From Beautiful Tilings to Improbable Crystals

“There are no quasicrystals only quasi-scientists.” Linus Pauling the two time Noble Laureate.

So, what are these quasicrystals, who discovered them and why did Linus Pauling disagree with their existence. We shall begin to answer these questions.

The crystallographic restriction theorem:

A 3 Dimensional Lattice (crystal) can only have 1, 2, 3, 4, or 6-fold rotational symmetry.

Figure 1: Axes of symmetry

Basically, N-fold symmetry happens when u rotate a crystal in an axis by 360˚/N and the corresponding axis is called N-fold axis.

We can see that 5-fold symmetry is conspicuously absent. Why did I not mention a 5-fold symmetry? Its because a 5-fold symmetry isn’t possible, its not too hard to see why:

Figure 2a: Triangular Tiling (3-fold symmetry)
Figure 2b: Square Tiling (2-fold/4-fold symmetry)
Figure 2c: Hexagonal tiling (6-fold symmetry)

We can see that the triangular tiling has a 3-fold symmetry, the square tiling has a 2-fold and 4-fold symmetry and the hexagonal tiling has a 6-fold symmetry.

What if we tile pentagons?

Figure 3a: three pentagons arranged about a point leave a gap
Figure 3b: Four pentagons arranged about a point intersect

This is because interior angle of a pentagon is 108˚ and 3*108˚ is less than 360 so there is a gap and if u put 4 it becomes more than 360 so they overlap, this is the problem with 5 fold symmetry. There are much more comprehensive mathematical proofs on this but I prefer to not get into them.

So, there is a problem with pentagonal tiling and 5-fold symmetry. But this didn’t stop Kepler from trying:


Figure 4: Kepler’s drawings published in his book Harmonice Mundi

These figures have a certain fivefold symmetry, it isn’t too hard to see why, look at the star in the centre (labelled C) around it there are five more stars around it (labelled 1, 2, 3, 4, 5 respectively). The stars (formally named as “pentagrams” by Kepler) form a pentagon with Star C at the centre. Now if u rotate a pentagon about its centre by an angle of 72° (360°/5) it repeats itself, i.e five fold symmetry therefore disproving the crystallographic restriction theorem. But, actually he didn’t disprove the crystallographic restriction theorem because it has an apparent 5-fold symmetry i.e it as a single unit exhibits a five fold symmetry. For example a pentagon as a single unit shows five fold symmetry, but for it to be a crystal it should go on forever, meaning it should tile the plane and the tiling should have a five fold symmetry not a part of the tiling. What I mean is that Kepler made this tiling which has an apparent five fold symmetry but he couldn’t continue the tiling to infinity and if he did, it didn’t have that five fold symmetry. Nonetheless, its still fascinating that Kepler found a pattern with a certain five fold symmetry even though not quite.

“We can make a crystal out of molecules which individually have five fold symmetry but we cannot expect the lattice to have five fold symmetry.”

Another fascinating thing about Kepler is that in his book Denive Sexangula (On the Six Cornered snowflake) he said “There must be a cause why snow has the shape of a six-cornered starlet, it cannot be chance why always six.” The cause is not to be looked for in the material, for vapour is formless and flows, but in an agent. his ‘agent’, he suspected, might be mechanical: the orderly stacking of frozen ‘globules’ that represent “the smallest natural unit of a liquid like water” essentially a water molecule. The fascinating part is that there was no atomic theory in Kepler’s time. But Kepler seemed to be on the verge of understanding it.

Periodic and aperiodic tilings:

Figure 5a: Square Periodic Tiling
Figure 5b: Parallelogram Periodic Tiling

A periodic tiling is one where you can outline a region of the tiling and tile the plane by translating copies of that region (without any rotation or reflection). If the tiling is periodic it in turn means that the pattern will repeat after a fixed number of tiles. Hence the name.

An Aperiodic tiling is when the pattern never repeats till infinity. There are an infinite number of tiles which can tile the plane periodically and non-periodically.

Figure 6: Example of aperiodic tiling

As you can see in figure 6, if we cut each square of a checkerboard (periodic) into quadrilaterals we get an aperiodic pattern, but there is an arrangement of these same quadrilaterals which is periodic.

Wang Tiles:

Hao Wang, a mathematician proposed a type of tile which is modelled visually by equal sized squares with a colour on each edge, the rule was that it can be arranged such that the sides touching each other should have the same colour and also no rotation or reflection, only translation.

Figure 7: A set of 13 Wang tiles

The question with Wang Tiles is that do they tile the plane and if they do in what way (periodic/aperiodic).

Wang conjectured that “if a set of tiles tile the plane, then there exists a periodic tiling of the same tiles.” He also observed that the conjecture would imply an existence of an algorithm to decide whether a finite set of Wang Tiles can tile the plane. The idea of matching adjacent colours of tiles comes from dominoes, so the Wang tiles came to be known as Wang dominoes and the algorithmic problem of whether a set of tiles can tile the plane came to be know as the “domino problem.”

Basically, the domino problem asks whether there is an effective procedure that settles the tiling problem for all the sets.

In 1964 Wang’s student, Robert Berger solved this algorithm and proved that the Wang’s conjecture was wrong. He found a set of 20,426 tiles which can tile the plane but only non-periodically.

The way he proved this is rather interesting, but it is way beyond the scope of this blog.

Robert Berger later reduced his set to 104 tiles, and Hans Lauchlii found a set only requiring 40 Wang Tiles. In 1971 Raphael M Robinson found a set of 6 tiles (Figure 8a) based on Wang tiles which can only tile the plane non-periodically (Figure 8b). If you see Figure 8b a little closely you will observe that it is aperiodic, it doesn’t repeat.It is ordered, but it doesn’t repeat.

Figure 8a: The six Robinson tiles
Figure 8b: A set of Wang Tiles tiling the plane aperioidically © Claudio Rocchini / CC BY-SA

Then came Robert Penrose who reduced the number of times to just two, only two.

Penrose Tilings:

Roger Penrose, a mathematician, initially found a tiling with only two tiles which can tile the plane but only aperiodically. That means that the tiling will go on till infinity but never repeat, it will go on till infinity but the pattern won’t repeat. Fascinating isn’t it. Before Penrose found the set of only two tiles he initially found a set of six tiles which tile only non-periodically. These six tiles are based on pentagons, before we get into that I want to come back to Kepler tiling that we were discussing about. As I discussed earlier tiling three pentagons about a point leaves a gap, that’s why we see stars (pentagrams, sorry Kepler) and octagons in that Kepler tiling. They are actually gaps in the pentagonal tiling. The Original Penrose tiling isn’t too different from the Kepler tiling, the way Penrose constructed the tiling is actually quite interesting.

Figure 9a: Penrose arranged pentagons and clearly noticed the gaps
Figure 9b: He also noticed that these pentagons are part of a larger pentagon

Penrose tiled pentagons around one another and he clearly noticed the gaps, but more importantly he noticed that these pentagons were part of a larger pentagon. This gave him an idea what if he broke the pentagon into smaller pentagons and so on.

The mathematical method of doing this is called substitution, without going into the mathematics and rules of what substitution is lets try to understand it intuitively.

Figure 10: Image of a Substitution Tiling (By Dirk Frettloeh – , CC BY-SA 3.0, Link)

Substitution tiling is basically blowing up the tiling and then breaking it (dissecting.. Yuck) and then blowing it up and then dissect it and so on and so forth. Now, use this substitution in the pentagonal tiling.

Figure 11: The construction of the Penrose Tiling using the Substitution Rules

The figure on the right is the Original Penrose Tiling, it has 6 tiles: Pentagon, Diamond, Star (Pentagram), 3/5th of star (Boat). Wait what, I said 6 tiles but I have mentioned only four. This is because, for ensuring an aperiodic tiling there are some matching rules. Thats why the colour of the tile is important too, hence the 6 tiles (A Red Pentagon, Blue Pentagon, Grey Pentagon, Yellow Diamond, Light Blue Boat and Green Pentagram).

The Second Penrose Tiling:

The 2nd Penrose Tiling is the Kite and Dart Tiling.

Figure 12: The Kite and Dart Tiles used in the Tiling (Geometry guy at English Wikipedia, CC BY-SA 3.0, via Wikimedia Commons)
Figure 13: The combinations possible with the Matching rules(Geometry guy at English Wikipedia, CC BY-SA 3.0, via Wikimedia Commons)

The matching rule: The red and green lines must form circular arcs (as shown in Fig 13) for forming an aperiodic tiling. Another matching rule presenting the same tiling is matching the white and black dots shown in Fig 12 which produces the same combination of tilings as in Fig 13.

Construction:

Figure 14: A Gif showing the Construction of the Kite-Dart Penrose Tiling with (Taktaal, CC BY-SA 4.0)

Fig 14 shows the construction of the tiling using deflations, the resulting tiling has a certain five-fold symmetry. Just bear in mind that i haven’t discussed what inflation and deflation i.e the substitutions in depth, because it isn’t needed in the topic I’m trying to discuss. There are a lot more stuff about inflation and deflation (Deflation of a sun, star etc.)

Quasicrystals:

Before we discuss a quasicrystal, we need to know the definition of a crystal.

A Crystal is a solid composed of atoms which are arranged periodically in all 3 Dimensions.

A Quasicrystal is a structure that is ordered but non periodic, so it is basically a 3D non periodic tiling. A normal crystal is supposed to be a structure that repeats, it is made up of unit cell but Quasicrystals aren’t made up of any unit. This is because the pattern never repeats in an aperiodic tiling so you can never find a unit that can finish the pattern. In any aperiodic pattern since the pattern never repeats you can never find out where you are on the pattern since the pattern never repeats you can zoom out trying to gather as much information as you want but you can never find where you are on the pattern for that you have to see the whole pattern which is impossible as it is infinite.

Mathematically many aperiodic tilings and quasicrystals were made but, no-one had found naturally occurring quasicrystals. The Opening Quote of this Blog: “There are no Quasicrystals only Quasi-scientists” (Linus Pauling). The reason Linus Pauling said this and highly disagreed with the concept of quasicrystals is because crystals are built by putting atoms together locally, but in the case of the Penrose Tiling there seemed to be some long range coordination in order for it to tile the plane, if you place a tile wrongly then you encounter some problems when tiling the plane it just wont tile after a point. So there is a problem because we need long range coordination a quasicrystal can never form.

In 1982, Dan Shechtman observed a distinct ten fold diffraction pattern, this was made during a routine investigation of a rapidly cooled alloy of aluminium and manganese.

Figure 15: The distinct 10 fold diffraction pattern observed by Dan Schechtman (©Materialscientist)

Schecthman got surprised on finding a diffraction pattern with 10-fold symmetry, because a diffraction pattern of a normal periodic crystal can only have 1, 2, 3, 4, 6 fold symmetry, just like the structure of a crystal which also prohibits 5 fold symmetry and anything above 6 fold symmetry. Schechtman got surprised on finding such a pattern which had more than 6 fold symmetry, he didn’t have an explanation for his result.

In the same year Shecthman took his results to Ilan Blech, who said that these diffraction patterns were seen before. This pattern was left unexplained until 1984, when Blech took a second look at the pattern. A common explanation for a 10 fold symmetry is the existence of twins, which makes a pseudo symmetry. What it means is that you take 10 twins (copies) of an alloy, each twin produces a normal diffraction pattern but the superposition of all the 10 diffraction patterns results in a pseudo five fold symmetry.

Figure 16: An example of a pseudo five-fold symmetry made with 10 identical twins of a Al-Fe periodic crystal.

Blech, first disproved the existence of twins in the Al-Mn alloy by a series of experiments. So no twins, Blech was left with only option an aperiodic crystal, a crystal which isnt restrained to form only 2, 3, 4, 6 fold rotational symmetrical diffraction pattern because it is aperiodic.

Blech trying to explain this pattern thought of a completely new structure made up of cells, without translational periodicity (aperiodic). Blech used a computer simulation to calculate the diffraction intensity from such a structure which doesn’t have translational symmetry but isn’t random. He named this structure “polyhedral”. The computer diffraction pattern that arose almost exactly matched the pattern observed by Blech. This gave them the courage to collectively write a paper titled “The Microstructure of Rapidly Solidified Al6Mn” he initially published it in the Journal Of Applied Physics (JAP) but it was rejected and he ended up publishing it in the Physical Review Letters(PRL).

Belch had termed the structure as polyhedral, what does that mean?

Figure 17: The five platonic solids which are the possible crystal structures for a quasicrystal

The five platonic solids shown in Fig 17 are the possible structures for quasicrystals, but before this paper it was well established that these platonic solids cannot be the structures for a crystal as they are full of 5 fold symmetries, which is non probable for a normal crystal but you can have these 5 fold symmetries in quasicrystals.

The publishing of this paper caused a lot of excitement in the scientific community. Not everyone was excited tho, as I mentioned previously Linus Pauling heavily disagreed with the existence of quasicrystals. In fact he spent the last decade of his life trying to prove that quasicrystals are nothing but just twinned periodic crystals but his efforts went in vain as all the models he proposed were disproved. By the time of his passing he was the only prominent opponent to quasicrystals.

I had said that the platonic solids were the possible structures for a quasicrystal but in-fact most of the quasicrystals found in nature are in the icosahedral phase.

Figure 18: Projection of the diffraction pattern using Petrie vectors to form a icosahedron (Jgmoxness)

A year after the paper was published, 12-fold symmetry was found in Ni-Cr particles. Soon after 8-fold symmetry was reported in V-Ni-Si and Cr-Ni-Si alloys. As time passed by hundreds of quasicrystals were reported, many stable quasicrystals were found too, opening the door to applications of the material. In 2011, Paul Steinhardt found a naturally occurring quasicrystal near the Khatykra river. The naturally quasicrystal phase, Al63Cu24Fe13, named icosahedrite was suspected to be of meteoric origin. Further research on the Khatykra meteorites revealed another naturally occurring quasicrystal(Al71Ni24Fe5), it was stable in the temperature range of 1100-1200K suggesting that quasicrystals were formed by rapid quenching of a meteorite during an impact-induced shock.

Schecthman, was awarded the Nobel prize in chemistry in 2011 for his work on quasicrystals, he led a shift in the understanding of structures. Coming to the topic of Nobel prizes, Robert Penrose was awarded a joint Nobel prize in Physics, that had nothing to do with the tilings but was awarded for his theoretical work on black holes.

As you know tiling of plane with pentagons is impossible, but you can tile it on a sphere forming a dodecadron.

Figure 19: A dodecahedral Ho-Mg-Zn quasicrystal, pentagons tiled on a sphere in 3 dimensions

After reading this long blog you might have a question, I had mentioned that there is some long range coordination required for the Penrose tiling, but then how did a quasicrystal form? The reason that quasicrystals can exist is because the matching rules for the curved lines on the tiles aren’t strong enough i.e you cant tile the plane without long range coordination, if you misplace a tile the tiling stops. But, if you put matching rules for the corners of the tiles, those rules are strong enough to ensure no long-range coordination.

Take Home message:

Quasicrystals could have been found long ago, after all they are materials within our reach. Then why weren’t they found? Due to the lack of self belief, they were scared to publish their discovery thinking the scientific community will criticise them forever.


“Never let anyone’s opinion stop you from doing anything”

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