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In 1843, Hamilton realized complex numbers needed not one more dimension but two. One real part, three imaginary - i, j, k. He carved the equation into a Dublin bridge on October 16th. That number system now runs inside every 3D graphics engine, every robotics controller, and every LLM... embedding space that maps language into high-dimensional geometry on GPU clusters today. Quaternion multiplication spins two perpendicular circles simultaneously - something that can only happen in four dimensions. i times j is k, but j times i is -k. Rotations don't commute, like a Rubik's cube. That's exactly why game engines and spacecraft use quaternions instead of matrices - no gimbal lock, no numerical drift. The same rotation math that every transformer uses when computing rotary position embeddings to encode token order in the complex plane. 2 minutes. Bookmark & watch today. The number system that powers every 3D engine and nobody taught you about.show more

Eron

11,127 subscribers

24,975 просмотров • 3 дней назад •via X (Twitter)

Наука и технологии Образование

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Комментарии: 9

Фото профиля Bart Christner

Bart Christner3 дней назад

Plagiarized content without attribution from Three Brown One Blue, Youtube, Sep 6, 2018. Visualizing the 4d numbers Quaternions.

Фото профиля Aaron M Kinder

Aaron M Kinder3 дней назад

That's what I said, I'm setting booby traps via @TheYTPro

Фото профиля David Minor

David Minor3 дней назад

This is not quite right complex numbers. Don’t need these, but Hamilton invented a whole Nother type of complex number which can have three or even more imaginary roots. Normal complex math and physics works just fine without them, but a lot of the matrix computations can be done using quaternions. That is the IJK but most physicist prefer to use standard Gaussian matrix arithmetic.

Фото профиля Jesse Palmer

Jesse Palmer3 дней назад

Another reason game engines use it as it takes less operations to multiply rotations

Фото профиля 菅野正人

菅野正人3 дней назад

なぜ単位ベクトルの始点が座標の原点にあるのですか？

Фото профиля Anish

Anish3 дней назад

very nice

Фото профиля supervinculum

supervinculum3 дней назад

Why you can't tie knots in four dimensions

Фото профиля ᒎᘴᓵᖶᓋ

ᒎᘴᓵᖶᓋ3 дней назад

Carving i² = j² = k² = ijk = -1 into a bridge is the most legendary commit message ever. Now it keeps robot arms and game cameras out of gimbal lock. Did Dublin ever ask him to revert? 🌉

Фото профиля Kin Altruism

Kin Altruism3 дней назад

Matrices don't have gimbal lock

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Lupen

24,677 просмотров • 3 месяцев назад

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Complex functions need 4 dimensions to graph - one input, one complex output. The four visualization methods solve this: plane deformations, Riemann surfaces (3D + color as 4th dimension), the Riemann sphere (infinity becomes a literal point, division by zero becomes a rotation), and vector fields. LLMs operate in that last mode - attention scores are dot products between query and key vectors, exactly the inner product that defines work along a curve in a vector field. The Riemann sphere is the wildest part. Möbius transformations become literal sphere rotations when you compactify the complex plane - what looks algebraically like division by zero is geometrically just moving the north pole. Softmax in GPU-parallelized attention does the same thing: maps unbounded scores onto a bounded surface. 12 minutes. Bookmark & watch today.

Eron

13,374 просмотров • 1 месяц назад

Carl Sagan filmed a free seven-minute explanation of the fourth dimension that has now been watched more than five million times. Universities charge tens of thousands of dollars a year for cosmology and geometry courses that never reach this material with the same clarity. He posted the entire sequence for nothing. Almost no one who casually debates higher dimensions has watched it to the end. The clip comes from Episode 10 of Cosmos, “The Edge of Forever,” recorded in 1980. Sagan begins by placing the viewer inside Flatland, a two-dimensional world inhabited by squares and circles who can move left-right and forward-back but have never heard of up or down. Into that plane he lowers a three-dimensional apple. The Flatlanders experience only a series of expanding and shrinking circles as the apple passes through their world. They never see the whole fruit, only successive slices, and they invent elaborate theories to explain the changing shapes that appear and vanish without cause. Sagan then constructs a cube by moving a square through a third dimension at right angles to itself, and asks what happens if the same operation is performed once more: take the cube and move it at right angles to all three existing directions. The result is a four-dimensional hypercube, a tesseract. He cannot show a true tesseract because both he and the viewer remain trapped in three dimensions, but he can display its three-dimensional shadow: two nested cubes with every vertex connected by lines. In the real four-dimensional object every edge would be equal and every angle a right angle; the distorted projection is simply the cost of losing a dimension. Every modern discussion of curved spacetime, finite but unbounded universes, and higher-dimensional geometry rests on the same progression he lays out with an apple and a cube. The sequence is free on YouTube. The willingness to sit through those seven minutes before speaking about the fourth dimension is a much rarer commodity than the confidence to argue without having watched it.

Carl Sagan filmed a free seven-minute explanation of the fourth dimension that has now been watched more than five million times. Universities charge tens of thousands of dollars a year for cosmology and geometry courses that never reach this material with the same clarity. He posted the entire sequence for nothing. Almost no one who casually debates higher dimensions has watched it to the end. The clip comes from Episode 10 of Cosmos, “The Edge of Forever,” recorded in 1980. Sagan begins by placing the viewer inside Flatland, a two-dimensional world inhabited by squares and circles who can move left-right and forward-back but have never heard of up or down. Into that plane he lowers a three-dimensional apple. The Flatlanders experience only a series of expanding and shrinking circles as the apple passes through their world. They never see the whole fruit, only successive slices, and they invent elaborate theories to explain the changing shapes that appear and vanish without cause. Sagan then constructs a cube by moving a square through a third dimension at right angles to itself, and asks what happens if the same operation is performed once more: take the cube and move it at right angles to all three existing directions. The result is a four-dimensional hypercube, a tesseract. He cannot show a true tesseract because both he and the viewer remain trapped in three dimensions, but he can display its three-dimensional shadow: two nested cubes with every vertex connected by lines. In the real four-dimensional object every edge would be equal and every angle a right angle; the distorted projection is simply the cost of losing a dimension. Every modern discussion of curved spacetime, finite but unbounded universes, and higher-dimensional geometry rests on the same progression he lays out with an apple and a cube. The sequence is free on YouTube. The willingness to sit through those seven minutes before speaking about the fourth dimension is a much rarer commodity than the confidence to argue without having watched it.

fomosapiens

73,505 просмотров • 2 месяцев назад

In 1958, the US Navy stood Frank Rosenblatt in front of a $2 million IBM computer and told reporters it was the embryo of a machine that would someday be conscious of its own existence. His entire idea fit on one line of math. In 1969, two MIT researchers published a single puzzle that killed the funding for a decade. The line of math survived anyway - it runs inside every neuron in every LLM deployed today, copied billions of times and stacked in layers. The formula is a weighted sum followed by a nonlinearity: step of W dot X plus B. Draw it geometrically and it's a straight line dividing space into ones and zeros. The learning rule nudges the weights whenever it's wrong. Minsky and Papert proved it could never solve XOR - four points on a square where no straight line separates the classes. The fix was stacking layers and training them with backpropagation, published in Nature in 1986 by Rumelhart, Hinton, and Williams. Hinton shared the 2024 Nobel Prize in Physics for it. Rosenblatt drowned in a boating accident in 1971 on his 43rd birthday, 15 years before the solution arrived. 5 minutes. Bookmark & watch today. The one line of 1958 math that every transformer, every LLM, and every GPU on earth still runs on.

In 1958, the US Navy stood Frank Rosenblatt in front of a $2 million IBM computer and told reporters it was the embryo of a machine that would someday be conscious of its own existence. His entire idea fit on one line of math. In 1969, two MIT researchers published a single puzzle that killed the funding for a decade. The line of math survived anyway - it runs inside every neuron in every LLM deployed today, copied billions of times and stacked in layers. The formula is a weighted sum followed by a nonlinearity: step of W dot X plus B. Draw it geometrically and it's a straight line dividing space into ones and zeros. The learning rule nudges the weights whenever it's wrong. Minsky and Papert proved it could never solve XOR - four points on a square where no straight line separates the classes. The fix was stacking layers and training them with backpropagation, published in Nature in 1986 by Rumelhart, Hinton, and Williams. Hinton shared the 2024 Nobel Prize in Physics for it. Rosenblatt drowned in a boating accident in 1971 on his 43rd birthday, 15 years before the solution arrived. 5 minutes. Bookmark & watch today. The one line of 1958 math that every transformer, every LLM, and every GPU on earth still runs on.

Deliriumm

11,638 просмотров • 28 дней назад

You can turn a sphere inside out without tearing, poking holes, or creating a single crease - and this is not a trick, it is a proven mathematical theorem. You cannot do the same thing to a circle. The reason is a topological invariant called the turning number, and it is the same invariant governing the geometry every LLM embedding space is built on. Stephen Smale proved eversion was possible in 1957. It took seven more years before anyone found a way to actually do it. The mechanism is corrugation. Divide the sphere into guide strips connected by wavy sections that let the surface pass through itself without pinching. Push the caps past each other before a crease forms. Twist them in opposite directions - the same way a belt with a loop untwists when you rotate each end half a turn. Then push the equator through and uncorrugate. Bowls, domes and saddles cancel in a pattern that stays invariant throughout - the same structure that makes high-dimensional transformer geometry behave nothing like 3D intuition. 21 minutes. Bookmark & watch today. The mathematical proof that breaks every intuition you have about surfaces.

You can turn a sphere inside out without tearing, poking holes, or creating a single crease - and this is not a trick, it is a proven mathematical theorem. You cannot do the same thing to a circle. The reason is a topological invariant called the turning number, and it is the same invariant governing the geometry every LLM embedding space is built on. Stephen Smale proved eversion was possible in 1957. It took seven more years before anyone found a way to actually do it. The mechanism is corrugation. Divide the sphere into guide strips connected by wavy sections that let the surface pass through itself without pinching. Push the caps past each other before a crease forms. Twist them in opposite directions - the same way a belt with a loop untwists when you rotate each end half a turn. Then push the equator through and uncorrugate. Bowls, domes and saddles cancel in a pattern that stays invariant throughout - the same structure that makes high-dimensional transformer geometry behave nothing like 3D intuition. 21 minutes. Bookmark & watch today. The mathematical proof that breaks every intuition you have about surfaces.

Eron

284,496 просмотров • 26 дней назад

In 2006, Netflix offered $1,000,000 to anyone who could improve their recommendation algorithm by 10%. Over 2,000 teams competed for three years. The team that won did not use more data. They used fewer dimensions. They found a basis - a small set of independent vectors that captured everything important about 100,000,000 movie ratings. The lead mathematician on the winning team: $2,800,000 a year. A machine learning engineer at Spotify building the same kind of system: $245,000 a year. This is MIT 18.06, Lecture 9 - Gilbert Strang. Free on YouTube. Most people think independence is obvious. Two vectors pointing in different directions. Then the definition. Independence means no combination of your vectors gives the zero vector - except the trivial one where all the coefficients are zero. That's it. That's the whole definition. But watch what it unlocks. Watch the moment Strang puts three vectors in a two-dimensional plane. He doesn't even tell you which three vectors. He just draws them. And immediately says: dependent. No question. No calculation. Why? Because three vectors in two-dimensional space means more columns than rows. More unknowns than equations. That always forces a free variable. A free variable always gives a non-zero solution to Ax = 0. And that non-zero solution is a combination of the columns that produces zero. Dependence. "Three vectors in the plane have to be dependent. That's the key fact." Then the basis. A basis is vectors that are independent and span the space. Not too few, not too many. Just right. The pivot columns of any matrix form a basis for the column space. Every other basis you can think of will have exactly the same number of vectors. Then the dimension. All bases for the same space have the same number of vectors. That number is the dimension. The rank of a matrix is the dimension of its column space. The number of free variables is the dimension of the null space. And rank plus null space dimension equals the total number of columns. "I don't take the dimension of A. I take the dimension of the column space of A. If you use those words right, it shows you've got the idea right." A data scientist at Netflix building recommendation engines: $230,000 a year. A quantitative researcher at Two Sigma finding independent factors in financial markets: $350,000 a year. A computer vision engineer at Apple using low-dimensional representations for face recognition: $260,000 a year. They all needed to know how many dimensions were really there. bookmark this and watch later - after this lecture every dataset you look at will feel like a matrix waiting to be reduced to its basis.

In 2006, Netflix offered $1,000,000 to anyone who could improve their recommendation algorithm by 10%. Over 2,000 teams competed for three years. The team that won did not use more data. They used fewer dimensions. They found a basis - a small set of independent vectors that captured everything important about 100,000,000 movie ratings. The lead mathematician on the winning team: $2,800,000 a year. A machine learning engineer at Spotify building the same kind of system: $245,000 a year. This is MIT 18.06, Lecture 9 - Gilbert Strang. Free on YouTube. Most people think independence is obvious. Two vectors pointing in different directions. Then the definition. Independence means no combination of your vectors gives the zero vector - except the trivial one where all the coefficients are zero. That's it. That's the whole definition. But watch what it unlocks. Watch the moment Strang puts three vectors in a two-dimensional plane. He doesn't even tell you which three vectors. He just draws them. And immediately says: dependent. No question. No calculation. Why? Because three vectors in two-dimensional space means more columns than rows. More unknowns than equations. That always forces a free variable. A free variable always gives a non-zero solution to Ax = 0. And that non-zero solution is a combination of the columns that produces zero. Dependence. "Three vectors in the plane have to be dependent. That's the key fact." Then the basis. A basis is vectors that are independent and span the space. Not too few, not too many. Just right. The pivot columns of any matrix form a basis for the column space. Every other basis you can think of will have exactly the same number of vectors. Then the dimension. All bases for the same space have the same number of vectors. That number is the dimension. The rank of a matrix is the dimension of its column space. The number of free variables is the dimension of the null space. And rank plus null space dimension equals the total number of columns. "I don't take the dimension of A. I take the dimension of the column space of A. If you use those words right, it shows you've got the idea right." A data scientist at Netflix building recommendation engines: $230,000 a year. A quantitative researcher at Two Sigma finding independent factors in financial markets: $350,000 a year. A computer vision engineer at Apple using low-dimensional representations for face recognition: $260,000 a year. They all needed to know how many dimensions were really there. bookmark this and watch later - after this lecture every dataset you look at will feel like a matrix waiting to be reduced to its basis.

Zyphor

14,720 просмотров • 2 месяцев назад