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A square wave jumps between two values and stays flat. Fourier showed it is built from sine waves alone: add the odd harmonics 3, 5, 7 and so on, each one that many times faster and that many times smaller. Every circle here is one harmonic and the last... show more

Math Pulse

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11,179 次观看 • 13 天前 •via X (Twitter)

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Sid the Argent13 天前

The slice is not the object nor the true objective motion.

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A square wave is a sum of odd sines, f(x) = (4/π) ∑ sin((2n−1)x)/(2n−1), and each term is a circle of radius 4/(π(2n−1)) turning 2n−1 times per period. Chained end to end, the height of the n-th joint is the partial sum Sₙ(x). The overshoot at each jump tends to 0.179, about 9 % of the jump, however many terms are added. At x = π/2 every arm points straight up or down and f = 1, which is Leibniz's series: π/4 = 1 − 1/3 + 1/5 − 1/7 + ⋯.

A square wave is a sum of odd sines, f(x) = (4/π) ∑ sin((2n−1)x)/(2n−1), and each term is a circle of radius 4/(π(2n−1)) turning 2n−1 times per period. Chained end to end, the height of the n-th joint is the partial sum Sₙ(x). The overshoot at each jump tends to 0.179, about 9 % of the jump, however many terms are added. At x = π/2 every arm points straight up or down and f = 1, which is Leibniz's series: π/4 = 1 − 1/3 + 1/5 − 1/7 + ⋯.

matharium

47,537 次观看 • 1 个月前

Each rotating circle contributes to the whole seahorse. Its outline is encoded in two Fourier series, one for the horizontal coordinate and one for the vertical. The guides from the two chains meet at the moving pen. With one harmonic per coordinate, the trace is an ellipse. Adding harmonics brings out the snout, jagged back and curled tail. At 100 harmonics per coordinate, the largest sampled pointwise error is about 0.45% of the outline’s height.

Each rotating circle contributes to the whole seahorse. Its outline is encoded in two Fourier series, one for the horizontal coordinate and one for the vertical. The guides from the two chains meet at the moving pen. With one harmonic per coordinate, the trace is an ellipse. Adding harmonics brings out the snout, jagged back and curled tail. At 100 harmonics per coordinate, the largest sampled pointwise error is about 0.45% of the outline’s height.

Philosophy Of Physics

35,148 次观看 • 27 天前

A square wave flipping between ±1 at multiples of π is the limit of f_N(t) = (4/π) ∑ sin((2k+1)t)/(2k+1) for k < N. Only odd harmonics appear, since f(t + π) = −f(t). Term k is a circle of radius (4/π)/(2k+1) turning 2k+1 times as fast as the first; the chain ends at height f_N(t). The sum overshoots each jump. Its first peak, at t = π/(2N), is 1.182 for N = 5 and 1.179 for N = 100, about 9 % of the jump. More terms narrow it, never remove it.

A square wave flipping between ±1 at multiples of π is the limit of f_N(t) = (4/π) ∑ sin((2k+1)t)/(2k+1) for k < N. Only odd harmonics appear, since f(t + π) = −f(t). Term k is a circle of radius (4/π)/(2k+1) turning 2k+1 times as fast as the first; the chain ends at height f_N(t). The sum overshoots each jump. Its first peak, at t = π/(2N), is 1.182 for N = 5 and 1.179 for N = 100, about 9 % of the jump. More terms narrow it, never remove it.

matharium

23,614 次观看 • 1 个月前

Sharp feather tips can be approximated by adding smooth sine and cosine waves. This hummingbird’s outline is described by two Fourier series, one for each coordinate. Their rotating chains position the pen where the horizontal and vertical guides meet. Low frequencies capture the broad shape; higher frequencies recover smaller details. The number below the bird is the highest harmonic retained in each coordinate. At 100, the largest sampled pointwise error is about 0.57% of the outline’s height.

Sharp feather tips can be approximated by adding smooth sine and cosine waves. This hummingbird’s outline is described by two Fourier series, one for each coordinate. Their rotating chains position the pen where the horizontal and vertical guides meet. Low frequencies capture the broad shape; higher frequencies recover smaller details. The number below the bird is the highest harmonic retained in each coordinate. At 100, the largest sampled pointwise error is about 0.57% of the outline’s height.

Philosophy Of Physics

38,515 次观看 • 27 天前

The X mark, drawn as one 20-corner outline with a thin slit cut into its thick stroke, is a sum of circles: z(t) = ∑ cₖ eⁱᵏᵗ, the cₖ from its discrete Fourier transform. With the N largest circles the widest gap to the outline is 49.9 % of its height at N = 1, 2.15 % at 64 and 0.49 % at 256. The tip's distance |z(t)| plays as pitch. Each corner adds one term to every circle, which caps them at |cₖ| ≤ 5.35/k² (k ≥ 1).

The X mark, drawn as one 20-corner outline with a thin slit cut into its thick stroke, is a sum of circles: z(t) = ∑ cₖ eⁱᵏᵗ, the cₖ from its discrete Fourier transform. With the N largest circles the widest gap to the outline is 49.9 % of its height at N = 1, 2.15 % at 64 and 0.49 % at 256. The tip's distance |z(t)| plays as pitch. Each corner adds one term to every circle, which caps them at |cₖ| ≤ 5.35/k² (k ≥ 1).

matharium

54,076 次观看 • 27 天前

The Pythagoras tree stands a right triangle on each square and a square on each of its legs, ten times over: 1023 squares. Its outline, with the triangular gaps joined in by thin bridges, is one closed curve and a sum of circles: z(t) = ∑ cₖ eⁱᵏᵗ. The widest gap to it is 49.7 % of the tree’s height at N = 1, 5.04 % at 128 and 1.39 % at 512, and the circles shrink like k^−1.32.

The Pythagoras tree stands a right triangle on each square and a square on each of its legs, ten times over: 1023 squares. Its outline, with the triangular gaps joined in by thin bridges, is one closed curve and a sum of circles: z(t) = ∑ cₖ eⁱᵏᵗ. The widest gap to it is 49.7 % of the tree’s height at N = 1, 5.04 % at 128 and 1.39 % at 512, and the circles shrink like k^−1.32.

matharium

34,892 次观看 • 25 天前

A polyrhythm is two even pulses sharing one span. One hand turns every 3 s and sounds each corner it crosses: the square's four, every 0.75 s, against a regular n-gon's n, every 3/n s. The two pulses meet gcd(n, 4) times a turn: once for 3:4, 5:4, 7:4, 9:4 and 15:4, twice for 6:4 and 10:4, and on every square beat for 8:4 and 20:4.

A polyrhythm is two even pulses sharing one span. One hand turns every 3 s and sounds each corner it crosses: the square's four, every 0.75 s, against a regular n-gon's n, every 3/n s. The two pulses meet gcd(n, 4) times a turn: once for 3:4, 5:4, 7:4, 9:4 and 15:4, twice for 6:4 and 10:4, and on every square beat for 8:4 and 20:4.

matharium

17,238 次观看 • 1 个月前

The outline of π, sampled at 2048 points, is a sum of circles: z(t) = ∑ cₖ eⁱᵏᵗ, the cₖ from its discrete Fourier transform. Circle k has radius |cₖ| and turns k times per lap. With the N largest, the widest gap to the outline is 41.6 % of the letter's height at N = 1, 1.37 % at 64, 0.36 % at 256. Finitely many circles make a smooth curve, and π has four sharp corners, so no chain is exact. The error stays below the sum of the radii left out.

The outline of π, sampled at 2048 points, is a sum of circles: z(t) = ∑ cₖ eⁱᵏᵗ, the cₖ from its discrete Fourier transform. Circle k has radius |cₖ| and turns k times per lap. With the N largest, the widest gap to the outline is 41.6 % of the letter's height at N = 1, 1.37 % at 64, 0.36 % at 256. Finitely many circles make a smooth curve, and π has four sharp corners, so no chain is exact. The error stays below the sum of the radii left out.

matharium

51,981 次观看 • 1 个月前

Faster than LIGHT TRAVEL is REAL and it's done through LONGITUDINAL WAVES!💥💥💥 In this clip, Eric Dollard and Chris Carson demonstrate how transverse waves move at the speed of light while the Longitudinal waves is an instantaneous transfer, aka infinitely faster than light travel. This is "the pressure wave through the aether/ether." "Transverse waves unfortunately are the one with loses. The Longitudinal waves is the one with NO losses that we don't even acknowledge exists. And that's also, I might add, that our minds transfer between each other, and that's the one that the cells use to communicate with each other, it's the one our nerves use to communicate with each other." -Chris Carson "It's my theory that the capacitors are the cell membrane of the cells next to each other and the coils is probably DNA. It is my guess."

Faster than LIGHT TRAVEL is REAL and it's done through LONGITUDINAL WAVES!💥💥💥 In this clip, Eric Dollard and Chris Carson demonstrate how transverse waves move at the speed of light while the Longitudinal waves is an instantaneous transfer, aka infinitely faster than light travel. This is "the pressure wave through the aether/ether." "Transverse waves unfortunately are the one with loses. The Longitudinal waves is the one with NO losses that we don't even acknowledge exists. And that's also, I might add, that our minds transfer between each other, and that's the one that the cells use to communicate with each other, it's the one our nerves use to communicate with each other." -Chris Carson "It's my theory that the capacitors are the cell membrane of the cells next to each other and the coils is probably DNA. It is my guess."

MaQaveli 🇺🇸🌎

32,422 次观看 • 29 天前

In this incredible demonstration from the Maritime Technical and Safety Institute in Japan, a large wave tank uses many synchronized wave-generating devices to create visible symbols and patterns on the surface of the water. Each paddle produces a small wave, but when many waves are emitted at specific timings, amplitudes, and phases, they begin to overlap. Where the wave crests reinforce one another, the water rises. Where a crest meets a trough, the motion cancels out. This is called interference, and it is one of the most fundamental principles of wave physics. By controlling the phase relationship between many individual wave sources, researchers can shape the surface of the water into temporary patterns, symbols, and directional flows. In other words, the visible image is not “drawn” onto the water. It emerges from the mathematics of waves interacting with one another. This is the same underlying principle behind acoustics, cymatics, ocean modeling, signal processing, holography, and even the way complex field patterns arise throughout nature. A single wave carries motion, but many waves, when organized in relationship, create structure. This is why water is such a powerful medium for understanding resonance. It makes invisible dynamics visible and shows us that form can emerge from rhythm, timing, frequency, and relational coherence. At a deeper level, this demonstration reveals something profound about reality itself… Patterns appear when many small movements become synchronized enough to behave as one field. The ocean, sound, light, and even the nervous system all do this, because life is vibrational in nature… Did this expand your perception?

In this incredible demonstration from the Maritime Technical and Safety Institute in Japan, a large wave tank uses many synchronized wave-generating devices to create visible symbols and patterns on the surface of the water. Each paddle produces a small wave, but when many waves are emitted at specific timings, amplitudes, and phases, they begin to overlap. Where the wave crests reinforce one another, the water rises. Where a crest meets a trough, the motion cancels out. This is called interference, and it is one of the most fundamental principles of wave physics. By controlling the phase relationship between many individual wave sources, researchers can shape the surface of the water into temporary patterns, symbols, and directional flows. In other words, the visible image is not “drawn” onto the water. It emerges from the mathematics of waves interacting with one another. This is the same underlying principle behind acoustics, cymatics, ocean modeling, signal processing, holography, and even the way complex field patterns arise throughout nature. A single wave carries motion, but many waves, when organized in relationship, create structure. This is why water is such a powerful medium for understanding resonance. It makes invisible dynamics visible and shows us that form can emerge from rhythm, timing, frequency, and relational coherence. At a deeper level, this demonstration reveals something profound about reality itself… Patterns appear when many small movements become synchronized enough to behave as one field. The ocean, sound, light, and even the nervous system all do this, because life is vibrational in nature… Did this expand your perception?

🧬Maxpein🧬

31,782 次观看 • 6 个月前

This is one of the best scenes from the movie Fury. 1 tank and 5 men against an SS battalion. "There's a wave coming, and we're the rock to break that wave."

This is one of the best scenes from the movie Fury. 1 tank and 5 men against an SS battalion. "There's a wave coming, and we're the rock to break that wave."

Paratrooper Brady

229,689 次观看 • 6 个月前

She is from a Brahmin family and lived with a Muslim boy Imtiaz. Family left her for that. Now she is alone with the child and he beats her on the road. She has complained many times and is tired. This is the result of ignoring every warning and choosing him over the family.

She is from a Brahmin family and lived with a Muslim boy Imtiaz. Family left her for that. Now she is alone with the child and he beats her on the road. She has complained many times and is tired. This is the result of ignoring every warning and choosing him over the family.

Arpit

35,778 次观看 • 22 天前

We simulated tonight’s World Series Game 3, Brewers at Rays, 10,000 times ⚾ Every pitch is a real one: the pitcher’s own mix for that count, where he really puts it, and what each hitter does with it. Here’s the most typical game, and the matchups that swing it 👇

We simulated tonight’s World Series Game 3, Brewers at Rays, 10,000 times ⚾ Every pitch is a real one: the pitcher’s own mix for that count, where he really puts it, and what each hitter does with it. Here’s the most typical game, and the matchups that swing it 👇

RotoBot App

616,083 次观看 • 13 天前

The Moore curve of order 4 is a loop of 256 unit steps through every cell of a 16 × 16 grid. Each coordinate has its own real Fourier series, and one chain of arms per axis draws it. The second half of the loop is the first half turned 180°, so every even harmonic is zero, and its mirror leaves x only sines and y only cosines. A curve that wiggles everywhere converges slowly: 32.3 % of the height at N = 1, still 0.47 % at N = 255.

The Moore curve of order 4 is a loop of 256 unit steps through every cell of a 16 × 16 grid. Each coordinate has its own real Fourier series, and one chain of arms per axis draws it. The second half of the loop is the first half turned 180°, so every even harmonic is zero, and its mirror leaves x only sines and y only cosines. A curve that wiggles everywhere converges slowly: 32.3 % of the height at N = 1, still 0.47 % at N = 255.

matharium

72,548 次观看 • 27 天前

Victoria Falls is approximately 1700m wide, and varies in height from 80-108 meters. It's one and a half times wider than Niagara Falls and is twice the height ... Victoria Falls ,Zimbabwe 🇿🇼. #VisitZimbabwe

Victoria Falls is approximately 1700m wide, and varies in height from 80-108 meters. It's one and a half times wider than Niagara Falls and is twice the height ... Victoria Falls ,Zimbabwe 🇿🇼. #VisitZimbabwe

EXPLOREREYES

220,058 次观看 • 3 年前

Discrete Fourier Transform by hand ✍️ ~ 12 steps walkthrough below Here is a little-known secret about the DFT and the inverse DFT: it is just matrix multiplication in both directions, one the transpose of the other, exactly like the forward pass and backpropagation I drew in other examples. Goal: recover which cosine waves a signal is made of, using nothing but multiplication and addition. = 1. Given = Three signals written as sums of cosines, and a fourth, X, that we do not know yet. = 2. Frequency matrix F = Let us write the coefficients as a matrix. Each signal is a row, each frequency a column, so A = cos(w) + 2cos(2w) becomes [1, 2, 0, 0]. = 3. Sample the waves = We read the four cosine waves at ten discrete time points. That word "discrete" is the whole difference between this and the continuous transform. = 4. Cosine matrix W = Let us write those samples as a matrix: each frequency a row, each time point a column. = 5. Frequency to time = We multiply F by W. That combines the four cosine waves in the proportions F specifies, and the result T is the three signals as they would look in time. = 6. Transpose = Let us stand each signal up as a column. = 7. Time to frequency = We multiply W by that transpose. Every cell is the dot product of one signal with one cosine wave, which measures how much of that wave the signal contains. Zero means none of it. = 8. Scale = Let us multiply by 2/n, with n = 10. The projections come out five times too large, and this is the correction. = 9. Transpose back = We turn it back around, and it is F again, exactly. That is the check: the transform recovered the coefficients we started from. = 10. Now solve for X = Let us run the same multiplication on the one signal whose recipe we never knew. = 11. Scale = We divide by 5 again. = 12. Transpose back = And X reads [0, 0, 3, 2], which says X = 3cos(3w) + 2cos(4w). Note: I originally drew this to show that the DFT is a special case of a convolution layer, its filters fixed to sine and cosine waves rather than learned. No wonder, then, that a convolution layer free to learn its own filters can be trained to process signals. 💾 Save this post!

Discrete Fourier Transform by hand ✍️ ~ 12 steps walkthrough below Here is a little-known secret about the DFT and the inverse DFT: it is just matrix multiplication in both directions, one the transpose of the other, exactly like the forward pass and backpropagation I drew in other examples. Goal: recover which cosine waves a signal is made of, using nothing but multiplication and addition. = 1. Given = Three signals written as sums of cosines, and a fourth, X, that we do not know yet. = 2. Frequency matrix F = Let us write the coefficients as a matrix. Each signal is a row, each frequency a column, so A = cos(w) + 2cos(2w) becomes [1, 2, 0, 0]. = 3. Sample the waves = We read the four cosine waves at ten discrete time points. That word "discrete" is the whole difference between this and the continuous transform. = 4. Cosine matrix W = Let us write those samples as a matrix: each frequency a row, each time point a column. = 5. Frequency to time = We multiply F by W. That combines the four cosine waves in the proportions F specifies, and the result T is the three signals as they would look in time. = 6. Transpose = Let us stand each signal up as a column. = 7. Time to frequency = We multiply W by that transpose. Every cell is the dot product of one signal with one cosine wave, which measures how much of that wave the signal contains. Zero means none of it. = 8. Scale = Let us multiply by 2/n, with n = 10. The projections come out five times too large, and this is the correction. = 9. Transpose back = We turn it back around, and it is F again, exactly. That is the check: the transform recovered the coefficients we started from. = 10. Now solve for X = Let us run the same multiplication on the one signal whose recipe we never knew. = 11. Scale = We divide by 5 again. = 12. Transpose back = And X reads [0, 0, 3, 2], which says X = 3cos(3w) + 2cos(4w). Note: I originally drew this to show that the DFT is a special case of a convolution layer, its filters fixed to sine and cosine waves rather than learned. No wonder, then, that a convolution layer free to learn its own filters can be trained to process signals. 💾 Save this post!

Tom Yeh

25,684 次观看 • 3 个月前

Discrete Fourier Transform by hand ✍️ ~ 12 steps walkthrough below Here is a little-known secret about the DFT and the inverse DFT: it is just matrix multiplication in both directions, one the transpose of the other, exactly like the forward pass and backpropagation I drew in other examples. Goal: recover which cosine waves a signal is made of, using nothing but multiplication and addition. = 1. Given = Three signals written as sums of cosines, and a fourth, X, that we do not know yet. = 2. Frequency matrix F = Let us write the coefficients as a matrix. Each signal is a row, each frequency a column, so A = cos(w) + 2cos(2w) becomes [1, 2, 0, 0]. = 3. Sample the waves = We read the four cosine waves at ten discrete time points. That word "discrete" is the whole difference between this and the continuous transform. = 4. Cosine matrix W = Let us write those samples as a matrix: each frequency a row, each time point a column. = 5. Frequency to time = We multiply F by W. That combines the four cosine waves in the proportions F specifies, and the result T is the three signals as they would look in time. = 6. Transpose = Let us stand each signal up as a column. = 7. Time to frequency = We multiply W by that transpose. Every cell is the dot product of one signal with one cosine wave, which measures how much of that wave the signal contains. Zero means none of it. = 8. Scale = Let us multiply by 2/n, with n = 10. The projections come out five times too large, and this is the correction. = 9. Transpose back = We turn it back around, and it is F again, exactly. That is the check: the transform recovered the coefficients we started from. = 10. Now solve for X = Let us run the same multiplication on the one signal whose recipe we never knew. = 11. Scale = We divide by 5 again. = 12. Transpose back = And X reads [0, 0, 3, 2], which says X = 3cos(3w) + 2cos(4w). Note: I originally drew this to show that the DFT is a special case of a convolution layer, its filters fixed to sine and cosine waves rather than learned. No wonder, then, that a convolution layer free to learn its own filters can be trained to process signals. 💾 Save this post!

Discrete Fourier Transform by hand ✍️ ~ 12 steps walkthrough below Here is a little-known secret about the DFT and the inverse DFT: it is just matrix multiplication in both directions, one the transpose of the other, exactly like the forward pass and backpropagation I drew in other examples. Goal: recover which cosine waves a signal is made of, using nothing but multiplication and addition. = 1. Given = Three signals written as sums of cosines, and a fourth, X, that we do not know yet. = 2. Frequency matrix F = Let us write the coefficients as a matrix. Each signal is a row, each frequency a column, so A = cos(w) + 2cos(2w) becomes [1, 2, 0, 0]. = 3. Sample the waves = We read the four cosine waves at ten discrete time points. That word "discrete" is the whole difference between this and the continuous transform. = 4. Cosine matrix W = Let us write those samples as a matrix: each frequency a row, each time point a column. = 5. Frequency to time = We multiply F by W. That combines the four cosine waves in the proportions F specifies, and the result T is the three signals as they would look in time. = 6. Transpose = Let us stand each signal up as a column. = 7. Time to frequency = We multiply W by that transpose. Every cell is the dot product of one signal with one cosine wave, which measures how much of that wave the signal contains. Zero means none of it. = 8. Scale = Let us multiply by 2/n, with n = 10. The projections come out five times too large, and this is the correction. = 9. Transpose back = We turn it back around, and it is F again, exactly. That is the check: the transform recovered the coefficients we started from. = 10. Now solve for X = Let us run the same multiplication on the one signal whose recipe we never knew. = 11. Scale = We divide by 5 again. = 12. Transpose back = And X reads [0, 0, 3, 2], which says X = 3cos(3w) + 2cos(4w). Note: I originally drew this to show that the DFT is a special case of a convolution layer, its filters fixed to sine and cosine waves rather than learned. No wonder, then, that a convolution layer free to learn its own filters can be trained to process signals. 💾 Save this post!

Tom Yeh

13,644 次观看 • 2 个月前