'Physics textbooks will have to be rewritten, as well as the definition of "cause and effect."'In my previous article Neutrino News And Neutrality it is manifest that if Neutrinos prove to be in fact travelling at faster than the speed of light then Einsteins theory of relativity is up for grabs.
This article chronicles the advances in understanding crystalline symmetry and how what was previously thought impossible, not only proved possible in theoretical terms but by a happy set of circumstances became a physical reality.
This is a story of how the impossible became possible. How, for centuries, scientists were absolutely sure that solids (as well as decorative patterns like tiling and quilts) could only have certain symmetries - such as square, hexagonal and triangular - and that most symmetries, including five-fold symmetry in the plane [two dimensions] and icosahedral symmetry in three dimensions (the symmetry of a soccer ball), were strictly forbidden. Then, about twenty years ago, a new kind of pattern, known as a 'quasicrystal,' was envisaged that shatters the symmetry restrictions and allows for an infinite number of new patterns and structures that had never been seen before, suggesting a whole new class of materials. By chance, solids with five-fold symmetry were discovered in the laboratory at about the same time. Even so, for nearly twenty years, many scientists continued to believe true quasicrystals were impossible because, they argued, such a pattern could only be formed with complex and physically unrealistic inter-atomic forces. In this talk, you will see simple, beautiful patterns and a series of geometrical toys and games that demonstrate, with subtlety and surprise, how this last conceptual barrier has been recently overcome - leading to new insights on how to grow perfect quasicrystals and inspire new technological applications.From the Perimeter Institute for Theoretical Physics http://www.perimeterinstitute.ca/index.php?option=com_content&task=view&id=551&Itemid=568&lecture_id=4126
This work which began in earnest some two hundred years ago, is important for more than one reason. Not only was the science of crystallography established, but most importantly the understanding of atoms being the basic building block for all matter stems from early work on crystals. Fundamental understandings about molecular structure, what is not possible and what is possible may be up for revision after Daniel Shechtman discovered this peculiar and previously thought impossible crystal structure. After many years of ridicule and disbelief among his scientific colleagues he has finally been vindicated with the award of the Nobel Prize in Chemistry.
By sending an electron wave through a molten metal "grate", the Israeli researcher was able to see how the wave was diffracted by the metals' atoms.Ironically, others interested in geometric shapes and symmetry mirrored in natural crystals were intrigued with quasi-symmetry. The picture at the top of this post is an example of man-made symmetry previously thought impossible to exist in nature. Below are two images side by side. The image on the left is a representation of what a diffraction test would reveal based on a structure similar to the one at the top of this page (if it was a real crystal structure). The test result on the right is of an actual diffraction result of what are now recognized as quasicrystals.
Under the microscope he observed that the new crystal was made up of perfectly ordered, but never repeating, units - a structure that is at odds with all other crystals that are regular and precisely repeating. BBC News
The point at which the abstract mathematical concepts of Paul J. Steinhardt and the evidence for the real quasi-crystal of Daniel Shechtman came together suddenly in a remarkable, jaw dropping meeting.
"It's one of those amazing moments you know when You are working on ideas that you think that are totally wild and suddenly you realize that it has some basis in reality"Prof. Paul J. Steinhardt
So, first the geometric shapes were toyed with and then their actuality in real matter was discovered. This is strongly reminiscent of the conspiracy of circumstances that research mathematician Leland McInnes spoke of when he said:
It is entirely all too common for mathematicians to embark on a purely theoretical exploration of the truly abstract and abstruse as a mental game, only to have, decades or centuries later, their work prove stunningly applicable to some very real world problem.Follow the Post: Mathematics and God.
Leland McInnes thought it astounding, (correctly in our view) that solutions to abstract maths problems periodically became answers to real world problems; sometimes centuries later. One wonders what his reaction would be to find that Steinhardt, who had come to the theoretical possibility of quasi-crystals, wanted to share this work at the exact same time that his acquaintance wanted to share what he had come to knowledge of, which was the actual quasi-crystal evidence by diffraction that Steinhardt demonstrated in theory. The two images above are what they both exchanged. What is also wonderfully coincidental is that contained within the hidden mathematical symmetry found, is the occurrence of the golden ratio, otherwise known as the Fibonacci sequence a pattern found in the most unlikely places all through the universe.
"The only way you could avoid that is if you had in mind the global picture of what the ideal tiling is like, and could look ahead"
“True quasi-crystals that would have this true, beautiful order may in fact be impossible, not only did they have the fact that they couldn’t make them in the laboratory, but they had two pretty solid theoretical arguments. The first was that in order to grow one of these structures, in order to actually fill the room with those tiles it looks like you need some kind of non-local interaction. What I mean by non-local interactions is the following: Suppose I left you… you come up afterward and you try to make the Penrose Tile with those chicken tiles ok?
Now you can do it in principle but what you’re going to find in practice is when you start to put, oh, five or ten of them together, you are going to find that you run into a conflict. A disagreement, what you tile down one end will disagree with what you tile down the other end and you are going to have to take some away and make a different choice and start again. In fact you’ll find you have to do this over and over and over again. The only way you could avoid that is if you had in mind the global picture of what the ideal tiling is like, and could look ahead and make sure when you put a tile over here it wasn’t going to conflict with the tile that you were laying over here; which implied some kind of long range interaction between the two. You’re... you’re using your eye in this case but what are the atoms and molecules doing when they make this structure? Are they somehow looking over long distances in order to decide whether to fit here or here in this way?
Well the atoms we are talking about are very simple atoms, metal atoms which as far as we know don’t have these long range interactions. So the fact that they seem, they seem to require these long interactions is a serious problem suggesting that maybe you can’t really get these perfect quasi-crystals. A second argument is that this structure is very complicated, you can already see that as you see the way these pieces going around the way these join together is rather complicated, you need several units how did these atoms figure out to make two and only two kinds of units that only fit together in certain ways, how do they figure out how to do that? It seems too complicated compared to a crystal which is a single repeating unit.
These are good arguments, and I now want to turn to them because it turns out that the subject of quasi-crystals didn’t just have one surprise, the one I’ve already presented you the fact that they exist, but in these issues there are further surprises to be found. Let me first of all talk about this issue of a non-local interaction, let me just first of all convince you that there really is a problem here. And we’ll do it by doing not the problem of tiling the plane cause that would take a lot of space we’ll think about a much simpler problem which is the problem of making a chain in one dimension. What I’ll call a Fibonacci Chain, a chain of long and short lengths which follows that same sequence of longs and shorts that we saw in the Penrose pattern.”In what followsSte inhardt says that in order to convince the sceptic that this is a very real problem not to be understated he follows a logical sequence of what it takes to make this Fibonacci Chain and what it would take logically to anticipate what should come next in the chain. Look at the words he employs to make his point:
“All of a sudden it makes a big difference which one we choose, in fact that one is not allowed, and why? Because if I make that choice there, well, the bottom choice there, what’ll happen is you’ll end up with a chain which goes….. … and… that you’ll never find in the ideal lattice….
So there’s a proof an absolute proof, I’ve just done it by hand waving, [laughter] but it’s an absolute proof, an absolute mathematical proof that you need infinite range
information in order to ensure that you can build an infinite, ah, chain perfectly. And of course in two dimensions or three dimensions we have a problem that’s much worse. Because when we build a tiling, a Penrose tiling we’re building a lattice of lines which follows a sequence in five different directions at the same time, so this looks really impossible. And in fact if I begin tiling according to Penrose’s rule that I add a first tile and then add a second tile following the joining rule, and a third and a fourth, a fifth etc. and just as I described with the chickens you’ll typically find after a handful or so of tiles I run into a problem, ok. And the only way to get rid of that problem is either to look ahead to avoid the problem ok, with a longer range interaction or I have to keep taking stuff away and replacing it, taking stuff away and replacing it. So the fact that one cannot make these highly perfect beautiful faceted large quasi crystals seems to be understood perfectly from this simple geometric argument. The only thing is, this argument turns out to be wrong. It’s quite surprising that it is wrong for example just when you think you understand what’s going on in this subject, that’s what I find so exciting about this subject, something surprising happens.” [Emphasis mine]
"Infinite range information"is what we might call foreknowledge, or omnipotent, omniscient predestination in theological terms! If I were to borrow terms used commonly in discussions of micro-biology and cosmology, the conditions necessary for their growth is irreducibly complex or of a specified complexity, far too complex as it happens for random activity.
It is our firm conviction that the reason mathematics, a shorthand form of logic and the foundation of much science, is so successful in understanding and making the universe intelligible to humanity is best explained by the idea that the universe was actually created by the use of mathematical principles. Maths is a way of logically expressing material properties in an abstract form. Or as Leland McInnes has stated: Maths is the art of abstraction.
In the Gospel of John in the New Testament we read:
In the beginning was the Logos, and the Logos was with God, and the Logos was God. The same was in the beginning with God. All things were made by him [Logos]; and without him was not any thing made that was made.(John 1:1-3)What is translated as "Word" in English loses some of its richness in meaning that John intended when he used the Greek word "Logos" Inherent in that word are the ideas of the logical, reasoning word. With this in mind- is it at all surprising that we should find inherent in all of creation an ever increasing complexity and yet it still yielding its secrets to intelligence? There has been no other sufficient answer other than that it is the result of an intelligent creator, whose stamp of logic is indelibly written in the mechanics of His world.
While all this is demonstrably true, we do acknowledge that the creator is not merely a supremely intelligent being, this is just one facet of the creator, we see in the beauty, sublimity and morality of the creation other aesthetic characteristics. So for those who hate maths there is really no need to bury your face in your hands and exclaim in a very downcast tone :
"Oh gosh, God likes maths"as one person did when the above was discussed! Perhaps along with other attributes we ought to add- a sense of humor.
As a footnote to this, I believe Steinhardt went on to explain that he has discovered a simple rule or formula for ensuring that these beautiful crystals conform to the complex patterns which seem to do away with the need for "longrange interaction" which is science speak for intelligent consciousness. That is really not my contention here, merely to point out the very strong bias against the idea of intelligent design- when something looks like it needed a creator to figure it out and which even had "mathematical proof" of its need for infinite longrange information the language of scepticism and incredulity is loud and clear. The fact that this complexity can be expressed in terms of a formula does nothing to detract from a creator. As Einstein has said his faith in science was underpinned by an overarching confidence that the Universe was intelligible and beautiful. Yet another formula discovered in fact adds to the theists confidence.
Some have argued that religion stultified science because whenever problems seemed insurmountable for the scientist the temptation to throw up her hands in despair and just declare God did it has kept science in the stone age. But this is patently not true, the opposite can be cited (as it was for Einstein), it is that order, design and intelligibility are so much a part of this universe that gives rise to the confidence that steadfast efforts would be rewarded. This is true irrespective of whether the scientific community have consciously realized it or not.