPhysics

17,272 görüntüleme • 7 ay önce

Mechanics is probably one of the first things we learn in physics. Newton, the falling apple, gravity, a few equations. It all seems pretty straightforward at first. But the more you learn about it, the more you realize just how enormous this field actually is. It’s not just about machines or moving objects. Mechanics helps us understand how planets orbit stars, how blood flows through our bodies, how molecules interact, and how our muscles work. And then there’s quantum mechanics, which is a whole different world. At that scale, things behave nothing like what we’re used to seeing. I find it fascinating that something as simple as studying motion can take us from a falling apple all the way to the strangest questions in modern physics. And there’s still so much we don’t understand. #Mechanics #Physics #edit #Animation #Engineering
0:16

Mechanics is probably one of the first things we learn in physics. Newton, the falling apple, gravity, a few equations. It all seems pretty straightforward at first. But the more you learn about it, the more you realize just how enormous this field actually is. It’s not just about machines or moving objects. Mechanics helps us understand how planets orbit stars, how blood flows through our bodies, how molecules interact, and how our muscles work. And then there’s quantum mechanics, which is a whole different world. At that scale, things behave nothing like what we’re used to seeing. I find it fascinating that something as simple as studying motion can take us from a falling apple all the way to the strangest questions in modern physics. And there’s still so much we don’t understand. #Mechanics #Physics #edit #Animation #Engineering

pitochk

231,500 görüntüleme • 2 gün önce

Having trouble with the linear algebra and vector/matrix math underlying 3D graphics, collision/physics, or signal processing? Skill issue, man! I neeeeeeeeeeeever fuck up the matrices or confuse coordinate systems or transform by transposes on accident or apply operations in the reverse order!!! Nope nope! NEVER! 🤪 Just do what I do and keep a scrapbook of your most epic linear algebra fails on your quest towards developing a visuospatial intuition (and eventually mastery) over this shit. 🤣 There are very few fields in computing where your screw-ups can wind up being so... spectacularly entertaining. May as well get a laugh out of it and have a good time while you get your ass handed to you by the matrices!
0:14

Having trouble with the linear algebra and vector/matrix math underlying 3D graphics, collision/physics, or signal processing? Skill issue, man! I neeeeeeeeeeeever fuck up the matrices or confuse coordinate systems or transform by transposes on accident or apply operations in the reverse order!!! Nope nope! NEVER! 🤪 Just do what I do and keep a scrapbook of your most epic linear algebra fails on your quest towards developing a visuospatial intuition (and eventually mastery) over this shit. 🤣 There are very few fields in computing where your screw-ups can wind up being so... spectacularly entertaining. May as well get a laugh out of it and have a good time while you get your ass handed to you by the matrices!

Falco Girgis

15,050 görüntüleme • 3 ay önce

Stop manually making flowcharts. Try this GitHub skill with Claude Code. I found an open-source tool that turns a plain description of a software system into a clean, interactive diagram. It's called Archify, and it has more than 66,000 stars on GitHub. You add it as a skill to an AI coding agent like Claude Code, Codex, Cursor, or OpenCode. Then you describe the system in the chat in one line. Archify turns that line into a single HTML file you can open, click through, and send to anyone. If you point your agent at an existing codebase, it reads the code and Archify maps how the pieces connect. I liked five things about it. 1. It makes five kinds of diagrams, including system maps and step-by-step workflows. 2. You can click any piece and trace what connects to it. 3. It checks the layout before it hands the diagram over, so arrows don't pile up and labels don't overlap. 4. It has dark and light themes and exports to PNG, SVG, or a short video. 5. You keep editing in plain chat, with requests like "add Redis" or "highlight the rollback path." You don't need a codebase to use it. A description in the chat is enough.
0:12

Stop manually making flowcharts. Try this GitHub skill with Claude Code. I found an open-source tool that turns a plain description of a software system into a clean, interactive diagram. It's called Archify, and it has more than 66,000 stars on GitHub. You add it as a skill to an AI coding agent like Claude Code, Codex, Cursor, or OpenCode. Then you describe the system in the chat in one line. Archify turns that line into a single HTML file you can open, click through, and send to anyone. If you point your agent at an existing codebase, it reads the code and Archify maps how the pieces connect. I liked five things about it. 1. It makes five kinds of diagrams, including system maps and step-by-step workflows. 2. You can click any piece and trace what connects to it. 3. It checks the layout before it hands the diagram over, so arrows don't pile up and labels don't overlap. 4. It has dark and light themes and exports to PNG, SVG, or a short video. 5. You keep editing in plain chat, with requests like "add Redis" or "highlight the rollback path." You don't need a codebase to use it. A description in the chat is enough.

Alex Veremeyenko

61,928 görüntüleme • 1 ay önce

When a nuclear reactor powers on for the first time, the core fills with a deep blue glow that feels almost unreal as it rises through the water in a quiet surge of energy. The light is Cherenkov radiation, created when charged particles move through the water faster than light can travel in that medium, a discovery made in the 1930s that revealed a new way to see high energy reactions. Every appearance of this glow is a reminder of how powerful atomic reactions can be, showing a rare moment where raw physics becomes something visible and strangely beautiful.
0:13

When a nuclear reactor powers on for the first time, the core fills with a deep blue glow that feels almost unreal as it rises through the water in a quiet surge of energy. The light is Cherenkov radiation, created when charged particles move through the water faster than light can travel in that medium, a discovery made in the 1930s that revealed a new way to see high energy reactions. Every appearance of this glow is a reminder of how powerful atomic reactions can be, showing a rare moment where raw physics becomes something visible and strangely beautiful.

Today In History

384,111 görüntüleme • 5 ay önce

🫯 A conflicting physical ontology ! One of the strangest properties in particle physics is the spin. The term spin is a historical misnomer. Electrons (𝑒⁻) do not physically rotate like a spinning top or a globe. Spin is a fundamental quantum property known as intrinsic angular momentum which exists as an inherent characteristic of the particle, much like its mass or electric charge. In quantum mechanics, electrons are treated as fundamental point-like particles with zero size, making the concept of physical 3D rotation nonsensical. A rotating electrical charge creates a magnetic field. An electron naturally possesses its own detectable magnetic field. Even though the electron isn't physically spinning, it acts exactly as if it were because of that. Moreover the spin of an 𝑒⁻ has orientation. As 𝑒⁻ carry a negative electric charge, their inherent spin turns them into tiny bar magnets. Two possible orientations exist in the natural state of the 𝑒⁻, spin up and spin down. When measured along a specific direction (called the z-axis) the 𝑒⁻ can only display one of the two orientation state, up or down. But the strangeness goes a bit further; when an 𝑒⁻ is misplaced or removed from its own orbit, it takes two full circles (720°) to return to its original quantum state. The spin isn’t motion, it’s an undeniable static, resilient force of this elementary particle, listed as one the fundamental constant of nature. 🔗
0:12

🫯 A conflicting physical ontology ! One of the strangest properties in particle physics is the spin. The term spin is a historical misnomer. Electrons (𝑒⁻) do not physically rotate like a spinning top or a globe. Spin is a fundamental quantum property known as intrinsic angular momentum which exists as an inherent characteristic of the particle, much like its mass or electric charge. In quantum mechanics, electrons are treated as fundamental point-like particles with zero size, making the concept of physical 3D rotation nonsensical. A rotating electrical charge creates a magnetic field. An electron naturally possesses its own detectable magnetic field. Even though the electron isn't physically spinning, it acts exactly as if it were because of that. Moreover the spin of an 𝑒⁻ has orientation. As 𝑒⁻ carry a negative electric charge, their inherent spin turns them into tiny bar magnets. Two possible orientations exist in the natural state of the 𝑒⁻, spin up and spin down. When measured along a specific direction (called the z-axis) the 𝑒⁻ can only display one of the two orientation state, up or down. But the strangeness goes a bit further; when an 𝑒⁻ is misplaced or removed from its own orbit, it takes two full circles (720°) to return to its original quantum state. The spin isn’t motion, it’s an undeniable static, resilient force of this elementary particle, listed as one the fundamental constant of nature. 🔗

𝓜𝒂𝒖𝒓𝒊𝔃𝒊𝒐 𝗜𝒃𝛼

22,933 görüntüleme • 3 ay önce

A straight line in an image can become a single point in another mathematical space. This is the fascinating geometry behind the Hough Transform, a classical computer vision technique that can detect lines even when the visible edges are broken, incomplete, or noisy. The idea is surprisingly clever. A single pixel could belong to infinitely many different lines. Instead of trying to connect pixels individually, the Hough Transform asks each edge pixel to vote for all the lines that could pass through it. The mathematics represents a line using two parameters, its perpendicular distance from the origin and the angle of its normal. ρ = x cos(θ) + y sin(θ) For each edge pixel, this equation produces a sinusoidal curve in parameter space. When several pixels lie along the same straight line, their curves intersect at a common location. That intersection is the important part. In a digital implementation, votes accumulate in discrete cells, and a prominent peak identifies a likely line in the original image. The method has an interesting history. Paul Hough introduced the underlying approach in the early 1960s, and Richard Duda and Peter Hart developed the influential angle-distance formulation in 1972. This mathematics became a useful tool in image processing, from detecting lines in scanned documents to identifying road markings and inspecting geometric structures in industrial images. There are practical limitations. Noise and unrelated edges can create false peaks, while the resolution of the parameter space and voting thresholds affect detection accuracy. But the central mathematical idea remains remarkably useful. Points that appear disconnected in an image can still agree on the same geometric structure. The Hough Transform turns that agreement into a voting problem, allowing a computer to find patterns without requiring every part of the original line to be visible. Happy Learning!!
0:14

A straight line in an image can become a single point in another mathematical space. This is the fascinating geometry behind the Hough Transform, a classical computer vision technique that can detect lines even when the visible edges are broken, incomplete, or noisy. The idea is surprisingly clever. A single pixel could belong to infinitely many different lines. Instead of trying to connect pixels individually, the Hough Transform asks each edge pixel to vote for all the lines that could pass through it. The mathematics represents a line using two parameters, its perpendicular distance from the origin and the angle of its normal. ρ = x cos(θ) + y sin(θ) For each edge pixel, this equation produces a sinusoidal curve in parameter space. When several pixels lie along the same straight line, their curves intersect at a common location. That intersection is the important part. In a digital implementation, votes accumulate in discrete cells, and a prominent peak identifies a likely line in the original image. The method has an interesting history. Paul Hough introduced the underlying approach in the early 1960s, and Richard Duda and Peter Hart developed the influential angle-distance formulation in 1972. This mathematics became a useful tool in image processing, from detecting lines in scanned documents to identifying road markings and inspecting geometric structures in industrial images. There are practical limitations. Noise and unrelated edges can create false peaks, while the resolution of the parameter space and voting thresholds affect detection accuracy. But the central mathematical idea remains remarkably useful. Points that appear disconnected in an image can still agree on the same geometric structure. The Hough Transform turns that agreement into a voting problem, allowing a computer to find patterns without requiring every part of the original line to be visible. Happy Learning!!

Tech with Mak

11,371 görüntüleme • 8 gün önce

Offset vector paths now available in Figma Draw. You can now take any flattened vector path and offset it to expand or retract it. It even lets you round out those sharp edges.
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Offset vector paths now available in Figma Draw. You can now take any flattened vector path and offset it to expand or retract it. It even lets you round out those sharp edges.

miggi from figgi

55,585 görüntüleme • 1 yıl önce