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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... show more

Philosophy Of Physics

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39,107 次观看 • 1 个月前 •via X (Twitter)

科学技术动物教育

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Peter MOUEZA🇲🇫 的头像

Peter MOUEZA🇲🇫1 个月前

By Fourier

Klaudivan 的头像

Klaudivan1 个月前

@threadreaderapp unroll please

Giovanni 的头像

Giovanni1 个月前

love that. wow

Master Steward of New London 的头像

Master Steward of New London1 个月前

This is how they program drawing automatons, by the way

Ariz Ashraf 的头像

Ariz Ashraf1 个月前

Basically the first few waves just sketch the rough outline of the bird, then the higher ones fill in the sharp feathers and finer bits. Gets pretty accurate once you add enough of them.

TheRoloway 的头像

TheRoloway1 个月前

Using Fourier series to redraw a hummingbird outline is a beautiful way to show how low frequencies hold the shape and the high ones chase the feathers.

Nícolas Abreu 的头像

Nícolas Abreu1 个月前

@grok me explique o que tá acontecendo de forma leiga e resumida.

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Sine and cosine are two projections of a single rotating point. As it moves around the unit circle, its vertical position traces the sine wave, while its horizontal position traces the cosine wave.

Sine and cosine are two projections of a single rotating point. As it moves around the unit circle, its vertical position traces the sine wave, while its horizontal position traces the cosine wave.

The Math Flow

14,899 次观看 • 11 天前

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

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 tip's height is the wave. With 1, 3, 9, 30 and 100 circles the tops flatten and the edges sharpen, but a 9 percent overshoot never goes away: the Gibbs phenomenon. Pitch follows the height.

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 tip's height is the wave. With 1, 3, 9, 30 and 100 circles the tops flatten and the edges sharpen, but a 9 percent overshoot never goes away: the Gibbs phenomenon. Pitch follows the height.

Math Pulse

11,179 次观看 • 17 天前

Each coordinate of the Koch snowflake's outline has its own real Fourier series, x(t) = ∑ (aₖ cos kt + bₖ sin kt), and one chain of arms per axis draws it where their guides cross. The snowflake's six-fold symmetry leaves only k ≡ ±1 (mod 6), so five arms in six are zero, and its mirror leaves x only sines and y only cosines. The error falls from 11.5 % of the height at N = 1 to 0.60 % at N = 193. At N = 1 the two arms are equal: a circle.

Each coordinate of the Koch snowflake's outline has its own real Fourier series, x(t) = ∑ (aₖ cos kt + bₖ sin kt), and one chain of arms per axis draws it where their guides cross. The snowflake's six-fold symmetry leaves only k ≡ ±1 (mod 6), so five arms in six are zero, and its mirror leaves x only sines and y only cosines. The error falls from 11.5 % of the height at N = 1 to 0.60 % at N = 193. At N = 1 the two arms are equal: a circle.

matharium

92,328 次观看 • 1 个月前

A lens can perform the Fourier transform! 🔍 This is the 4F system: the first lens converts an image into its frequency components, and the second lens reconstructs it. By adding an aperture at the Fourier plane, we can filter out high frequencies. #Optics #Physics #b3d

A lens can perform the Fourier transform! 🔍 This is the 4F system: the first lens converts an image into its frequency components, and the second lens reconstructs it. By adding an aperture at the Fourier plane, we can filter out high frequencies. #Optics #Physics #b3d

Hans Chiu

220,741 次观看 • 1 年前

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 • 1 个月前

The mating call of an alligator The vibrations you see radiating in the water are faraday waves produced by vibrating their lungs at low frequencies and the scutes on their back This communicates his desire for company some distance through the water

The mating call of an alligator The vibrations you see radiating in the water are faraday waves produced by vibrating their lungs at low frequencies and the scutes on their back This communicates his desire for company some distance through the water

Science girl

784,430 次观看 • 3 年前

I believe this is a projectile launched from a cloaked Pleiadian craft which is used to eliminate undesired frequencies in a specific location. ------- #NEIOH - "This Is A Laser Focused Frequency Sent From A Pleiadian Craft. The Energy Is Powerful And What Is Seen Is The Trail From A Small Pod Like Container That Will Dissolve Quickly When The Target Is Hit. This Is A Secret Operation Of The Galactic Federation. The Mothership Is Miles Away And The Fleet Of Crafts Has Been Dispersed In Many Directions With Each Projecting Frequencies Of Light. This Speed Can Only Be Attained By The Direction From Our Crafts. The Purpose Is To Eliminate Areas Of Frequencies Where Dark Forces Have Their Own Agenda." Malvern Hills, England - August 5, 2025

I believe this is a projectile launched from a cloaked Pleiadian craft which is used to eliminate undesired frequencies in a specific location. ------- #NEIOH - "This Is A Laser Focused Frequency Sent From A Pleiadian Craft. The Energy Is Powerful And What Is Seen Is The Trail From A Small Pod Like Container That Will Dissolve Quickly When The Target Is Hit. This Is A Secret Operation Of The Galactic Federation. The Mothership Is Miles Away And The Fleet Of Crafts Has Been Dispersed In Many Directions With Each Projecting Frequencies Of Light. This Speed Can Only Be Attained By The Direction From Our Crafts. The Purpose Is To Eliminate Areas Of Frequencies Where Dark Forces Have Their Own Agenda." Malvern Hills, England - August 5, 2025

Kab

339,131 次观看 • 1 年前

In 1859, Riemann wrote one sentence about where a function equals zero. 10 trillion cases checked, not a single counterexample - and it's still unproven. Worth $1 million. The zeros of the zeta function encode the exact position of every prime number as a sum of waves - the same Fourier decomposition that every transformer uses when its positional encoding layers convert token positions into sine and cosine frequencies. Each zero at 1/2 + iγ produces a wave in log x with height growing like √x and frequency γ. The first zero at 14.13i repeats every time x gets multiplied by 1.56. Add 100 pairs of waves and the prime count at 100 converges from 98.16 to within 0.4 of truth. If a single zero sits off the critical line, its wave would grow faster than √x and eventually swamp every other correction. In 2025, an AI agent called Gauss completed a machine-checked proof of the prime number theorem in Lean - verified line by line. But the full hypothesis needs an idea no human or AI has found yet. 10 minutes. Bookmark & watch today. The $1 million sentence that decides where every prime number lives.

In 1859, Riemann wrote one sentence about where a function equals zero. 10 trillion cases checked, not a single counterexample - and it's still unproven. Worth $1 million. The zeros of the zeta function encode the exact position of every prime number as a sum of waves - the same Fourier decomposition that every transformer uses when its positional encoding layers convert token positions into sine and cosine frequencies. Each zero at 1/2 + iγ produces a wave in log x with height growing like √x and frequency γ. The first zero at 14.13i repeats every time x gets multiplied by 1.56. Add 100 pairs of waves and the prime count at 100 converges from 98.16 to within 0.4 of truth. If a single zero sits off the critical line, its wave would grow faster than √x and eventually swamp every other correction. In 2025, an AI agent called Gauss completed a machine-checked proof of the prime number theorem in Lean - verified line by line. But the full hypothesis needs an idea no human or AI has found yet. 10 minutes. Bookmark & watch today. The $1 million sentence that decides where every prime number lives.

Eron

31,159 次观看 • 14 天前

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

Is anyone interested in seeing more posts about the bells, their history, and the power of their frequencies?🔔

Is anyone interested in seeing more posts about the bells, their history, and the power of their frequencies?🔔

FLAT OUT TRUTH

373,891 次观看 • 1 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

A square is constructed using its Fourier series (represented by sums of circles rotating around circles), then the higher order (smaller, faster rotation) circles are removed in turn. Credit: generomics

A square is constructed using its Fourier series (represented by sums of circles rotating around circles), then the higher order (smaller, faster rotation) circles are removed in turn. Credit: generomics

Mathematica

137,863 次观看 • 7 个月前

This shows ~1,200 years of seawater erosion of the Khafre Pyramid, compressed into one minute. For the first ~500 years a high ocean stand holds about 300 ft above the base; its waves lap the soft Tura limestone casing and dissolve it (carbonic acid) at roughly 0.08 ft/year — about one inch of perpendicular recession annually, and ~14× faster than the slow dissolution on submerged surfaces below. As the sea recedes, it then pauses at successively lower stands, each held ~75–100 years, and each carves a distinct horizontal band where the casing is stripped to the bare core. Loosened blocks slough off at their own pace and fall down each face, piling into talus that mounds toward the center of all four sides. The smooth white cap above the highest waterline is never reached and survives intact.

This shows ~1,200 years of seawater erosion of the Khafre Pyramid, compressed into one minute. For the first ~500 years a high ocean stand holds about 300 ft above the base; its waves lap the soft Tura limestone casing and dissolve it (carbonic acid) at roughly 0.08 ft/year — about one inch of perpendicular recession annually, and ~14× faster than the slow dissolution on submerged surfaces below. As the sea recedes, it then pauses at successively lower stands, each held ~75–100 years, and each carves a distinct horizontal band where the casing is stripped to the bare core. Loosened blocks slough off at their own pace and fall down each face, piling into talus that mounds toward the center of all four sides. The smooth white cap above the highest waterline is never reached and survives intact.

Sterling Cooley

642,546 次观看 • 5 个月前

David Icke on The Law of attraction 🧲 The Law of Attraction teaches us that the energy we radiate into the universe is what we attract back into our lives. By focusing on positive thoughts and emotions, we can manifest abundance, joy, and success. You are a magnet for all that you desire! Everything in the universe is made up of energy, including our thoughts and emotions. The Law of Vibration states that similar frequencies attract each other. By raising our vibrations through positive thoughts, gratitude, and self-love, we align ourselves with the higher frequencies of love, abundance, and well-being...

David Icke on The Law of attraction 🧲 The Law of Attraction teaches us that the energy we radiate into the universe is what we attract back into our lives. By focusing on positive thoughts and emotions, we can manifest abundance, joy, and success. You are a magnet for all that you desire! Everything in the universe is made up of energy, including our thoughts and emotions. The Law of Vibration states that similar frequencies attract each other. By raising our vibrations through positive thoughts, gratitude, and self-love, we align ourselves with the higher frequencies of love, abundance, and well-being...

Wisdom Walk

17,034 次观看 • 8 个月前

Pay attention to how Jake Barber describes the feeling and the emotions of the divine feminine energy. "Tuned in," "frequency," "possessed," "spirit"—they are using (Skywatchers) these frequencies to summon UAPs or higher entities. The same frequencies were, and are, used by the CIA in MK ULTRA, Project Stargate, and the Gateway Experience. The same frequencies are released during earthquakes, which is why many people witness lights or orbs. These frequencies were reproduced in the ancient religions, rituals, drumming, rites, etc., and the temples were designed to produce these frequencies. (1/3)🧵

Pay attention to how Jake Barber describes the feeling and the emotions of the divine feminine energy. "Tuned in," "frequency," "possessed," "spirit"—they are using (Skywatchers) these frequencies to summon UAPs or higher entities. The same frequencies were, and are, used by the CIA in MK ULTRA, Project Stargate, and the Gateway Experience. The same frequencies are released during earthquakes, which is why many people witness lights or orbs. These frequencies were reproduced in the ancient religions, rituals, drumming, rites, etc., and the temples were designed to produce these frequencies. (1/3)🧵

Open Minded Approach

154,725 次观看 • 1 年前

Primes Numbers Are Not Random Noise Prime numbers look scattered when you meet them one by one, but their disorder has structure. In this animation, every integer is placed on a growing spiral, primes ignite as sharp golden events, and the hidden oscillations come from the first Riemann zeta zeros. The scene is based on the von Mangoldt prime signal Λ(n) and the explicit formula for ψ(x), where primes appear as impulses and the zeta zeros behave like frequencies correcting the smooth drift of x. #PrimeNumbers #RiemannHypothesis #NumberTheory #Mathematics #MathAnimation

Primes Numbers Are Not Random Noise Prime numbers look scattered when you meet them one by one, but their disorder has structure. In this animation, every integer is placed on a growing spiral, primes ignite as sharp golden events, and the hidden oscillations come from the first Riemann zeta zeros. The scene is based on the von Mangoldt prime signal Λ(n) and the explicit formula for ψ(x), where primes appear as impulses and the zeta zeros behave like frequencies correcting the smooth drift of x. #PrimeNumbers #RiemannHypothesis #NumberTheory #Mathematics #MathAnimation

Mathelirium

16,800 次观看 • 5 个月前

Joseph Fourier’s 1823 engraving, traced as 57 ink shapes joined by thin bridges into one closed outline, 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 42.0 % of the portrait’s height at N = 1, 3.16 % at 128 and 0.92 % at 512, and the circles shrink like k^−1.47. The tip’s distance |z(t)| plays as pitch.

Joseph Fourier’s 1823 engraving, traced as 57 ink shapes joined by thin bridges into one closed outline, 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 42.0 % of the portrait’s height at N = 1, 3.16 % at 128 and 0.92 % at 512, and the circles shrink like k^−1.47. The tip’s distance |z(t)| plays as pitch.

matharium

108,520 次观看 • 1 个月前

there is no distortion, waveshaping, or clipping effect on these sawtooth chords. It's just an additional 500 pure sine waves, their frequencies set to the sum/differences of harmonics in the chords.

there is no distortion, waveshaping, or clipping effect on these sawtooth chords. It's just an additional 500 pure sine waves, their frequencies set to the sum/differences of harmonics in the chords.

sudara

16,821 次观看 • 2 个月前

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 • 1 个月前