In 1798 Joseph Fourier, a 30-year-old professor at the École Polytechnique in Paris, received an urgent message from the minister of the interior informing him that his country required his services, and that he should “be ready to depart at the first order.” Two months later, Fourier set sail from Toulon as part of a 25,000-strong military fleet under the command of General Napoleon Bonaparte, whose unannounced objective was the invasion of Egypt.

Fourier was one of 167 eminent scholars, the savants, assembled for the Egyptian expedition. Their presence reflected the French Revolution’s ideology of scientific progress, and Napoleon, a keen amateur mathematician, liked to surround himself with colleagues who shared his interests.

It is said that when the French troops reached the Great Pyramid at Giza, Napoleon sat in the shade underneath, scribbled a few notes in his jotter and announced that there was enough stone in the pyramid to build a wall 3 meters high and a third of a meter thick that would almost perfectly encircle France.

Gaspard Monge, his chief mathematician, confirmed that the General’s estimate was indeed correct. The Great Pyramid has sides of length 751 feet and a height of 479 feet. France is roughly a rectangle 480 miles north to south by 435 miles east to west. With these figures, Napoleon’s estimate is only 3 percent off.

Excerpted from The Grapes of Math: How Life Reflects Numbers and Numbers Reflect Life

On Fourier’s return from Egypt, Napoleon appointed him prefect of the Alpine department of Isère, based in Grenoble. Always a man of fragile health, with extreme sensitivity to cold, Fourier never left home without an overcoat, even in the summer, often making sure a servant carried a second coat for him in reserve. He kept his rooms baking hot at all times.

In Grenoble, his academic research was also preoccupied with heat. In 1807 he published a groundbreaking paper, On the Propagation of Heat in Solid Bodies. In it he revealed a remarkable finding about sinusoids.

What's So Special About Sinusoids?

The sinusoid is what’s called a “periodic wave,” an entity in which a curve repeats itself again and again along the horizontal axis. The sinusoid is the simplest type of periodic wave because the circle, which generates it, is the simplest geometrical shape. Yet even though it is such a basic concept, the wave models many physical phenomena. The world is a carnival of sinusoids.

Fourier’s famous theorem states that every periodic wave can be built up by adding sinusoids together. The result is surprising. Fourier’s contemporaries met it with disbelief. Many waves look nothing at all like sinusoids, such as the square wave, illustrated below. The square wave is made up of straight lines, whereas the sinusoid is continuously curved. Yet Fourier was right: We can build a square wave with only sinusoids.

Here’s how. In the illustration below there are three sine waves: the basic wave, a smaller sine wave with three times the frequency and a third of the amplitude, and an even smaller sine wave with five times the frequency and a fifth of the amplitude. We can write these three waves as sin x, sin 3x/3, and sin 5x/5.

In the illustration below, I have started to add these waves together. We see the basic wave, sin x. The sum sin x + sin 3x/3 is a wave that looks like a row of molar teeth. The sum sin x + sin 3x/3 + sin 5x/5 is a wave that looks like the filaments of a light bulb. If we carry on adding terms of the series: sin x + sin 3x/3 + sin 5x/5 + sin 7x/7 + … we will get closer and closer to the square wave. At the limit, adding an infinity of terms, we will have the square wave.

It is stunning that such a rigid shape can be constructed using only undulating wiggles. Any periodic wave consisting of jagged lines, smooth curves, or even a combination, can be built up with sinusoids.

The sum of sinusoids that make up a wave is called its “Fourier series.” It’s a remarkably useful concept since it allows us to understand a continuous wave in terms of discrete signals. The terms in the series for the square wave, for example, can now be represented by the bar chart below.

The horizontal axis represents the frequencies of the constituent sinusoids, and the vertical axis their amplitudes. Each bar stands for a sinusoid, and the leftmost bar is the sinusoid that has the “fundamental” frequency. This type of graph is known as the “frequency spectrum,” or “Fourier transform,” of the wave.

Fourier’s theorem was one of the most significant mathematical results of the 19th century because phenomena in many fields—from optics to quantum mechanics, and from seismology to electrical engineering—can be modeled by periodic waves. Often, the best way to investigate these waves is to break them down into simple sinusoids.

How You Could Play a Symphony Using Only Tuning Forks

The science of acoustics, for example, is essentially an application of Fourier’s discoveries. Sound is the vibration of air molecules. The molecules oscillate in the direction of travel of the sound, forming alternate areas of compression and rarefaction. The variation in air pressure at any point over time is a periodic wave.

The sound wave and frequency spectrum of a clarinet.

As you can see in the illustration to the right, the clarinet wave is jagged and complicated. Fourier’s theorem tells us, however, that we can break it down into a sum of sinusoids, whose frequencies are all multiples of the “fundamental” frequency of the first term. In other words, the wave can be represented as a spectrum of frequencies with different amplitudes.

Remember, the jagged wave and the bar chart in the illustration represent exactly the same sound, but in each image the information is encoded differently. For the wave, the horizontal axis is time, whereas on the bar chart the horizontal axis is frequency. Sound engineers say that the wave is in the “time domain,” and the transform is in the “frequency domain.”

The frequency domain also provides us with all the information we need to re-create the sound of a clarinet using only tuning forks. Each bar in the bar chart represents a sinusoid oscillating at a fixed frequency. The sound wave made by a tuning fork is a sinusoid. So, in order to re-create the sound of a clarinet, all that is required is to play a selection of tuning forks at the correct frequencies and amplitudes described by the bar chart.

Likewise, the frequency spectrum of a violin would provide us with instructions on how to use tuning forks to produce the sound of a violin. The difference in timbre between middle C played on the clarinet and the same pitch played on the violin is the result of the same set of tuning forks oscillating with different relative amplitudes.

A consequence of Fourier’s theorem is that it is theoretically possible to play the symphonies of Beethoven with tuning forks, in a way that is audibly indistinguishable from an orchestra.

Why a Harmonica Is Like a Picket Fence

When a fire engine passes Dolby Laboratories in San Francisco, employees clasp their ears—especially the “golden ears,” those members of the staff with exceptional hearing—hoping to protect their auditory faculties. Dolby built its reputation on noise reduction systems for the music and film industries, and it now creates sound quality software for consumer electronic devices, using technology based entirely on sinusoids.

The benefit of being able to switch a sound wave from the time domain to the frequency domain is that some jobs that are really difficult in one domain become much simpler in the other. All sound played out of digital devices—such as your TV, phone and computer—is stored as data in the frequency domain, rather than the time domain.

“The wave form is like a noodle,” Brett Crockett, senior director of research sound technology, told me. “You can’t grab it.” Frequencies are much easier to store because they are a set of discrete values. It also helps that our ears cannot hear all frequencies. “[Ears] don’t need the whole picture,” Crockett added.

Dolby’s software turns sound waves into sinusoids, and then strips out nonessential sinusoids so that the best possible sound can be recorded and stored with the least possible information. When the information is played back as sound, the spectrum of remaining frequencies is reconverted into a wave in the time domain.

It sounds easy, but in practice the task of filleting sinusoids from the frequency spectrum is exceedingly complex. One of the hardest sounds to get right is the harmonica, because its frequency spectrum looks like a picket fence—the amplitudes of the different frequencies are at the same height, forcing you to delete frequencies you can hear.

For all Dolby’s state-of-the-art know-how, the piece of music its software struggles most to re-create faithfully is “Moon River,” Henry Mancini’s hauntingly beautiful 1961 song. Brett Crockett’s golden ears judge new Dolby technology based on how faithfully it plays a harmonica riff recorded more than half a century ago.

Excerpted from The Grapes of Math: How Life Reflects Numbers and Numbers Reflect Life* by Alex Bellos. Copyright © 2014 by Alex Bellos. Reprinted by permission of Simon & Schuster, Inc.*