Matpic
mathematics, in pictures
Generator

A square wave from 5 sine waves

A generator in the analysis library, called 43 times across 11 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

6 cosines, and a curve with no tangent anywhere. Partial sums of a sum of cosines whose amplitudes shrink geometrically and whose frequencies grow faster. Each term adds finer detail; the curve converges and its slopes do not.

The same corner, magnified 1,000 times

The same corner, magnified 1,000 times. Four windows on the same curve, each a tenth the width of the last. The curve looks the same at every magnification, which is what having no tangent anywhere amounts to.

A plucked string, released

A plucked string, released. A string plucked into a tent shape at 0.5, drawn at times 0, 0.1, 0.25, 0.5, 0.75, 1: the corner splits into two corners that run apart, reflect off the ends upside down and meet again.

One tent, as heat and as a string

One tent, as heat and as a string. The same tent profile evolved by the heat equation and by the wave equation, at times 0, 0.02, 0.1, 0.3. The heat profile rounds and sinks; the string's corner splits and travels.

Two travelling halves add up to the string

Two travelling halves add up to the string. d'Alembert's solution for a string plucked at 0.2: half the odd periodic extension moving right and half moving left, with their sum on the interval from 0 to 1 at time 0.3.

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.

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 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.

 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