Change and calculus

1822

Everything is waves

Joseph Fourier

Can any sound, a violin, a voice, a door slamming, be built from pure tones?

Studying how heat spreads, Fourier claimed that almost any shape can be built by adding up simple waves. Many mathematicians doubted it. Today it is how music, images and radio signals are stored and sent.

First, make a guess

How many spinning circles do you need before a heart looks like a heart?

How they figured it out

Joseph Fourier went with Napoleon to Egypt in 1798 and later ran a French province while studying how heat spreads through solid bodies. In 1807 he claimed that any temperature profile, even one with sharp corners, could be written as a sum of smooth sine waves.

Lagrange, among others, objected that this could not possibly be true in general. Fourier kept refining it and published his book in 1822. Mathematicians spent the next century working out exactly when he was right, which turned out to be almost always. Fourier is also often credited with first describing what we now call the greenhouse effect.

The equation

f(x)=∑ncn einxf(x) = \sum_{n} c_n\, e^{i n x}

Any repeating pattern is a stack of pure waves, each with its own size.

Everyday echoThe equaliser bars in a music app are a Fourier transform.

See it in real life

How your phone recognises a song

Song-recognition apps chop a few seconds of music into its frequencies and match the pattern against millions of songs. Noise-cancelling headphones, MP3 files, guitar tuners and Wi-Fi all run on Fourier’s idea.

Where engineers use it

  • Audio and images

    MP3 and JPEG keep the strongest frequency components and drop the rest.

  • Vibration analysis

    Engineers find failing bearings by spotting new peaks in a machine’s spectrum.

  • Heat flow

    Fourier invented the series to solve heat conduction, still used to design heat sinks.

Where it came from

  • Circles on circles c. AD 150

    Analogy only

    Circles riding on circles can trace almost any repeating path, just as stacked waves can build almost any shape.

    Where the comparison stops: Ptolemy fitted circles by hand to match the sky, with no theory of why or when it works. Fourier’s mathematics says exactly when such sums succeed.

  • Euler’s formula 1748

    Built on

    Fourier’s waves are most neatly written with Euler’s formula.

What it led to

  • The first image of a black hole April 2019

    Same technique

    Each pair of telescopes measures one Fourier ingredient of the image.

    Where the comparison stops: Only a scattered set of ingredients is measured, so rebuilding the picture needs extra assumptions, tested carefully by several independent teams.