The Infinite Pattern That Never Repeats

A portion of this video was sponsored by LastPass. This video is about a pattern people thought was impossible and a material that wasn't supposed to exist. The story begins over 400 years ago in Prague. I'm now in Prague and the Czech Republic, which is perhaps my favorite European city that I've visited so far. I'm going to visit the Kepler Museum because he's one of the most famous scientists who lived and worked around Prague.

I want to tell you five things about Johannes Kepler that are essential to our story.

Number one: Kepler is most famous for figuring out that the shapes of planetary orbits are ellipses. But before he came to this realization, he invented a model of the solar system in which the planets were on nested spheres separated by the Platonic solids. What are the Platonic solids? Well, they are objects where all of the faces are identical and all of the vertices are identical, which means you can rotate them through some angle and they look the same as they did before. So the cube is an obvious example. Then you also have the tetrahedron, the octahedron, the dodecahedron, which has 12 pentagonal sides, and the icosahedron, which has 20 sides. And that's it! There are just five platonic solids, which was convenient for Kepler because, in his day, they only knew about six planets. So this allowed him to put a unique Platonic solid between each of the planetary spheres. Essentially, he used them as spacers.

He carefully selected the order of the Platonic solids so that the distances between planets would match astronomical observations as closely as possible. He had this deep abiding belief that there was some geometric regularity in the universe, and of course, there is—just not this.

Two: Kepler's attraction to geometry extended to more practical questions, like how do you stack cannonballs so they take up the least space on a ship's deck? By 1611, Kepler had an answer: hexagonal close packing and the face-centered cubic arrangement are both equivalently and optimally efficient, with cannonballs occupying about 74 percent of the volume they take up. Now this might seem like the obvious way to stack spheres—I mean, it is the way that oranges are stacked in the supermarket—but Kepler hadn't proved it; he just stated it as fact, which is why this became known as Kepler's conjecture.

Now it turns out he was right, but it took around 400 years to prove it. The formal proof was only published in the journal form of mathematics in 2017.

Three: Kepler published his conjecture in a pamphlet called "De Niva Sexangula," on the six-cornered snowflake, in which he wondered, "There must be a definite cause why, whenever snow begins to fall, its initial formations invariably display the shape of a six-cornered starlet, for if it happens by chance, why do they not fall just as well with five corners or with seven? Why always with six?" In Kepler's day, there was no real theory of atoms or molecules or how they self-arrange into crystals, but Kepler seemed to be on the verge of understanding this. I mean, he speculates about the smallest natural unit of a liquid like water—essentially a water molecule—and how these tiny units could stack together mechanically to form the hexagonal crystal, not unlike the hexagonal close-packed cannonballs.

Four: Kepler knew that regular hexagons can cover a flat surface perfectly with no gaps. In mathematical jargon, we say the hexagon tiles the plane periodically. You know that a tiling is periodic if you can duplicate a portion of it and continue the pattern only through translation, with no rotations or reflections. Periodic tilings can also have rotational symmetry. A rhombus pattern has two-fold symmetry because if you rotate it 180 degrees, one half turn, the pattern looks the same as it did before. Equilateral triangles have three-fold symmetry, squares have four-fold symmetry, and hexagons have six-fold symmetry. But those are the only symmetries you can have: two, three, four, and six. There is no five-fold symmetry; regular pentagons do not tile the plane, but that didn't stop Kepler from trying.

See this pattern right here? He published it in his book "Harmonices Mundi" or "Harmony of the World." It has a certain five-fold symmetry but not exactly, and it's not entirely clear how you would continue this pattern to tile the whole plane. There are an infinite number of shapes that can tile the plane periodically. The regular hexagon can only tile the plane periodically. There are also an infinite number of shapes that can tile the plane periodically or non-periodically. For example, isosceles triangles can tile the plane periodically, but if you rotate a pair of triangles, well then the pattern is no longer perfectly periodic. A sphinx tile can join with another, rotate it at 180 degrees, and tile the plane periodically, but a different arrangement of these same tiles is non-periodic.

This raises the question: Are there some tiles that can only tile the plane non-periodically? Well, in 1961, Hao Wang was studying multi-colored square tiles. The rules were touching edges must be the same color, and you can't rotate or reflect tiles, only slide them around. Now the question was: If you're given a set of these tiles, can you tell if they will tile the plane? Wang's conjecture was that if they can tile the plane, well they can do so periodically. But it turned out Wang's conjecture was false. His student, Robert Berger, found a set of 20,426 tiles that could tile the plane but only non-periodically.

Think about that for a second! Here we have a finite set of tiles—okay, it's a large number, but it's finite—and it can tile all the way out to infinity without ever repeating the same pattern. There's no way even to force them to tile periodically, and a set of tiles like this that can only tile the plane non-periodically is called an aperiodic tiling. Mathematicians wanted to know if there were aperiodic tilings that required fewer tiles. Well, Robert Berger himself found a set with only 104. Donald Knuth got the number down to 92, and then in 1969, you had Raphael Robinson who came up with six tiles—just six—that could tile the entire plane without ever repeating.

Then along came Roger Penrose, who would ultimately get the number down to two. Penrose started with a pentagon; he added other pentagons around it and, of course, noticed the gaps. But this new shape could fit within a larger pentagon, which gave Penrose an idea. What if he took the original pentagons and broke them into smaller pentagons? Well now, some of the gaps start connecting up into rhombus shapes; other gaps have three spikes. But Penrose didn't stop there. He subdivided the pentagons again. Now some of the gaps are large enough that you can use pentagons to fill in part of them, and the remaining holes you're left with are just rhombuses, stars, and a fraction of a star that Penrose calls a "kitescape."

You can keep subdividing indefinitely and you will only ever find these shapes. So, with just these pieces, you can tile the plane aperiodically with an almost five-fold symmetry.

The fifth thing about Johannes Kepler is that if you take his pentagon pattern and you overlay it on top of Penrose's, the two match up perfectly.

Once Penrose had his pattern, he found ways to simplify the tiles. He distilled the geometry down to just two tiles: a thick rhombus and a thin rhombus. The rules for how they can come together can be enforced by bumps and notches or by matching colors, and the rules ensure that these two single tiles can only tile the plane non-periodically. Just two tiles go all the way out to infinity without ever repeating.

Now, one way to see this is to print up two copies of the same Penrose pattern and one on a transparency and overlay them on top of each other. Now the resulting interference you get is called a moiré pattern. Where it is dark, the patterns are not aligned. You can see there are also some light spots, and that's where the patterns do match up. And as I rotate around, you can see the light spots move in and get smaller, and then at a certain point they move out and get bigger.

What I want to do is try to enlarge one of these bright spots and see how big of a matching section I can find. Oh yes, yes! It's like all of a sudden everything is illuminated. I love it! So these patterns are perfectly matching up here, here, here, here, and here—but not along these radial lines, and that is why they look dark.

So what this shows us is that you can't ever match any section perfectly to one beneath it. There will always be some difference. So my favorite Penrose pattern is actually made out of these two shapes, which are called kites and darts, and they have these very particular angles. The way they're meant to match up is based on these two curves. You can see there's a curve on each piece, and so you have to connect them so that the curves are continuous. And that's the rule that allows you to build an aperiodic tiling from these two pieces.

So, I laser cut thousands of these pieces, and oh, I'm gonna try to put them together and make a huge Penrose! Oh man! Come on! If you stare at a pattern of kites and darts, you'll start to notice all kinds of regularities like stars and suns, but look closer and they don't quite repeat in the way you'd expect them to. These two tiles create an ever-changing pattern that extends out to infinity without repeating. Does this mean there is only one pattern of kites and darts and every picture that we see is just a portion of that overall singular pattern?

Well, the answer is no! There are actually an uncountably infinite number of different patterns of kites and darts that tile the entire plane. And it gets weirder: If you were on any of those tilings, you wouldn't be able to tell which one it is. I mean, you might try to look further and further out, gather more and more data, but it's futile because any finite region of one of these tilings appears infinitely many times in all of the other versions of those tilings.

I mean, don't get me wrong, those tilings are also different in an infinite number of ways, but it's impossible to tell that unless you could see the whole pattern, which is impossible. There's this kind of paradox to Penrose tilings where there's an uncountable infinity of different versions, but just by looking at them, you could never tell them apart.

Now what if we count up all the kites and darts in this pattern? Well, I get 440 kites and 272 darts. Does that ratio ring any bells? Well, if you divide one by the other, you get 1.618—that is the golden ratio!

So why does the golden ratio appear in this pattern? Well, as you know, it contains a kind of five-fold symmetry, and of all the irrational constants, the golden ratio φ is the most five-ish of the constants. I mean, you can express the golden ratio as 0.5 plus 5 to the power of 0.5 times 0.5. The golden ratio is also heavily associated with pentagons. I mean, the ratio of the diagonal to an edge is the golden ratio, and the kite and dart pieces themselves are actually sections of pentagons. Same with the rhombuses. So they actually have the golden ratio built right into their construction.

The fact that the ratio of kites to darts approaches the golden ratio, an irrational number, provides evidence that the pattern can't possibly be periodic. If the pattern were periodic, then the ratio of kites to darts could be expressed as a ratio of two whole numbers—the number of kites to darts in each periodic segment.

And it goes deeper! If you draw on the tiles not curves but these particular straight lines, well now when you put the pattern together, you see something interesting: they all connect up perfectly into straight lines. There are five sets of parallel lines. This is a kind of proof of the five-fold symmetry of the pattern, but it is not perfectly regular.

Take a look at any one set of parallel lines; you'll notice there are two different spacings. Call them long and short. From the bottom, we have long, short, long, short, long, long—wait, that breaks the pattern! These gaps don't follow a periodic pattern either, but count up the number of longs and shorts in any section. Here I get 13 shorts and 21 longs, and you have the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. And the ratio of one Fibonacci number to the previous one approaches the golden ratio.

Now the question Penrose faced from other scientists was: Could there be a physical analog for these patterns? Do they occur in nature, perhaps in crystal structure? Penrose thought that was unlikely. The very nature of a crystal is that it is made up of repeating units, just as the fundamental symmetries of the shapes that tile the plane had been worked out much earlier. The basic unit cells that compose all crystals were well established—there are 14 of them—and no one had ever seen a crystal that failed to fit one of these patterns.

And there was another problem: Crystals are built by putting atoms and molecules together locally, whereas Penrose tilings, well, they seem to require some sort of long-range coordination. Take this pattern, for example. You could put a dart over here and continue to tile out to infinity—no problems! Or you could put a kite over here on the other side—again, no problems! But if you place the kite and dart in here simultaneously, well then this pattern will not work.

I mean, you can keep tiling for a while, but when you get to somewhere around here, well, it's not gonna work. You can put a dart in there, which completes the pattern nicely, but then you get this really awkward shape there, which is actually the shape of another dart. But if you put that one in there, then the lines don't match up; the pattern doesn't work.

So how could this work as a crystal? I mean, both of these tiles obey the local rules, but in the long term, they just don't work.

In the early 1980s, Paul Steinhardt and his students were using computers to model how atoms come together into condensed matter—that is, essentially solid material at the smallest scales—and he found that locally they like to form icosahedrons. But this was known to be the most forbidden shape because it is full of five-fold symmetries. So the question they posed was: How big can these zicosahedrons get? They thought maybe 10 atoms or 100 atoms, but inspired by Penrose tilings, they designed a new kind of structure, a 3D analog of Penrose tilings now known as a quasi-crystal.

They simulated how x-rays would diffract off such a structure, and they found a pattern with rings of 10 points reflecting the five-fold symmetry. Just a few hundred kilometers away, completely unaware of their work, another scientist, Dan Schechtman, created this flaky material from aluminum and manganese. When he scattered electrons off his material, this is the picture he got—it almost perfectly matches the one made by Steinhardt.

So if Penrose tilings require long-range coordination, then how do you possibly make quasi-crystals? Well, I was talking to Paul Steinhardt about this, and he told me if you just use the matching rules on the edges, those rules are not strong enough, and if you apply them locally, you run into problems like this: you misplace tiles.

But he said if you have rules for the vertices, the way the vertices can connect with each other, those rules are strong enough locally so that you never make a mistake, and the pattern can go on to infinity. One of the seminal papers on quasi-crystals was called "De Niva Quinquangula" on the pentagonal snowflake, in a shout-out to Kepler.

Now not everyone was delighted at the announcement of quasi-crystals, a material that up until then people thought totally defied the laws of nature. Double Nobel Prize winner Linus Pauling famously remarked, "There are no quasi-crystals, only quasi-scientists." But, uh, Schechtman got the last laugh. He was awarded the Nobel Prize for Chemistry in 2011, and quasi-crystals have since been grown with beautiful dodecahedral shapes.

They are currently being explored for applications from non-stick electrical insulation and cookware to ultra-durable steel. The thing about this whole story that fascinates me the most is: What exists that we just can't perceive because it's considered impossible?

I mean, the symmetries of regular geometric shapes seemed so obvious and certain that no one thought to look beyond them—that is until Penrose. And what we found are patterns that are both beautiful and counterintuitive, and materials that existed all along that we just couldn't see for what they really are.

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