Stack it, shift it, shake it, break it

Patterns you already own

The pencil in your drawer holds the star of the show. Its lead is graphite: sheets of carbon arranged in a honeycomb pattern, stacked like pages, each sheet one atom thick. The sheets sit 0.335 nanometres apart, about three million of them to a millimetre, and a million piled up would stand about as tall as three pages of a paperback. Each stroke you make shears tiny stacks of them onto the paper.

The sketchpad

Five patterns sit on the bench: the square, the triangle, the honeycomb, the kagome, a weave of triangles and six-sided holes borrowed from Japanese basketry, and a five-fold pattern built from two diamond shapes that never repeats. Choose one and set how far apart the points sit, then stack up to four layers. AA places each layer directly on top of the last. AB slides every second layer half a step across, the way graphite stacks in your pencil. Twist turns each new layer by a small angle, and that is where things get strange. The Motion controls send a wave through the whole thing.

The flyscreen trick

When the lattice starts to sing

Nothing in a solid sits still. In a steel rail on a cold morning every atom is jiggling about its place, and the jiggles are not random: they travel. Nudge one atom and its neighbours feel it a moment later, and the disturbance runs off through the pattern as a wave. Set Motion on the sketchpad to Along the wave and the points shuffle back and forth in the direction the wave is heading, bunching and thinning like a slinky given a shove. Set it to Across the wave and they swing sideways instead, like a skipping rope flicked at one end. A solid carries both. Air only carries the first, because air has nothing to hold a sideways swing against.

You can feel the difference in an earthquake. Two waves leave the same rupture at the same instant. The along-the-wave one runs through crust at roughly six kilometres a second; the across-the-wave one manages about three and a half. A hundred kilometres from the source they arrive about twelve seconds apart: the first is the jolt, the second is the roll. Richard Oldham noticed in 1906 that the second arrival goes missing on the far side of the planet, and that absence is how the Earth's outer core was read as liquid. A sideways swing has nothing to grip in a liquid, so it simply stops.

Physicists count these waves the way they count light. One wave of a given note, carrying a fixed parcel of energy, gets handled as a single countable thing and given a name: a phonon, which is a sound wave in a crystal counted like a particle. The word is a nod to the photon, and the trick is the same trick.

There is a shortest wave the pattern can hold, and the sketchpad shows it. Slide Wavelength down to two spacings and neighbouring atoms swing in exact opposition, one up while the next goes down. Ask for anything shorter and there are no atoms in between to carry it. That floor puts a ceiling on the highest note a crystal can ring at. In silicon the ceiling lands near 15.6 million million wobbles a second, read straight off a single sharp line in the light a laser instrument bounces back from a clean wafer. The highest note a person can hear is about 20,000 wobbles a second, so a silicon crystal tops out around 800 million times higher than your ears do.

Nothing is perfect, and that is mostly the point

A drawn lattice repeats forever without a slip. A real crystal never does, and the faults are where most of the useful behaviour lives. The bench below holds one square sheet of atoms and lets you put four kinds of fault into it.

A missing atom

The simplest fault is an empty site: somewhere the pattern says an atom belongs and there is none. They are not rare, and they are not damage. Heat a metal and the count climbs steeply, because at any temperature above absolute zero there is a bargain to be struck between the energy an empty site costs and the disorder it buys. Careful dilatometry on aluminium, comparing how much a bar grows against how much its lattice spacing grows, puts the count near one site in ten thousand empty right at the melting point of 660 degrees Celsius, which is 933 kelvin counting up from absolute zero, the scale that bargain is actually struck on. One in ten thousand is a full stadium of fifty thousand people with five seats spare.

In a piece of aluminium the size of a pinhead, one cubic millimetre, one in ten thousand comes to about six thousand million million empty sites: roughly 750,000 vacancies for every person alive. Those empty sites are how most atoms move through a solid. An atom the size of its neighbours cannot get past them in a full lattice, so it waits for a hole and steps sideways into it, and the hole steps the other way. That shuffle is how heat treatment works, how two metals weld without melting, and how salt slowly finds its way through a glaze. Small atoms do not wait: carbon in iron and hydrogen in palladium slip through the gaps between atoms in a lattice with every site filled, which is how case hardening puts carbon into a steel surface without melting anything.

A squeezed-in atom

Next is the opposite: an extra atom shoved into a space the pattern never left for it. It costs more energy than an empty site, so at rest there are fewer of them, but they arrive in pairs the moment something energetic passes through. Knock an atom off its site and you have made a hole and a squeezed-in atom in one stroke. That pair is the first page of radiation damage, in the wall of a reactor, in the sensor of a satellite, in a chip flown at altitude. Watch the drawn sheet bulge around it: the strain reaches several atoms out in every direction, which is why one extra atom is felt by thousands.

A line of mismatch, which is why metal bends

Here is the one that changed engineering. Push a whole plane of atoms across the plane beneath it and you would have to break every bond along it at the same instant. Work that out for copper and the stress comes to something near 4.5 gigapascals, a unit of pressure: about 23 utes resting on a patch the size of your thumbnail. Then take a soft copper wire and pull it: it gives way at around 10 megapascals, about the weight of one adult on that same thumbnail, 450 times less. For thirty years that gap was the loudest unanswered question in the study of metals.

In 1934 Geoffrey Taylor in Cambridge, Egon Orowan in Berlin and Michael Polanyi in Berlin published the same answer separately within months of each other. The plane does not move all at once. An extra half-plane of atoms is wedged in from one side, and the wedge travels, one column of bonds at a time, like a rumple travelling under a rug you are straightening with your foot. You never lift the rug. You only ever move the rumple.

Choose Line of mismatch above and slide it across, or press Bend the metal and watch it go. Count the columns above the dashed line and below it: there is one more above. When the wedge reaches the far edge, the top half of the crystal has slipped exactly one atom's width over the bottom half, and the crystal has a step on its side. Stack a few million of those steps and you have bent a paperclip.

For twenty-two years it stayed an idea on paper. Then in 1956, at the Cavendish Laboratory in Cambridge, Peter Hirsch, Robert Horne and Michael Whelan had a foil of aluminium in a Siemens electron microscope when the condenser aperture came out. The extra flood of electrons warmed the foil, the lines of mismatch, which metallurgists call dislocations, started to move, and they filmed them. The paper went into Philosophical Magazine that year with the pictures in it.

The lines are dense. Soft annealed metal carries roughly ten thousand million of them crossing every square metre, which works out at about ten kilometres of line packed into a single cubic centimetre. Work that metal hard, hammer it or roll it or bend it back and forth, and the count climbs a hundred thousand times: about a million kilometres of line in that same cubic centimetre, two and a half times the distance to the Moon. They tangle, and tangled lines cannot move, and a metal whose lines cannot move is a hard metal. That is the whole of work hardening, and it is why a paperclip snaps on the fourth bend instead of the first.

A stray atom, put there on purpose

The last fault is the deliberate one, and a whole industry rests on it. Pure silicon is a mediocre conductor. Swap out one silicon atom for a boron atom or a phosphorus atom here and there and it becomes a material whose conductivity you can switch with a voltage. The doses run from about one stray atom in a thousand million at the lightest to about one in five hundred at the heaviest; a typical device body sits near one in five million, which is one odd person out in a crowd the size of Greater Sydney, which passed 5.4 million in 2024.

For that one-in-five-million to mean anything, everything else has to be cleaner still. Electronic-grade silicon is refined until unwanted atoms sit below one in a thousand million, then the wanted ones are put back in by hand. A cubic centimetre of it holds about fifty thousand million million million atoms; if each were a grain of rice the pile would bury the whole of Australia 130 metres deep. Out of that pile, one grain in every five million is the one put there on purpose, and those are the grains the switching runs on.

The turns a repeating pattern is allowed

Every pattern on the bench has two kinds of hidden move in it. A mirror line is a line you could fold the pattern along and have both halves land on each other. A rotation centre is a pin you could stick through the pattern and spin it by some fraction of a full turn and have it land on itself. The viewer below finds both by trying: it takes the pattern, applies the move, and checks whether every point has landed on a point. Nothing is asserted, and nothing is looked up.

• half turn
• third of a turn
• quarter turn
• fifth of a turn
• sixth of a turn
• mirror line

Work through the four repeating patterns and a short list comes back every time: half turns, third turns, quarter turns, sixth turns. Never a fifth. Never a seventh. That is not a shortage of imagination on the part of the tool, and it is not a shortage of clever tilers either. It is arithmetic, and it has been known since the nineteenth century. Evgraf Fedorov in Russia and Arthur Schoenflies in Germany both finished the count in 1891: there are exactly seventeen ways to repeat a pattern across a flat surface, and 230 ways to repeat one through a solid. That count rests on one premise: the pattern has to repeat. Every wallpaper in every shop, every tiled floor in every station, every crystal that repeats: seventeen and 230. Drop the repeating, and the count is a different one. Which is the next section.

Then a solid did it anyway

On 8 April 1982, at what was then the National Bureau of Standards in Gaithersburg, Maryland, Dan Shechtman of the Technion put a fast-cooled alloy of aluminium and manganese, 86 atoms of aluminium to every 14 of manganese, in front of an electron beam. Shooting electrons through a crystal and catching what comes out the far side gives a pattern of bright spots whose symmetry is the crystal's symmetry, which is the everyday way of reading a structure. What came back had ten-fold symmetry. He wrote in his notebook beside sample 1725: ten fold, with three question marks.

The reaction was hard. He was asked to leave his research group. Linus Pauling, twice a Nobel laureate and one of the most respected structural chemists alive, said in public: "There is no such thing as quasicrystals, only quasi-scientists." The measurement did not change. Shechtman published with Ilan Blech, Denis Gratias and John Cahn in Physical Review Letters in November 1984, other laboratories grew the same phases, and in 2011 he was given the Nobel Prize in Chemistry on his own.

Here is the part worth being precise about, because it is easy to tell wrong. The tiling rule did not fall over. Five-fold rotation really is impossible in a pattern that repeats. What fell over was the older assumption underneath it: that an ordered solid has to repeat at all. A quasicrystal is a solid whose atoms sit in a strict, completely determined order that never comes back to the same arrangement twice, no matter how far you walk. It has the order without the repeat, and once you drop the repeat the fifth turn is allowed.

Press Five-fold in the viewer above and watch what the tool returns. One fifth-turn centre, in the middle, and five mirrors through it. Then the last line: no repeat found. Slide that pattern over itself by any distance in any direction and it never lands back on itself. It is built from two diamond shapes, a fat one and a thin one, in a scheme Roger Penrose set out in 1974 and Nicolaas de Bruijn gave a clean recipe for in 1981; the version drawn here is built by de Bruijn's method, five families of parallel lines crossed at 72 degrees and turned inside out.

The definition of a crystal had to be rewritten to hold it. In 1992 the International Union of Crystallography's commission on aperiodic crystals settled on a new one: a crystal is any solid whose diffraction pattern comes back as sharp separate spots. Ordered, not repeating. That sentence is what a measurement did to a definition.

Into the third dimension

Flat patterns are half the story. A real crystal fills space, and the smallest block you can copy in all three directions to build the whole thing is called its unit cell. Four cells cover most of the metals in a hardware shop, and the difference between them comes down to one question: how do you stack balls in a box?

Johannes Kepler asked exactly that in 1611, in a short book he wrote as a New Year's gift about why snowflakes have six arms. In the same few pages he looked at how greengrocers pile cannonballs and fruit, said the way they do it is the tightest possible, and moved on. Nobody could prove him right for 387 years. Thomas Hales announced a proof in 1998 that leaned on a great deal of computer checking; a version checked all the way down by machine was finished in 2014 and published in 2017. The answer is 74.05 per cent full, and a quarter of any crate of oranges is air.

How full each box is

Two of the four hit exactly the same number, 74.05 per cent, and they are not the same arrangement. Face-centred cubic stacks its close-packed sheets in a three-step cycle before repeating; hexagonal close packed uses a two-step cycle. Same density, different order, and the difference shows up in how the metal behaves under a hammer. Face-centred metals have plenty of directions for a line of mismatch to slide along, so aluminium and copper draw into wire and fold into cans. Hexagonal metals have far fewer, so magnesium and zinc will take a bend and then crack, and titanium sheet has to be worked warm.

Iron does something better than pick one. Below about 912 degrees Celsius its atoms sit body-centred; heat it past that and they rearrange themselves into face-centred; cool it and they go back. Carbon dissolves far more easily in the face-centred form than the body-centred one, so heating steel, then cooling it at a chosen rate, traps the carbon in ways that make the same bar soft or hard to order. Every blacksmith who ever quenched a blade was using a change of unit cell, several thousand years before anyone could name it.

What the sketchpad draws and what it leaves out

The benches above draw geometry. They place points, join near neighbours, colour the layers, apply a wave, insert a fault and test for symmetry. Not one of them weighs a force or an energy, so none of them can tell you whether a stack would hold together, conduct, insulate or fall apart. That depends on which atoms sit on the points and what their electrons do, and it is a different and much heavier calculation.

Real crystal prediction adds that calculation on top of the geometry. For each candidate arrangement, a computer works out how comfortably the electrons would settle in; arrangements that settle into low energy are the ones that could survive as real crystals. GNoME, a crystal-hunting program built at Google DeepMind, ran that maths across millions of candidate arrangements, and Amil Merchant and colleagues reported in Nature in November 2023 that about 380,000 of them look stable enough to be worth making. Cooking one a day, and cooking is lab talk for making the real crystal in a furnace, would keep a kitchen busy for more than a thousand years. Each of those candidates began the way your sketch did: as an arrangement of points, put forward and then tested.

So have a guess. Stack a kagome AB. Twist a square by nine degrees. Run a wave two spacings long through a four-layer triangle and lean it right over. Put a line of mismatch through a square sheet and push it out the far side. Some of the shapes worth cooking may be ones nobody has drawn yet. The bench will not tell you whether your shape holds together; it hands you the part a crystal starts with, which is the pattern.

Keep going

Sources

• Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras and P. Jarillo-Herrero, "Unconventional superconductivity in magic-angle graphene superlattices", Nature 556, 43, April 2018 · arxiv.org/abs/1803.02342
• D. Shechtman, I. Blech, D. Gratias and J. W. Cahn, "Metallic phase with long-range orientational order and no translational symmetry", Physical Review Letters 53, 1951, November 1984 · link.aps.org/doi/10.1103/PhysRevLett.53.1951
• The Nobel Prize in Chemistry 2011, awarded to Dan Shechtman for the discovery of quasicrystals; the notebook entry of 8 April 1982 and the alloy composition are held by the National Institute of Standards and Technology · nist.gov/nist-and-nobel/dan-shechtman
• R. Penrose, "The role of aesthetics in pure and applied mathematical research", Bulletin of the Institute of Mathematics and its Applications 10, 266, 1974; and N. G. de Bruijn, "Algebraic theory of Penrose's non-periodic tilings of the plane", Indagationes Mathematicae 43, 39, 1981, which is the recipe the five-fold pattern here is built from.
• P. J. Lu and P. J. Steinhardt, "Decagonal and quasi-crystalline tilings in medieval Islamic architecture", Science 315, 1106, February 2007 · science.org/doi/10.1126/science.1135491
• L. Bindi, P. J. Steinhardt, N. Yao and P. J. Lu, "Natural quasicrystals", Science 324, 1306, June 2009.
• International Union of Crystallography, Commission on Aperiodic Crystals, report of 1992, which replaced the repeating-lattice definition of a crystal with a diffraction one.
• G. I. Taylor, "The mechanism of plastic deformation of crystals", Proceedings of the Royal Society A 145, 362, 1934; E. Orowan, Zeitschrift für Physik 89, 634, 1934; M. Polanyi, Zeitschrift für Physik 89, 660, 1934. Three separate papers in one year, each proposing the same travelling wedge.
• P. B. Hirsch, R. W. Horne and M. J. Whelan, "Direct observations of the arrangement and motion of dislocations in aluminium", Philosophical Magazine 1, 677, 1956, from the Cavendish Laboratory in Cambridge.
• J. Kepler, Strena Seu de Nive Sexangula (On the Six-Cornered Snowflake), 1611, where the sphere-packing guess first appears; T. C. Hales and colleagues, "A formal proof of the Kepler conjecture", Forum of Mathematics Pi 5, e2, 2017 · cambridge.org, Forum of Mathematics Pi 5, e2
• A. Merchant, S. Batzner, S. S. Schoenholz, M. Aykol, G. Cheon and E. D. Cubuk, "Scaling deep learning for materials discovery", Nature 624, 80, published 29 November 2023, the GNoME paper · nature.com/articles/s41586-023-06735-9
• Equilibrium vacancy counts in aluminium near melting, from differential dilatometry, which compares the growth of a bar against the growth of its lattice spacing; and room-temperature thermal conductivity figures for natural single-crystal diamond of 2,000 to 2,200 watts per metre per kelvin against copper's 400.