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I am a mathematician. My (limited) understanding is that quasicrystals are structured as parts of aperiodic tilings of $\mathbb{R}^3$. Such tilings were already known when Shechtman first studied his alloy. So, I was wondering why there were so many accomplished chemists who resisted the idea of quasicrystals.

I know that the diffraction pattern that Shechtman obtained had a symmetry of order 5, which contradicts the crystallographic restriction theorem, but the crystallographic restriction theorem assumes that the crystal can be modelled as a discrete lattice, i.e. that it has translational symmetries, i.e. that it is periodic. Thus, it does not say anything against the existence of quasicrystals as I understand them.

Were chemists perhaps convinced that, for some chemical (rather than mathematical) reason, any crystal-like material must have a periodic structure?

Edit: I am afraid that I should elaborate further. I am aware of line groups, frieze groups, rod groups, wallpaper groups, layer groups, crystallographic groups, and Bravais lattices, and the history of both mathematical crystallography and tilings. What I am asking is this: Robert Ammann obtained a 3-dimensional aperiodic tiling in 1975. Dan Shechter published his paper in ... 1982? Is that correct? So, my question is, how is it possible for very intelligent people, including Linus Pauling, to still hold onto their assumption that crystals had to be periodic? Was there an actual chemical reason for them to think so?

I would like to thank everyone who is trying to answer this.

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    $\begingroup$ Anything new is controversial. The idea of non-translational long range order was new. True, it was known in math (thanks to Penrose), but that didn't matter much to chemistry. Those math people always come up with some weird stuff, you know. $\endgroup$ – Ivan Neretin Jun 23 at 11:38
  • $\begingroup$ @EdV And I think that OP asks if there was actual chemical reason and not if a not so great chemist was just stubborn. $\endgroup$ – Mithoron Jun 23 at 11:52
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    $\begingroup$ There is perhaps also something to be said about the doubts that a 5-fold symmetric tiling which had really good diffraction patterns would emerge 'naturally' - the long range interactions seemed at odds with experience in fairly simple systems, much less all the pain and agony needed to get good crystals of, e.g., proteins, with similar long-range interactions needed. $\endgroup$ – Jon Custer Jun 23 at 13:06
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    $\begingroup$ I see there the parallel to the Special/General Relativity theory or Quantum Mechanics/electrodynamics. A "crazy" circle of "crazy" evidence ->, "crazy" hypothesis/theory -> "crazy" predictions -> "crazy" evidence . Minds of many scientists refuse to accept such ideas at the time, but "reasonable" ideas have no clue. $\endgroup$ – Poutnik Jun 23 at 13:47
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    $\begingroup$ @GeorgeKontogeorgiou - well, of course those materials were metal alloys, so chemists probably were negatively predisposed to begin with. I was a grad student at the time in materials science, and there was no negative predisposition, just surprise. It really didn't take long for folks to get comfortable with the the results since they were not that difficult to replicate. And then folks jumped in to figure it out. As I recall Neil Ashcroft did a lot of work making it mainstream. $\endgroup$ – Jon Custer Jun 23 at 22:15
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So, my question is, how is it possible for very intelligent people, including Linus Pauling, to still hold onto their assumption that crystals had to be periodic?

Linus Pauling advocated for a hypothesis called twinning in which grains of periodic lattice were joined together at an angle, such that you got non-standard symmetries on a large scale that did not exist at the atomic scale. His argument was that there were plausible alternative hypotheses that had not yet been ruled out by the evidence.

While Pauling probably pushed his certainty too far, there is a sense in which this is how science is supposed to work. Confidence in scientific conclusions is only justified when a determined attempt to refute it by competent and motivated opponents has been made and has failed. A hypothesis gains in confidence only as all the alternative hypotheses get rejected. Thus, for any hypothesis to be justifiably accepted, you need systematic sceptics to try to attack it and pull it apart. It's like evolution by natural selection - only the fittest hypotheses survive. So in this, Pauling was acting as a good scientist, trying to pick holes in the theory, and challenge it for as long as it could be challenged.

You should also consider the psychology of people like Pauling. There is a tendency in human social interaction to fit in with the crowd, and follow the intellectual fashion. That's fine for learning and applying what's already known, but is an impediment to discovery. Scientific revolutions are made by mavericks who can stand against the consensus, and who can stick to their own beliefs against all the slings and arrows that the community can throw at them. Pauling was likely as successful in science as he was because he had such a stubborn, contrarian, self-confident personality. Of course, while such a personality is very useful to persist and push your theory against dogmatic opposition when you happen to be right, it can be a drawback for the individual on the occasions that you are wrong. Although, as I said, science needs such people to continually challenge and test the consensus, it is a reminder that we are all fallible, even the cleverest of us. And personality is not the same thing as intelligence.

It's common for a field to take some time to get used to new ideas. Mathematicians took a long time to accept negative numbers and complex numbers. Physicists spent many years after the invention of General Relativity rejecting the possibility of black holes - even Einstein once wrote a paper proving them to be impossible! The German physicist Max Planck said that science advances one funeral at a time. Or more precisely:

A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.

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    $\begingroup$ I think I commented earlier to prevent such answer as that, wasn't too effective, I guess. $\endgroup$ – Mithoron Jun 23 at 22:08
  • $\begingroup$ @Mithoron, the answer to the question was that Pauling advocated 'twinning'. That's 'an actual chemical reason', as requested. Why would you want to prevent such an answer? $\endgroup$ – Nullius in Verba Jun 23 at 22:25
  • $\begingroup$ I think the paragraph about 'psychology'—while perhaps an interesting point for reflection—is not needed, particularly on a site where Q and A are supposed to be factual rather than than speculative. In a way, the question sets itself up for anecdotal / speculative answers, but psychoanalysis may be taking it a bit too far. In my opinion, this paragraph doesn't really add much to your answer; the rest about scientific process etc. is more valuable. $\endgroup$ – orthocresol Jun 23 at 22:52
  • $\begingroup$ @orthocresol The psychology is supposed to be addressing a more general point about the philosophy of science. Not only did Pauling have 'an actual chemical reason' for his opposition, but he was right to do so, and probably made the great achievements in chemistry that he did precisely because he stuck to his guns. This is probably a more important reason, in scientific terms, than the merely technical one. The importance of sceptical opposition to science is not anecdotal/speculative. And social dynamics are important to understanding why this behaviour constantly happens in science. $\endgroup$ – Nullius in Verba Jun 23 at 23:00
  • $\begingroup$ Hmm, well, ok. I honestly don't think it's answering the question (the 'chemical reason' OP sought is in your first paragraph and ends there), but I can also see why you feel it's important to include it. But let's leave this discussion; it's your answer, after all. $\endgroup$ – orthocresol Jun 23 at 23:06
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The basic problem was that chemists assumed that crystals had to be periodic.

It is probably easier to understand than you think. The chemistry of crystallography and the mathematics of symmetry are tightly related. And chemists always started trying to understand crystal structures based on the fundamental assumption that crystals had to be periodic in their structure. If it isn't periodic it isn't a regular structure, is it?

The problem wasn't helped by the mathematics either. Even for the simpler problem of tesselating the plane many mathematicians assumed that tilings had to be periodic. So, for example, we know that only triangles, rectangles, and hexagons can tile the plane but pentagons cannot (complicated shapes can but the symmetry falls into a small number of categories. I'm simplifying a lot!) Strictly speaking there are 17 possible "wallpaper" tilings with different regular symmetry.

That infinite tilings of the plane without periodicity could exist only emerged in the 1960s and Penrose didn't find the very simple ones we are now familiar with until the 1970s. Before that point chemists, who work in 3D not 2D, didn't even have simple examples of tilings that could exist without being periodic in 2D so they assumed that the same principle applied in 3D crystals. Given the theorems of how regular 3D symmetry can be built in the 3D case they assumed that crystals with, for example, 5-fold symmetry were impossible.

But as the mathematical theory of aperiodic tilings developed, it became more obvious that the false assumption was perfect periodicity. The 2D Penrose tilings show that a plane can be tiled in a way that does not precisely repeat. But does have an approximate symmetry (5-fold in the standard Penrose patterns).

Less than 10 years after the Penrose tiles were discovered, chemists started to recognise the 3D equivalent in crystal diffraction experiments. The big discovery was by Shechtman in the mid 1980s (which eventually won him a Nobel). The realisation that aperiodic structures could form well-defined diffraction patterns led to a change in the very definition of a crystal.

What had held chemists back was the assumption that crystals had to be periodic. But the real-world discovery of this didn't fall too far behind the mathematical theory so we can't just blame chemists for not noticing earlier examples in their diffraction patterns: they didn't have any theory to make sense of them given their assumptions about what makes a crystal.

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  • $\begingroup$ @OscarLanzi I'm not sure that example works to create non-periodic plane tilings. And the first math example of a non-periodic plane timing was 1961 (a side effect of a proof of a different tiling problem) and had ~20k different tiles in it. $\endgroup$ – matt_black Jun 23 at 12:44
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    $\begingroup$ I am afraid that I should elaborate further. I am aware of line groups, frieze groups, rod groups, wallpaper groups, layer groups, crystallographic groups, and Bravais lattices, and the history of both mathematical crystallography and tilings. What I am asking is this: Robert Ammann obtained a 3-dimensional aperiodic tiling in 1975. Dan Shechter published his paper in ... 1982? Is that correct? So, my question is, how is it possible for very intelligent people, including Linus Pauling, to still hold onto their assumption that crystals had to be periodic? $\endgroup$ – George Kontogeorgiou Jun 23 at 12:46
  • $\begingroup$ See the history in Wikipedia: en.wikipedia.org/wiki/Quasicrystal $\endgroup$ – matt_black Jun 23 at 12:46
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    $\begingroup$ I mean, if this was just their mathematical intuition, it strikes me odd that they held onto it for some years. I would expect Dan Shechter to reference Ammann's work or something, and then other chemists would be like "oh, interesting, so there is nothing really compelling a crystal to be periodic, so maybe Shechter is onto something here. This is why I inquired whether there was some chemical reason, why they thought that a crystal should be periodic. $\endgroup$ – George Kontogeorgiou Jun 23 at 12:51
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    $\begingroup$ matt_black Perhaps it wasn't a long resistance, but it was quite strong. Dan Shechtman offered a few examples of people who were initially very hostile to the idea. Let us take the extreme case of Linus Pauling. Why do you believe that he so vehemently resisted the idea of quasicrystal? Do you know whether he had any arguments except for "we have always assumed crystals to be periodic, so this must be so, even if it's not mathematically necessary"? $\endgroup$ – George Kontogeorgiou Jun 23 at 13:00
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To extend matt_black's answer by a a few illustrations, imagine you want to tile the floor of your bathroom and have tiles of 2, 3, 4, 5, and 6-fold symmetry and you want to stick to one type only.

As soon as you select the tiles of five-fold symmetry, you will leave space in-between the tiles revealing the background:

enter image description here

These patterns are a simplification, but the same symmetry principles applies to more complicated tiles, too, e.g. the by Angelo Gavezotti:

enter image description here

(credit to IUCr teaching pamphlet #21)

or some of the tessellations by M.C. Escher:

enter image description here

(credit)

Note how these tiles' orientation are symmetry related. For the classical pattern, you were allowed to move in integer steps from one equivalent position to the other (e.g., reptile of same colour) only by translation. And this distance of translation would apply to all yellow, red, and green reptiles to meet again a yellow, red, or green reptile.

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    $\begingroup$ Thank you for the answer, but my question indicates that I know both the crystallographic restriction theorem and about translational symmetries. My point is, I also know that not all tilings must admit translational symmetries and obey the crystallographic restriction theorem, i.e. there are aperiodic tilings. And people of that time also knew this. So why were some chemists, including some accomplished ones, so hostile to the idea of quasicrystals? $\endgroup$ – George Kontogeorgiou Jun 23 at 13:12
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    $\begingroup$ @GeorgeKontogeorgiou It wasn't for giving a proper answer, rather than an extension of the other by a few illustrations. Don't underestimate the power of dogma within communities in lines of «you belong to our school / line of thought if you follow concept A, refrain from touching concept B, and strongly disagree to this newly concept X by our (perhaps personally disliked) competitors». It takes courage to admit if a perspective previously taken was incomplete (or wrong) - irrespective if in the past, or in present time, irrespective if in science, or society. $\endgroup$ – Buttonwood Jun 23 at 13:36

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