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Concept

Spectrum

The list of how much of each frequency a signal holds, which describes it as completely as its shape over time does. Reading it is how a wave is decomposed, and the fact that convolution multiplies spectra is why it settles so many problems.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

Partial sums of the square wave. Approximations using 1, 3, 7, 21 terms; the corners sharpen but a fixed overshoot remains.

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 · Fourier series The target, multiplied by one harmonic at a time. Four panels, each showing the square wave multiplied by a single sine. The areas cancel exactly except against the harmonics the wave actually contains.

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.

analysis · Fourier series The spectrum of a pulse train, as the period grows. The same pulse repeated at three different intervals, with its spectrum below each. The lines move closer together as the period lengthens and the curve they lie on does not move at all.

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 · Fourier series A square profile of heat, spreading. The same profile at four times, each drawn from the same harmonics with each one damped by the exponential of minus its frequency squared times the time. The corners go first.

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 · Fourier series 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.

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 · Fourier series Two different graphs with the same adjacency matrix eigenvalues. a star with four arms: adjacency matrix eigenvalues 2, 0³, −2; a square and a lone point: adjacency matrix eigenvalues 2, 0³, −2. The characteristic polynomials are identical.

Two graphs the eigenvalues cannot tell apart

A graph's matrix has eigenvalues, and they count a surprising amount of the drawing: its edges, its triangles, every closed walk of every length. They do not count everything. A star with four arms and a square beside a lone point have the same eigenvalues exactly, although one of them is in two pieces — and on six points ten of the 156 graphs have a twin of this kind.

algebra · Linear maps The 11 trees on 7 points, and how many labellings each has. All trees on 7 points with their numbers of symmetries and of distinct labellings; the labellings add to 16807.

Almost every tree can be turned over

Cayley's n^(n−2) counts trees with labels on their points. Take the labels off and the count has no formula, because a symmetric shape absorbs labellings: the star on seven points can be labelled only seven ways, the one asymmetric shape 5,040. The bookkeeping that reconciles the two counts says something unexpected — almost every tree, labelled or not, can be turned over onto itself, where almost every graph cannot.

discrete · Labelled trees

Named alongside it

The objects these essays reach for when they reach for this one.

Fourier analysisHarmonicsPeriodicityConvergenceEigenfunctionExhaustive searchHeat equationIrreversibilityOrthogonalityAutomorphismCanonical formCharacteristic polynomial

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