We never had any intention of using the weapon. But they were such a terribly troublesome race. They insisted on seeing us as the “enemy” despite all our efforts at reassurance. When they fired their entire nuclear stockpile at our ship, orbiting high above their blue planet, our patience simply ran out.

The weapon was simple but effective. It squeezed out all the empty space from matter.

As the commander of our Sirian expedition examined the shimmering metallic cube, barely 1 centimetre across, he shook his primary head despairingly. Hard to believe that this was all that was left of the “human race”!

If the idea of the entire human race fitting into the volume of a sugar cube sounds like science fiction, think again. It is a remarkable fact that 99.9999999999999 per cent of the volume of ordinary matter is empty space. If there were some way to squeeze all the empty space out of the atoms in our bodies, humanity would indeed fit into the space occupied by a sugar cube.

The appalling emptiness of atoms is only one of the extraordinary characteristics of the building blocks of matter. Another, of course, is their size. It would take 10 million atoms laid end to end to span the width of a single full stop on this page, which raises the question, how did we ever discover that everything is made of atoms in the first place?

The idea that everything is made of atoms was actually first suggested by the Greek philosopher Democritus in about 440 BC.1 Picking up a rock—or it may have been a branch or a clay pot—he asked himself the question: “If I cut this in half, then in half again, can I go on cutting it in half forever?” His answer was an emphatic no. It was inconceivable to him that matter could be subdivided forever. Sooner or later, he reasoned, a tiny grain of matter would be reached that could be cut no smaller. Since the Greek for “uncuttable” was “a-tomos,” Democritus called the hypothetical building blocks of all matter “atoms.”

Since atoms were too small to be seen with the senses, finding evidence for them was always going to be difficult. Nevertheless, a way was found by the 18th-century Swiss mathematician Daniel Bernoulli. Bernoulli realised that, although atoms were impossible to observe directly, it might still be possible to observe them indirectly. In particular, he reasoned that if a large enough number of atoms acted together, they might have a big enough effect to be obvious in the everyday world. All he needed was to find a place in nature where this happened. He found one—in a “gas.”

Bernoulli imagined a gas like air or steam as a collection of billions upon billions of atoms in perpetual frenzied motion like a swarm of angry bees. This vivid picture immediately suggested an explanation for the “pressure” of a gas, which kept a balloon inflated or pushed against the piston of a steam engine. When confined in any container, the atoms of a gas would drum relentlessly on the walls like hailstones on a tin roof. Their combined effect would be to create a jittery force that, to our coarse senses, would seem like a constant force pushing back the walls.

But Bernoulli’s microscopic explanation of pressure provided more than a convenient mental picture of what was going on in a gas. Crucially, it led to a specific prediction. If a gas were squeezed into half its original volume, the gas atoms would need to fly only half as far between collisions with the container walls. They would therefore collide twice as frequently with those walls, doubling the pressure. And if the gas were squeezed into a third of its volume, the atoms would collide three times as frequently, trebling the pressure. And so on.

Exactly this behaviour was observed by the English scientist Robert Boyle in 1660. It confirmed Bernoulli’s picture of a gas. And since Bernoulli’s picture was of tiny grainlike atoms flying hither and thither through empty space, it bolstered the case for the existence of atoms. Despite this success, however, definitive evidence for the existence of atoms did not come until the beginning of the 20th century. It was buried in an obscure phenomenon called Brownian motion.

Brownian motion is named after Robert Brown, a botanist who sailed to Australia on the Flinders expedition of 1801. During his time down under, he classified 4,000 species of antipodean plants; in the process, he discovered the nucleus of living cells. But he is best remembered for his observation in 1827 of pollen grains suspended in water. To Brown, squinting through a magnifying lens, it seemed as if the grains were undergoing a curious jittery motion, zigzagging their way through the liquid like drunkards lurching home from a pub.

Brown never solved the mystery of the wayward pollen grains. That breakthrough had to wait for Albert Einstein, aged 26 and in the midst of the greatest explosion of creativity in the history of science. In his “miraculous year” of 1905, not only did Einstein overthrow Newton, supplanting Newtonian ideas about motion with his special theory of relativity, but he finally penetrated the 80-year-old mystery of Brownian motion.

The reason for the crazy dance of pollen grains, according to Einstein, was that they were under continual machine-gun bombardment by tiny water molecules. Imagine a giant inflatable rubber ball, taller than a person, being pushed about a field by a large number of people. If each person pushes in their own particular direction, without any regard for the others, at any instant there will be slightly more people on one side than another. This imbalance is enough to cause the ball to move erratically about the field. Similarly, the erratic motion of a pollen grain can be caused by slightly more water molecules bombarding it from one side than from another.

Einstein devised a mathematical theory to describe Brownian motion. It predicted how far and how fast the average pollen grain should travel in response to the relentless battering it was receiving from the water molecules all around. Everything hinged on the size of the water molecules, since the bigger they were the bigger would be the imbalance of forces on the pollen grain and the more exaggerated its consequent Brownian motion.

The French physicist Jean Baptiste Perrin compared his observations of water-suspended “gamboge” particles, a yellow gum resin from a Cambodian tree, with the predictions of Einstein’s theory. He was able to deduce the size of water molecules and hence the atoms out of which they were built. He concluded that atoms were only about one 10-billionth of a metre across—so small that it would take 10 million, laid end to end, to span the width of a full stop.

Atoms were so small, in fact, that if the billions upon billions of them in a single breath were spread evenly throughout Earth’s atmosphere, every breath-sized volume of the atmosphere would end up containing several of those atoms. Put another way, every breath you take contains at least one atom breathed out by Albert Einstein—or Julius Caesar or Marilyn Monroe or even the last Tyrannosaurus Rex to walk on Earth!

What is more, the atoms of Earth’s “biosphere” are constantly recycled. When an organism dies, it decays and its constituent atoms are returned to the soil and the atmosphere to be incorporated into plants that are later eaten by animals and humans. “A carbon atom in my cardiac muscle was once in the tail of a dinosaur,” writes Norwegian novelist Jostein Gaarder in Sophie’s World.

Brownian motion was the most powerful evidence for the existence of atoms. Nobody who peered down a microscope and saw the crazy dance of pollen grains under relentless bombardment could doubt that the world was ultimately made from tiny, bulletlike particles. But watching jittery pollen grains—the effect of atoms—was not the same as actually seeing atoms. This had to wait until 1980 and the invention of a remarkable device called the scanning tunnelling microscope (STM).

The idea behind the STM, as it became known, was very simple. A blind person can “see” someone’s face simply by running a finger over it and building up a picture in their mind. The STM works in a similar way. The difference is that the “finger” is a finger of metal, a tiny stylus reminiscent of an old-fashioned gramophone needle. By dragging the needle across the surface of a material and feeding its up-and-down motion into a computer, it is possible to build up a detailed picture of the undulations of the atomic terrain.2

Of course, there is a bit more to it than that. Although the principle of the invention was simple, there were formidable practical difficulties in its realisation. For instance, a needle had to be found that was fine enough to “feel” atoms. The Nobel Prize committee certainly recognised the difficulties. It awarded Gerd Binnig and Heinrich Rohrer, the IBM researchers behind the STM, the 1986 Nobel Prize for Physics.

Binnig and Rohrer were the first people in history to actually “see” an atom. Their STM images were some of the most remarkable in the history of science, ranking alongside that of Earth rising above the gray desolation of the Moon or the sweeping spiral staircase of DNA. Atoms...