A square wave from 5 sine waves
fourier is one function. Everything below came out of it during this
build, at parameters taken from the essays rather than invented for this page — so a figure
here is the same figure a reader meets in an essay, and if the generator changes, this page
changes with it.
At its defaults
show: "weierstrass"
show: "zoom"
show: "spectrum"
show: "progression"
show: "gibbs"
What it checks while it draws
Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.
- the 0-th line of the period-2 train sits on the envelope ×94
- harmonic 1 has decayed by exp(−1²t) at t = 0.005 ×18
- sine 1 against sine 2 integrates to nothing ×12
- the 1-term curve is sampled fast enough to be the function rather than an alias ×5
- sine 1 against sine 1 integrates to π ×4
- the lines of the period-2 train are 1/T apart ×4
- the projection onto sine 1 is the harmonic this family draws ×3
- the projection onto sine 2 cancels to nothing ×2
- a longer period puts the lines closer together ×1
- and its finest oscillation is more than three pixels wide ×1
- and the slopes grow faster than the amplitudes shrink, which is the whole condition ×1
- and the terms it leaves out are below a twentieth of what the window shows ×1
- between 5 and 121 harmonics carry the profile ×1
- between 5 and 61 harmonics ×1
- between two and four harmonics, each between 1 and 12 ×1
- between two and four periods, each positive ×1
- between two and six times, none of them negative ×1
- by the last frame the profile is a single sine to within two percent ×1
- each window is between a hundred-thousandth and two wide ×1
- each window is drawn from between 200 and 4000 samples ×1
- each window is sampled fast enough for the terms it draws ×1
- every term count is a whole number between 1 and 200 ×1
- magnifying ten times does not flatten the curve ten times ×1
- more terms is closer, away from the jump ×1
- the amplitudes shrink ×1
- the coefficients of the sawtooth wave fall like 1/m^1 ×1
- the coefficients of the square wave fall like 1/m^1 ×1
- the coefficients of the triangle wave fall like 1/m^2 ×1
- the construction is the one with no tangents ×1
- the curve drawn is the sum of the harmonics ×1
- the deepest zoom still has something to look at ×1
- the fine detail fades faster than the coarse shape ×1
- the frequencies multiply by an odd whole number ×1
- the pulse fits inside every period drawn ×1
- the spectrum has something in it ×1
- the steepest chord gets steeper as the spacing shrinks ×1
- the view is one the family draws ×1
- the wave is one the family draws ×1
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
A curve with a corner at every point
Continuity means a curve can be drawn without lifting the pen. Differentiability means it has a tangent. The first was assumed to nearly imply the second until 1872, when Weierstrass exhibited a curve that is continuous everywhere and has a tangent nowhere — and it is a sum of cosines.
NumberOne sum, squared two ways
Add the p-th roots of unity, each taken with a plus or a minus according to whether its index is a square. The walk that results closes on a point at distance √p from the origin — and squaring that one number, evaluated two different ways, is the reciprocity law.
AnalysisA square wave built entirely out of round ones
Add enough sine waves together and flat tops and vertical cliffs appear from nothing. Almost — there is a 9% overshoot that never goes away, and it is not a bug.
AnalysisThe corners go first
Fourier was not decomposing waves for the pleasure of it. He was solving the flow of heat, and the whole apparatus exists because each harmonic fades at a rate set by the square of its frequency — which is why a sharp profile smooths instantly and why the flow cannot be run backwards.
ProbabilityThe shape that averaging leaves alone
Adding independent quantities blurs their distributions together, and rescaling restores the width. Almost every shape is changed by that operation. Exactly one is returned unaltered, and that is why sums of unrelated things keep arriving at it.
AnalysisThe staircase that is not the diagonal
A staircase can be made to follow a quarter circle as closely as anyone likes. Its length is 2 at every stage and the arc's length is 1.5708, and no amount of refinement closes the gap — which is a fact about length rather than about staircases.
ProbabilityThe walk that becomes a curve
Shrink the steps of a random walk and it disappears. Shrink them while stretching the time in the right proportion — space by the square root of whatever time is divided by — and something is left behind, which is a curve nobody could draw.
AnalysisWhen the period grows without bound
A repeating signal has a spectrum of separate lines. Stretch the gap between repeats and the lines crowd together while the curve they sit on stays exactly where it is — and at infinite period the lines are gone and the curve is the whole answer.
AnalysisWhere the coefficients come from
The recipe for a square wave has a four over pi in front and a one over three on the second term, and the first rung of this ladder used them without saying where they came from. They come from multiplying by one harmonic and taking the area.