Generator

A square wave from 5 sine waves

A generator in the analysis library, called 55 times across 13 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

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.

With nothing chosen

A square wave from 5 sine waves. The sum of the first 5 harmonics, compared with the square wave it approaches.

6 cosines, and a curve with no tangent anywhere

The same corner, magnified 1,000 times

The copies each grid makes of the disc

Samples needed, square grid against the best

The winning lattices are nearly hexagonal

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 partial sum of Q₂₅₆ at 0 up to frequency 0 ×7
the 1-term curve is sampled fast enough to be the function rather than an alias ×5
the partial sum of Q₂ at 0 up to frequency 0 ×5
sine 1 against sine 1 integrates to π ×4
the lines of the period-2 train are 1/T apart ×4
harmonic 1 of the heat profile has decayed by exp(−m²π²t) ×3
the projection onto sine 1 is the harmonic this family draws ×3
between 6 and 40 harmonics ×2
the projection onto sine 2 cancels to nothing ×2
113 frequencies need 169 samples on a square grid and 124 on the best lattice ×1
a longer period puts the lines closer together ×1
a missing harmonic has a node at the pluck ×1
a ramp width up to 0.2 ×1
a time from 0 to 2 ×1
and every Lₙ past ten is within 0.02 of the logarithm ×1
and it is never negative ×1
and its finest oscillation is more than three pixels wide ×1
and so is its absolute area ×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
at every tick the two sines agree ×1
at three turns a second the eight-spoked wheel stands still, just below it turns backwards ×1
between 5 and 121 harmonics carry the profile ×1
between two and eight times, each from 0 to 4 ×1
between two and five times, each from 0 to 2 ×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 block's peak stands higher than the last ×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
every winning lattice is within nine degrees and five per cent of hexagonal, and within a degree and a half from W = 10 on ×1
halfway through, the shape is the pluck inverted and mirrored ×1
in three dimensions the best lattice needs well under two-thirds of the cubic grid's samples ×1
its signed area is 1 ×1
L₁ is 1/3 + 2√3/π ×1
Lₙ − (4/π²) ln N has settled to its limit ×1
Lₙ rises with N ×1
magnifying ten times does not flatten the curve ten times ×1
more terms is closer, away from the jump ×1
N from 2 to 40 ×1
one to three kernel sizes from 1 to 60 ×1
one to three pluck positions, each away from the ends ×1
reconstruction fails below twice the top frequency and succeeds above it ×1
the amplitudes shrink ×1
the apparent turn is within half a spoke gap ×1
the area of |D₁₂| by quadrature matches Fejér's closed form ×1
the area of |D₄| by quadrature matches Fejér's closed form ×1
the averaged kernel is the mean of the first N + 1 Dirichlet kernels ×1
the best lattice always beats the square grid and settles near 2√3/π samples a frequency ×1
the best lattice's 124 samples give the pattern back exactly; a square grid of 121 does not ×1
the cardinal series rebuilds the signal between the samples ×1
the closed-form b₁ is the projection of the tent onto sin(1πx) ×1
the closed-form b₂ is the projection of the tent onto sin(2πx) ×1
the closed-form b₅ is the projection of the tent onto sin(5πx) ×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 cubic grid needs 2/√3, √2, 2, 16 and 2²⁴ times as many samples in dimensions 2, 3, 4, 8 and 24 ×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 folded frequency lies between nought and half the rate ×1
the frequencies multiply by an odd whole number ×1
the function never leaves [−1, 1] ×1
the motion repeats after a time of 2 ×1
the partial sum at 0 comes within a tenth of Lₙ and never exceeds it ×1
the pluck is at least a twentieth from either end ×1
the pulse fits inside every period drawn ×1
the range runs to between 100 and 100,000 ×1
the rings outrun the grid inside the square ×1
the signed area of Dₙ is 1 ×1
the spectrum has something in it ×1
the steepest chord gets steeper as the spacing shrinks ×1
the string keeps a slope at least half the pluck's steepest at every time ×1
the travelling halves and 400 terms of the series agree to within the series' own tail ×1
the view is one the family draws ×1
the wave is one the family draws ×1
the Wilbraham–Gibbs integral ×1
two to four blocks — the fourth sits at 2⁶⁴ ×1

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Analysis

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.

Analysis

A grid that packs the copies

Sampling a pattern on a grid makes copies of its frequencies, one for every point of a lattice, and the pattern can be read back from the samples only if the copies do not collide. For a pattern whose frequencies fill a disc the square grid wastes room between its copies. Searching every lattice of samples for the fewest that work finds a nearly hexagonal one every time — 124 samples where the square grid needs 169 — and the saving is the hexagonal packing's over the square, 13.4% in the limit. In three dimensions it is 29%, and in twenty-four the cubic grid needs 16,777,216 times as many samples as the best.

Analysis

A plucked string keeps its corners

Heat smooths a sharp profile at once, because each harmonic decays at a rate set by the square of its frequency. Change one time derivative into two and nothing decays at all: each harmonic swings for ever, the corner of a pluck splits in two and runs along the string, and after one period the shape comes back exactly. The same sines, the same coefficients — and a flow that loses nothing.

Analysis

A wave known only at its ticks

Read a sine wave nine cycles a second at ten instants a second and the readings describe a wave of one cycle a second instead. Read a wave with nothing faster than 3.7 cycles a second at more than 7.4 instants a second and the readings determine it exactly. Both facts are the same fold of the frequency axis, and it is why wagon wheels turn backwards on film.

Number

One 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.

Analysis

A 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.

Analysis

The 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.

Analysis

The ripples that make a series run away

Adding up the first N terms of a Fourier series is the same as averaging the function against one fixed wiggly curve. Its area is always one, but the area of its absolute value grows like the logarithm of N, without limit — and that single number is enough to force a continuous function, with no jump and no corner anywhere, whose Fourier series diverges at a point. Averaging the partial sums removes the negative ripples, and with them the whole problem.

Probability

The 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.

Analysis

The 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.

Probability

The 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.

Analysis

When 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.

Analysis

Where 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 essay that built a square wave from sines used them without saying where they came from. They come from multiplying by one harmonic and taking the area.

The whole library · What the figures prove