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00:00A portion of this video was sponsored by LastPass.
00:04This video is about a pattern people thought was impossible, and a material that wasn't
00:09supposed to exist.
00:11The story begins over 400 years ago in Prague.
00:15I'm now in Prague, in the Czech Republic, which is perhaps my favorite European city
00:20that I've visited so far.
00:21I'm going to visit the Kepler Museum, because he's one of the most famous scientists who
00:25lived and worked around Prague.
00:26I want to tell you five things about Johannes Kepler that are essential to our story.
00:34Number one.
00:35Kepler is most famous for figuring out that the shapes of planetary orbits are ellipses.
00:40But before he came to this realization, he invented a model of the solar system in which
00:45the planets were on nested spheres, separated by the platonic solids.
00:52What are the platonic solids?
00:53Well, they are objects where all of the faces are identical, and all of the vertices are
00:59identical.
01:00Which means you can rotate them through some angle, and they look the same as they did before.
01:06So the cube is an obvious example.
01:09Then you also have the tetrahedron, the octahedron, the dodecahedron, which has 12 pentagonal sides,
01:19and the icosahedron, which has 20 sides.
01:23And that's it.
01:24There are just five platonic solids.
01:26Which was convenient for Kepler, because in his day they only knew about six planets.
01:31So this allowed him to put a unique platonic solid between each of the planetary spheres.
01:36Essentially, he used them as spacers.
01:39He carefully selected the order of the platonic solids, so that the distances between planets
01:44would match astronomical observations as closely as possible.
01:48He had this deep abiding belief that there was some geometric regularity in the universe.
01:54And of course there is, just not this.
01:572.
01:58Kepler's attraction to geometry extended to more practical questions like, how do you
02:02stack cannonballs so they take up the least space on a ship's deck?
02:07By 1611, Kepler had an answer.
02:09Hexagonal close packing and the face-centered cubic arrangement are both equivalently and
02:15optimally efficient, with cannonballs occupying about 74% of the volume they take up.
02:21Now this might seem like the obvious way to stack spheres.
02:24I mean, it is the way that oranges are stacked in the supermarket.
02:28But Kepler hadn't proved it.
02:29He just stated it as fact, which is why this became known as Kepler's conjecture.
02:35Now it turns out he was right, but it took around 400 years to prove it.
02:40The formal proof was only published in the journal Forum of Mathematics in 2017.
02:453.
02:47Kepler published his conjecture in a pamphlet called Deneva Sixangula on the six-cornered
02:53snowflake, in which he wondered, there must be a definite cause why, whenever snow begins
02:58to fall, its initial formations invariably display the shape of a six-cornered starlet.
03:03For if it happens by chance, why do they not fall just as well with five corners or with
03:08seven?
03:09Why always with six?
03:11In Kepler's day, there was no real theory of atoms or molecules or how they self-arrange
03:16into crystals.
03:18But Kepler seemed to be on the verge of understanding this.
03:21I mean, he speculates about the smallest natural unit of a liquid like water, essentially a water
03:27molecule, and how these tiny units could stack together mechanically to form the hexagonal
03:33crystal.
03:34Not unlike the hexagonal close-packed cannonballs.
03:394.
03:41Kepler knew that regular hexagons can cover a flat surface perfectly, with no gaps.
03:46In mathematical jargon, we say the hexagon tiles the plane periodically.
03:50You know that a tiling is periodic if you can duplicate a portion of it and continue the
03:55pattern only through translation, with no rotations or reflections.
04:01Periodic tilings can also have rotational symmetry.
04:03A rhombus pattern has two-fold symmetry because if you rotate it 180 degrees, one half turn,
04:09the pattern looks the same as it did before.
04:13Equilateral triangles have three-fold symmetry.
04:16Squares have four-fold symmetry.
04:19And hexagons have six-fold symmetry.
04:22But those are the only symmetries you can have.
04:25Two, three, four, and six.
04:28There is no five-fold symmetry.
04:31Regular pentagons do not tile the plane.
04:35But that didn't stop Kepler from trying.
04:37See this pattern right here?
04:39He published it in his book Harmonicis Mundi, or Harmony of the World.
04:44It has a certain five-fold symmetry, but not exactly.
04:49And it's not entirely clear how you would continue this pattern to tile the whole plane.
04:56There are an infinite number of shapes that can tile the plane periodically.
05:01The regular hexagon can only tile the plane periodically.
05:05There are also an infinite number of shapes that can tile the plane periodically or non-periodically.
05:10For example, isosceles triangles can tile the plane periodically.
05:14But if you rotate a pair of triangles, well then the pattern is no longer perfectly periodic.
05:20A sphinx tile can join with another, rotate it at 180 degrees, and tile the plane periodically.
05:27But a different arrangement of these same tiles is non-periodic.
05:33This raises the question, are there some tiles that can only tile the plane non-periodically?
05:39Well in 1961, Hao Wang was studying multicolored square tiles.
05:44The rules were, touching edges must be the same color, and you can't rotate or reflect tiles.
05:50Only slide them around.
05:52Now the question was, if you're given a set of these tiles, can you tell if they will
05:57tile the plane?
05:58Wang's conjecture was that if they can tile the plane, well they can do so periodically.
06:04But it turned out Wang's conjecture was false.
06:08His student, Robert Berger, found a set of 20,426 tiles that could tile the plane, but
06:15only non-periodically.
06:18Think about that for a second.
06:20Here we have a finite set of tiles, okay it's a large number, but it's finite, and it can
06:25tile all the way out to infinity without ever repeating the same pattern.
06:31There's no way even to force them to tile periodically.
06:35And a set of tiles like this, that can only tile the plane non-periodically, is called
06:41an aperiodic tiling.
06:43And mathematicians wanted to know, were there aperiodic tilings that required fewer tiles?
06:50Well, Robert Berger himself found a set with only 104.
06:54Donald Knuth got the number down to 92.
06:57And then in 1969, you had Raphael Robinson who came up with six tiles, just six, that could
07:03tile the entire plane without ever repeating.
07:07Then along came Roger Penrose, who would ultimately get the number down to two.
07:14Penrose started with a pentagon.
07:16He added other pentagons around it, and of course noticed the gaps.
07:20But this new shape could fit within a larger pentagon, which gave Penrose an idea.
07:27What if he took the original pentagons and broke them into smaller pentagons?
07:32Well now, some of the gaps start connecting up into rhombus shapes.
07:37Other gaps have three spikes.
07:39But Penrose didn't stop there.
07:41He subdivided the pentagons again.
07:43Now, some of the gaps are large enough that you can use pentagons to fill in part of them.
07:49And the remaining holes you're left with are just rhombuses, stars, and a fraction of a star
07:53that Penrose calls a justice cap.
07:56You can keep subdividing indefinitely, and you will only ever find these shapes.
08:01So with just these pieces, you can tile the plane aperiodically with an almost 5-fold symmetry.
08:10The fifth thing about Johannes Kepler is that if you take his pentagon pattern and you overlay
08:16it on top of Penrose's, the two match up perfectly.
08:23Once Penrose had his pattern, he found ways to simplify the tiles.
08:27He distilled the geometry down to just two tiles, a thick rhombus and a thin rhombus.
08:34The rules for how they can come together can be enforced by bumps and notches, or by matching
08:39colors.
08:40And the rules ensure that these two single tiles can only tile the plane non-periodically.
08:46Just two tiles go all the way out to infinity without ever repeating.
08:51Now one way to see this is to print up two copies of the same Penrose pattern, and one
08:57on a transparency, and overlay them on top of each other.
09:01Now the resulting interference you get is called a moire pattern.
09:05Where it is dark, the patterns are not aligned.
09:08But you can see there are also some light spots, and that's where the patterns do match up.
09:14And as I rotate around, you can see the light spots move in and get smaller, and then at
09:21a certain point they move out and get bigger.
09:25And what I want to do is try to enlarge one of these bright spots and see how big of a matching
09:32section I can find, oh yes, yes!
09:38It's like all of a sudden everything is illuminated.
09:42I love it.
09:45So these patterns are perfectly matching up here, here, here, here, and here, but not along
09:50these radial lines.
09:52And that is why they look dark.
09:55So what this shows us is that you can't ever match any section perfectly to one beneath
10:01it.
10:02There will always be some difference.
10:06So my favorite Penrose pattern is actually made out of these two shapes, which are called
10:12kites and darts.
10:14And they have these very particular angles.
10:17And the way they're meant to match up is based on these two curves.
10:21You can see there's a curve on each piece.
10:23And so you have to connect them so that the curves are continuous.
10:27And that's the rule that allows you to build an aperiodic tiling from these two pieces.
10:33So I laser cut thousands of these pieces.
10:39And I'm going to try to put them together and make a huge Penrose tiling.
10:45Oh man, come on.
10:51If you stare at a pattern of kites and darts, you'll start to notice all kinds of regularities
10:56like stars and suns.
10:59But look closer and they don't quite repeat in the way you'd expect them to.
11:06These two tiles create an ever-changing pattern that extends out to infinity without repeating.
11:14Does this mean there is only one pattern of kites and darts?
11:19And every picture that we see is just a portion of that overall singular pattern?
11:26Well the answer is no.
11:28There are actually an uncountably infinite number of different patterns of kites and
11:35darts that tile the entire plane.
11:37And it gets weirder.
11:39If you were on any of those tilings, you wouldn't be able to tell which one it is.
11:45I mean you might try to look further and further out, gather more and more data, but it's futile.
11:50Because any finite region of one of these tilings appears infinitely many times in all
11:57of the other versions of those tilings.
11:59I mean don't get me wrong, those tilings are also different in an infinite number of ways,
12:04but it's impossible to tell that unless you could see the whole pattern, which is impossible.
12:12There's this kind of paradox to Penrose tilings where there's an uncountable infinity of different
12:19versions, but just by looking at them, you could never tell them apart.
12:29Now what if we count up all the kites and darts in this pattern?
12:33Well I get 440 kites and 272 darts.
12:38Does that ratio ring any bells?
12:41Well if you divide one by the other you get 1.618.
12:46That is the golden ratio.
12:50So why does the golden ratio appear in this pattern?
12:54Well as you know it contains a kind of five-fold symmetry.
12:58And of all the irrational constants, the golden ratio phi is the most five-ish of the constants.
13:05I mean you can express the golden ratio as 0.5 plus 5 to the power of 0.5 times 0.5.
13:14The golden ratio is also heavily associated with pentagons.
13:17I mean the ratio of the diagonal to an edge is the golden ratio.
13:22And the kite and dart pieces themselves are actually sections of pentagons.
13:27Same with the rhombuses.
13:28So they actually have the golden ratio built right into their construction.
13:32The fact that the ratio of kites to darts approaches the golden ratio, an irrational number,
13:38provides evidence that the pattern can't possibly be periodic.
13:41If the pattern were periodic, then the ratio of kites to darts could be expressed as a ratio of two
13:47whole numbers.
13:48The number of kites to darts in each periodic segment.
13:51And it goes deeper.
13:52If you draw on the tiles, not curves, but these particular straight lines,
13:58well now when you put the pattern together, you see something interesting.
14:02They all connect up perfectly into straight lines.
14:07There are five sets of parallel lines.
14:10This is a kind of proof of the five-fold symmetry of the pattern.
14:15But it is not perfectly regular.
14:18Take a look at any one set of parallel lines.
14:21You'll notice there are two different spacings.
14:24Call them long and short.
14:26From the bottom we have long, short, long, short, long, long.
14:31Wait, that breaks the pattern.
14:34These gaps don't follow a periodic pattern either.
14:38But count up the number of longs and shorts in any section.
14:42Here I get 13 shorts and 21 longs.
14:45And you have the Fibonacci sequence.
14:491, 1, 2, 3, 5, 8, 13, 21, 34, and so on.
14:54And the ratio of one Fibonacci number to the previous one approaches the golden ratio.
15:05Now the question Penrose faced from other scientists was,
15:08could there be a physical analog for these patterns?
15:11Do they occur in nature, perhaps in crystal structure?
15:15Penrose thought that was unlikely.
15:17The very nature of a crystal is that it is made up of repeating units.
15:21Just as the fundamental symmetries of the shapes that tile the plane had been worked out much earlier,
15:27the basic unit cells that compose all crystals were well established.
15:31There are 14 of them, and no one had ever seen a crystal that failed to fit one of these patterns.
15:37And there was another problem.
15:39Crystals are built by putting atoms and molecules together locally.
15:44Whereas Penrose tilings, well they seem to require some sort of long range coordination.
15:49Take this pattern for example.
15:50You could put a dart over here and continue to tile out to infinity, no problems.
15:57Or you could put a kite over here on the other side, again, no problems.
16:04But if you place the kite and dart in here simultaneously, well then this pattern will not work.
16:14I mean, you can keep tiling for a while, but when you get to somewhere around here,
16:21well, it's not going to work.
16:23You can put a dart in there, which completes the pattern nicely.
16:26But then you get this really awkward shape there, which is actually the shape of another dart.
16:32But if you put that one in there, then the lines don't match up.
16:36The pattern doesn't work.
16:38So how could this work as a crystal?
16:40I mean, both of these tiles obey the local rules, but in the long term, they just don't work.
16:46In the early 1980s, Paul Steinhardt and his students were using computers to model how
16:51atoms come together into condensed matter.
16:54That is essentially solid material at the smallest scales.
16:58And he found that locally, they like to form icosahedrons.
17:02But this was known to be the most forbidden shape because it is full of five-fold symmetries.
17:08So the question they posed was, how big can these icosahedrons get?
17:13They thought maybe 10 atoms or 100 atoms.
17:16But inspired by Penrose tilings, they designed a new kind of structure,
17:20a 3D analog of Penrose tilings now known as a quasicrystal.
17:24And they simulated how x-rays would diffract off such a structure.
17:28And they found a pattern with rings of 10 points, reflecting the five-fold symmetry.
17:33Just a few hundred kilometers away, completely unaware of their work,
17:37another scientist, Dan Schechtman, created this flaky material from aluminum and manganese.
17:43And when he scattered electrons off his material, this is the picture he got.
17:49It almost perfectly matches the one made by Steinhardt.
17:56So if Penrose tilings require long-range coordination,
18:01then how do you possibly make quasicrystals?
18:05Well, I was talking to Paul Steinhardt about this, and he told me,
18:08if you just use the matching rules on the edges, those rules are not strong enough.
18:14And if you apply them locally, you run into problems like this.
18:16You misplace tiles.
18:18But he said if you have rules for the vertices, the way the vertices can connect with each other,
18:24those rules are strong enough locally so that you never make a mistake.
18:29And the pattern can go on to infinity.
18:32One of the seminal papers on quasicrystals was called
18:35Deneva Quinquangula on the pentagonal snowflake in a shout out to Kepler.
18:42Now, not everyone was delighted at the announcement of quasicrystals,
18:45a material that up until then people thought totally defied the laws of nature.
18:50Double Nobel Prize winner Linus Pauling famously remarked,
18:54There are no quasicrystals, only quasi-scientists.
19:01But Schechtman got the last laugh.
19:03He was awarded the Nobel Prize for Chemistry in 2011.
19:07And quasicrystals have since been grown with beautiful dodecahedral shapes.
19:12They are currently being explored for applications from non-stick electrical insulation and cookware
19:18to ultra-durable steel.
19:20And the thing about this whole story that fascinates me the most is
19:23what exists that we just can't perceive because it's considered impossible.
19:27I mean, the symmetries of regular geometric shapes seemed so obvious and certain
19:32that no one thought to look beyond them.
19:34That is, until Penrose.
19:36And what we found are patterns that are both beautiful and counterintuitive,
19:41and materials that existed all along that we just couldn't see for what they really are.
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