Take any wave, square, sawtooth, triangle, any shape at all.
Fourier proved you can build it from simple sine waves.
Add enough of them together and you can approximate any periodic function you like.
This seems like a mathematical curiosity.
But it unlocked the physics of heat, the engineering of electronics, and the mathematics of infinity itself.
Last time, we met Gaspar Monge, the revolutionary geometer who invented descriptive geometry
and founded the École Polytechnique.
We saw how a military secret became the foundation of modern technical education.
Welcome back to Men of Mathematics.
Today, we meet Joseph Fourier, orphan, revolutionary, Egyptian adventurer,
and the mathematician who was arrested, but survived.
Perhaps his mathematical abilities made him too valuable
schools and nearly losing his head during the terror. He was arrested, but survived.
Perhaps his mathematical abilities made him too valuable to guillotine.
In 1798, Napoleon invited Fourier to join the Egyptian expedition. For three years,
Fourier served as secretary of the Institut d'Egypte, organizing the scientific survey of Egypt
and surviving adventure, plague, and military defeat.
The experience marked him forever.
Back in France, he kept his rooms uncomfortably hot.
He could never get warm enough after Egypt.
As prefect of Iser, a regional administrator,
Fourier had time to think about heat.
How does temperature change over time?
How does heat flow through a metal bar?
He derived the heat equation.
The rate of temperature change over time equals a constant times the curvature of the temperature distribution.
To solve the heat equation, Fourier made a stunning claim.
Any function can be expressed as an infinite sum of sines and cosines.
A square wave from smooth sines, a sawtooth from periodic curves.
It seemed impossible.
Lagrange himself objected.
But Fourier was right.
Each sine component decays at a different rate under the heat equation.
High-frequency components, sharp features, decay fastest.
Low-frequency components, gradual variations, persist longest.
This explains why a heated rod smooths out its temperature.
The mathematics of heat and the mathematics of waves are secretly the same For non functions Fourier extended his idea to the Fourier transform which converts any signal between the time domain when things happen and the frequency domain
which frequencies are present. Fourier analysis is everywhere. MP3 audio compression stores
frequencies, not samples. JPEG image compression uses Fourier cousins in 2D. MRI machines reconstruct
images from frequency data. Telecommunications separates radio stations by frequency. And
quantum mechanics uses wave-particle duality with Fourier. Fourier's results were correct,
but his proofs weren't rigorous by modern standards. Making Fourier series mathematically
precise, occupied the greatest minds of the 19th century. Dirichlet established conditions for
convergence. Riemann developed integration theory. Cantor invented set theory from studying Fourier
series. And Labesco created modern integration. The quest to understand Fourier series drove the
development of modern analysis, Fourier made another discovery. The Earth's atmosphere acts
as an insulating layer, trapping heat from the sun. This was the first recognition of what we now call
the greenhouse effect. Fourier also insisted that physical equations must have consistent dimensions.
You can add meters to seconds This principle of dimensional homogeneity now standard in physics was pioneering in 1822 Fourier masterwork published in 1822 is considered one of the great scientific books of all time
Lord Kelvin called Fourier's theorem, one of the most beautiful results of modern analysis.
Joseph Fourier discovered that any function can be decomposed into simple waves,
signs and cosines at different frequencies.
This insight, born from studying heat flow,
transformed mathematics, physics, and engineering,
from audio compression to medical imaging,
from telecommunications to quantum mechanics.
Fourier's idea echoes everywhere.
There's something poetic about Fourier's hit the notification bell.
New episodes release every week.
I'll see you next time.