Talk:Crystallographic restriction theorem

Statement of theorem

Oddly, there was no separate mathematical theorem: the theorem was mixed with mentioning an application (crystal) and a discussion of its usefulness ("force of the theorem"). I tried to formulate the theorem itself, that should be there. If you think you can improve it go ahead.--Patrick 08:34, 7 November 2005 (UTC)

I do hope you will patiently read my full response. I want you to understand that neither my writing nor my reverting are knee-jerk spasms.

The text, reproduced here for convenience:

• The crystallographic restriction theorem in its basic form is the observation that the rotational symmetries of a crystal are limited to 2-fold, 3-fold, 4-fold, and 6-fold. This is strictly true for the mathematical formalism, but in the physical world quasicrystals occur with other symmetries, such as 5-fold.
• In mathematical terms, a crystal is modeled as a discrete lattice, generated by a list of independent finite translations. Because we insist on a lower bound on the spacing between lattice points, any rotational symmetry of the lattice must belong to a finite group. The force of the theorem is that not all finite groups are compatible with a discrete lattice; in any dimension, we will have only a finite number of compatible groups.

And later, after the necessary definitions:

• The crystallographic restriction in general form states that Ord_N consists of those positive integers m such that ψ(m) ≤ N.

So there it is in black and white. The first paragraph begins "…the rotational symmetries of a crystal are limited…" and the second paragraph continues "…a crystal is modeled as a discrete lattice…". This is followed by a lattice proof, a matrix proof, and a formal description for all dimensions (without proof).

Here is what you inserted between the paragraphs:

• In mathematical terms: for objects with a discrete symmetry group, translational symmetry in as many independent directions as the dimension of the space is compatible with rotational symmetry of orders 2, 3, 4 and 6 only.

If this is intended as a statement of a mathematical theorem, it is false.

I added 2D/3D.--Patrick 09:20, 8 November 2005 (UTC)

If this is intended as a comprehensible sentence in the English language, it also fails.

Follow the links if any terms are not clear, or say what it unclear. Anyway, the new formulation for 2D/3D is simpler.--Patrick 09:20, 8 November 2005 (UTC)

Furthermore, it is thrust between the two paragraphs, interrupting the logical continuity of the full statement they make together.

I moved it. --Patrick 09:20, 8 November 2005 (UTC)

Therefore it is muddled on three different levels simultaneously: mathematics, sense, and structure. Hence my terse edit summary as I removed it: "rv muddle".

Those who truly need a formal mathematical statement have all they need in the "Higher dimensions" section. Those with a practical interest in wallpaper or crystallography, say, should also be satisfied. This article has a wider audience than mathematicians interested in proofs. I include two proofs, not as a mathematical exercise, but because I believe they are important to help readers understand the restriction and why it applies, and to serve as stepping stones to the general theorem.

One obstacle in teaching is that we tend to explain for people who think like us; yet people absorb and understand ideas in a variety of ways, and their interests in those ideas vary, too. Here I'm trying to speak to a high school student who is researching a class assignment on wallpaper, a rockhound who is curious about crystal symmetry, and a post-doc in mathematics who is tackling an unfamiliar area. One may think verbally, one visually, one algebraically, and so on. So I begin with a tangible item from the real world, a crystal. I present a lattice proof using a picture. I present an entirely different proof — still fairly concrete — using matrix properties. Then I evolve that matrix version to a generalized number theory version.

I think about what I say, and equally important, what I deliberately omit. I choose my nouns, verbs, voice (preferably active), examples, and pictures. If necessary, I create pictures. I organize to present a clear structure with an easy-to-follow logical flow. And I delight in other editors who take equal care — alas, too few.

I appreciate your interest in improving the article, but I believe your efforts in this case are misguided, and do more harm than good. I don't believe additional text is necessary. I'd be especially curious to hear views from crystallographers and assorted mathematicians. --KSmrq^T 11:19, 7 November 2005 (UTC)

I still think a clear formulation of the mathematical theorem would be useful. This could be put for 2D and 3D in the section "Dimensions 2 and 3" before the two proofs, and for higher dimensions at the start of that section. My formulation applies for 2D and 3D, so my formulation needs improvement in that it should mention this. What else is not clear about it?

"In mathematical terms, a crystal is modeled as a discrete lattice" can be put between a statement about crystals and a mathematical theorem, but it should not be mixed up with the latter.

"Any rotational symmetry of the lattice must belong to a finite group." is not clear, symmetry groups with translational symmetry are infinite.

Please do not have the attitude "This is my article, I made it perfect, so nobody should touch it."

--Patrick 15:05, 7 November 2005 (UTC)

Let's take the last point first. My professional experience leads me to believe peer review is a Good Thing for greenhorns and graybeards alike. We all benefit, especially readers. Which brings me to my precise words above:

• And I delight in other editors who take equal care — alas, too few.

That's a far cry from "…nobody should touch it". What is true is that I do not appreciate sloppy reading and sloppy writing. I believe I have made a strong case that your writing made the article worse. If I ever come to feel your edits consistently make articles stronger, I will happily invite your edits by posting a message on your talk page. I sincerely regret that that day has not yet come.

It almost looks as if articles can be edited by invitation only. That is not how Wikipedia works.--Patrick 09:29, 8 November 2005 (UTC)

Distorting my words doesn't help build my confidence in your judgment. --KSmrq^T 11:41, 8 November 2005 (UTC)

Working backward, I agree you have found a nit one could pick, in that the phrase "belong to" could allow the rotations to be a part of the larger group that includes the translations. I'll reword it. Thanks.

Next, again you want to insist on a precise mathematical statement of a theorem at a point where I think it will do more harm than good. If I can find a way to work it in without losing the audience I'm trying to include, I will. The location you now propose is much more appropriate than the spot you originally chose.

My concern is that a precise statement requires some precise preliminaries, which threaten to be an interruption and distraction. You can see this in the higher dimensional case where I do give a formal statement. It takes two paragraphs to set up the machinery to be able to make the statement. That statement cannot begin the section, because nobody would know the meaning of Ord or ψ.

The theorem is necessarily fairly long due to some definitions, but the explanation with example of 4D rotations in general need not be put within the formulation of the theorem (as you say at the bottom, it is not even only about rotations), and it is about order of isometries, matrices need not be mentioned in the theorem.--Patrick 11:04, 8 November 2005 (UTC)

Finally, you ask what's not clear about your formulation. That's a fair question. Perhaps if I can write a statement that is clear, the comparison may help show you what I mean. I hope you will understand that I have neither the time nor the enthusiasm to give a detailed critique of all your writing. If you intend to write in volume, you really owe it to yourself and your readers to take a writing course where someone will give you the kind of feedback you need.

It is unacceptable to delete text without even be willing to be concrete about your objections. If you think you can improve it, you can edit it.--Patrick 09:29, 8 November 2005 (UTC)

Now I want to add something. We have had our differences on various pages, sometimes heated. I want you to know that I deeply appreciate your willingness to have a civil discussion here rather than a war. I may vigorously disagree with editorial decisions you make, but I do hope in future we can proceed in this more productive fashion. Thank you. --KSmrq^T 07:15, 8 November 2005 (UTC)

My mistake. I guess you really do want a revert war. So be it. --KSmrq^T 11:41, 8 November 2005 (UTC)

"I do hope in future we can proceed in this more productive fashion" sounded good, but what is going on now again???--Patrick 13:48, 8 November 2005 (UTC)

I appreciate the work you have done in the past on this artcle, but you really should not obstruct its further development. I have taken your comment into account and point-by-point responded above to your comments. Be specific if there is any part of my edits you do not like. Just reverting everything I did is, well, you know how that is called.--Patrick 13:58, 8 November 2005 (UTC)

4D examples

4D examples with isometries of order 5, 8, 10, and 12 would be interesting.--Patrick 10:47, 8 November 2005 (UTC)

Compromise?

I tried to read carefully the article and the talk page. I suggest a compromise. I do like Patrick's formulation, but I think it is kind of intimidating especially given that everything else in this article is rather elementary. I tried to thus put it at the bottom as a higher-math perspective.

In this way no content is lost, and people can get the more abstract perspective, but not at the expense of sacrificing a good read for people like me who have no idea of what a discrete isometry is. By the way, I am not an expert, so if the compromise sticks, please do edit that new section. I tried to convey in there some motivation for the isometries formulation, some more might be needed. Oleg Alexandrov (talk) 00:57, 9 November 2005 (UTC)

Clearer introduction

I rephrased the introduction to make it more precise both for the novice and the expert reader. For one thing, I did not like the caveat that the theorem doesn't have to apply to finite crystals. Yes, all crystals are finite, but they are plenty big enough for the theorem to hold. Greg Kuperberg 17:32, 14 February 2007 (UTC)

Reverted. And your statement that all crystals are "big enough" is mathematical nonsense. The physical observation of crystal symmetry is different from the mathematical proof. A symmetry group that includes translation must be infinite. Quantum chemistry uses the tools of mathematics for explanations, but the universe does what it pleases. When we discover quasicrystals, we do not falsify any theorems. --KSmrq^T 00:04, 15 February 2007 (UTC)

Lattice proof reverts

Someone please rewrite the proof. Its writtern only for those who already know it and doesn't do any good in creating content which can provide comprehensible information to people who do not have apriori knowledge.

And anyone but KSmrq should take this responsibility, as after reading the discussion, it seems he is too obsessed with his writing and starts personal & abusive language for anyone who tries to make a suggestion. Poornenduc (talk) 17:27, 25 December 2007 (UTC)

KSmrq, maybe we understand the word "displacement" there differently. I'm not a mathemathician, and as the paragraph started asking to consider an 8-fold rotation, I understand the "displacement" mentioned as "the act of displacing" (by means of the rotation considered).

Reading it this way (which I would say is the natural one, since the paragraph talks about a rotation, and it's the way I understood it -being not a mathemathician and having no prior knowledge of this topic), the sentence "that same displacement is repeated everywhere in the lattice" is false, since in the same rotation every lattice point gets displaced differently. (Displacing the same way every point would be a translation, not a rotation).

About the second sentence I added, I did because I didn't understand why the collected vectors should have to point to lattice points. So when I discovered why I added it (making the proof clearer, in my opinion). --euyyn 03:26, 1 March 2007 (UTC)

You don't understand the words and you're not sure of the mathematics but you think you can improve it? Not such a good idea. I have added one extra word, which may help; but I've got a minute so I'll explain the lattice proof to you personally, and at greater length than is customary.

The heart of the theorem and the proof is the interaction between translational symmetry and rotational symmetry. We are trying to have both. Consider lattice point A, and lattice point A₁ displaced from it by vector v. At any other lattice point, B, symmetry says that B+v must also be a lattice point, B₁. This is true for any lattice, regardless of rotational symmetry.

Now suppose we have 8-fold rotational symmetry. If displacement v exists, then rotated copies of it exist (and exist everywhere in the lattice). Beginning with v horizontal from A to A₁, rotation gives a regular octagon of lattice points surrounding A, call them A₁ through A₈.

Focus attention on the displacement from A₁ to A₂, call it vector u. Trivial geometry shows that u is about 80% shorter than v. But now we can apply rotational symmetry to u to generate a smaller octagon, then use one of its edges, and so on until the displacment vector is as short as we wish.

That's a problem: We demand that lattice points should be separated by finite displacements, a demand that this process violates. Thus the 8-fold rotational symmetry forces a contradiction. In fact, such diminution happens for any n-fold rotational symmetry when n is greater than 6.

Note carefully that this problem does not arise with 4-fold and 6-fold rotational symmetry. The construction is a little different for the 5-fold contradiction, but the shrinkage problem is still the heart of the demonstration. In the 8-fold case we have the luxury of merely spinning u, but in the 5-fold case we do not. If we "simplify" the 8-fold description, as I have done here, we merely postpone the explanation we will need in the 5-fold description. So instead we use the same method for both, which leads to overall greater simplicity.

Please do not keep trying to revise the proof; instead, revise your understanding. The proof already says precisely (and concisely) what I have here explained at greater length. Wrestling with proofs until you thoroughly understand what they say is part of the process of learning mathematics. Your attempted alternations inject your confusion, not additional clarity. --KSmrq^T 18:03, 1 March 2007 (UTC)

"You don't understand the words" <- ?? displacement You really could do with a little humility. I've already explained why the current wording leads to assume the first sense of the word is used. You can read what I wrote above if you want to know.

"you're not sure of the mathematics" <- I'm quite sure of the mathematics, what I wasn't sure of was the correctness of your proof, due to its bad wording. This is why I had to discover it by myself, despite having a Wikipedia article with the proof.

"you think you can improve it?" <- Yeh. Indeed, since the current wording didn't serve me completely, I'm quite sure a less ambiguous one would improve it very much. An essential step to be able to improve something is to not believe it being flawless. So I think I'm currently in a better position than you to improve the comprehensibility of the proof. Of course you are still in a better position to improve my grammar, correct my typpos, change "axis" by "centre" (if you can figure out that was what I meant), etc.

"[proof]" <- If you had read more carefuly what I wrote you may have infered that I already grasp both proofs, yours and mine. Maybe the problem is you don't still understand mine (and that would be why you thought I was confused).

"If we "simplify" the 8-fold description, as I have done here, we merely postpone the explanation we will need in the 5-fold description." <- I also liked better your original explanation, provided that it is made understandable.

"The proof already says precisely (and concisely) what I have here explained at greater length." <- Sure. It's the ambiguity of its wording what makes it hard to understand to someone who doesn't already know the proof.

"Wrestling with proofs until you thoroughly understand what they say is part of the process of learning mathematics." <- But that's not one of the objectives of an encyclopedic article. You're free to train mathematical abilities of students at Wikiversity or wherever. The reader of this part of the article only wants to know why 5-fold and >6-fold rotational symmetries are not possible. We should avoid puzzling people.

"Your attempted alternations inject your confusion, not additional clarity." <- I had no confussion, you just didn't realize we were using different meanings for the same word. I indeed explained it here... well, you know, that text above what you wrote.

Since your proof is the one both of us understand I have no problem keeping it. But its wording must be changed. Do it yourself if you don't want me trying to do it. But I think that it's more difficult for you to do so since you knew the proof beforehand, while I have memory of which parts of the explanation were hard to follow.

You've not yet explained (and I don't understand) the reversion of my formatting edit. --euyyn 02:11, 2 March 2007 (UTC)

Brace yourself, I intend to speak bluntly.

There seems to be quite a lot you do not understand, with the common theme that you ignore what is right in front of you. Each paragraph of the proof already begins with an italicized parenthetical guide to its focus, an aid beyond the content of the text, yet that strikes you as TOO SUBTLE!!. Your edit summary "Added visual clues about the structure of the proof" wrongly suggests that none previously existed, when the fact is that you only hyper-emphasized what was already there. You also ignore the synergy between the guides in the proof and the guides in the figure caption; both use "compatible" and "incompatible", and both use italics. In fact, you seem to ignore the figure altogether. If you can assure me that you are legally blind, I'll try to sympathize (though my friends with handicaps rarely look for sympathy); in any event, the problem is not the article.

You complain that your assumed meaning for "displacement" led you to question the correctness of the proof, and blame the wording of the article. Ironically, on the very dictionary page you invoke above is the American Heritage definition 1c,

A vector or the magnitude of a vector from the initial position to a subsequent position assumed by a body.

The article is consistent and correct, not only using "displacement" with the same meaning in the "compatible" paragraph (which does not start by mentioning a rotation), but also including an illustration that explicitly depicts the meaning. Yet with all these items visibly contradicting your assumption, you persist, and grumble that the proof is to blame for your difficulties. It is not.

Finally, you try to amend the proof, inserting text meant to "help" others like you. I have read your attempts, and your "explanations" here repeatedly. My conclusion is that you probably still do not understand the proof, and in any event your additions are jumbled nonsense. We never move a centre of rotation in the proof. We rotate exactly once, to generate the initial polygon. From then on we need only deal with translation vectors ("displacements") between lattice points, and we need only translate such vectors from one lattice position to another. The "5-fold" proof could hardly be more explicit, and I have pointed out to you that the 8-fold proof deliberately uses sliding instead of turning to minimize new ideas. That's one more thing you ignore.

So please, do everyone a favor. Study the article all you like. Write private versions for yourself if it will help clarify your thinking. But don't even think about trying to improve this for others, because you persistently do the opposite.

I rarely speak so bluntly; blunt speech tends to cause people to "defend" themselves rather than take the message to heart. I do so only because I lack the wisdom or patience or insight to see some other way of telling you the necessary facts. I feel sure your confusion was genuine, as is your wish to help. It is a noble, generous impulse, and at some other time, with some other article, you may succeed; but not now and not here.

If I may suggest, perhaps you would prefer W. Barlow's proof, as presented by Coxeter in Introduction to Geometry, second edition, pp. 60–61. Its approach seems more compatible with your mindset than the one we are discussing, which is based on

W. Scherrer (1946), Die Einlagerung eines regulären Vielecks in ein Gitter, Elemente der Mathematik 1, pp. 97–98. [1]

Although Scherrer speaks of including a regular polygon in a lattice, his brief note is covertly just another way of presenting the crystallographic restriction theorem. I learned of it through a web page titled The Centre For Conceptual Sculpture; but note that the pentagon figure there is incorrect. --KSmrq^T 11:53, 4 March 2007 (UTC)

You should try some relaxation methods, or risk to have heart diseases for something as stupid as the wording of a Wikipedia article. I hope my silence has helped you to calm down.

Let's talk first about the formatting edit. I did put the word "visual" before "clues" in my edit summary just to state I wasn't modifying any of the previously existant (as you well say) textual guides. Anyway not understanding an edit summary isn't a reason to revert the edit, nor is it the anger caused by misunderstanding it, so I assume that wasn't your reason. The fact is that when the reader reaches the first part of the proof he wonders what's that parenthesis' intent, so he stops reading and has a look at the next paragraphs to confirm that it's indeed some kind of structure. If we mark the parenthesis visually, in a way that's clearly distinguishable from the rest of the text, no such stop is needed. Another way to achieve this is to make the structure of the proof visually explicit, by placing bullets before each part.

I didn't change the words "compatible" and "incompatible", and I didn't remove the italics. That makes me wonder if you actually took a look at my edit before reverting it. The sinergy with the figure caption is supported mainly by the use of the words "compatible" and "incompatible": The use of italics, since they are used in the rest of the parenthesis too, isn't so striking to the reader.

If you, for some strange reason, dislike any kind of visually explicit aid to the reader besides the use of italics (which doesn't help much to make something stand out until the moment you read it - and you claim to have had writing lessons?), we can at least introduce textually the structure of the proof to the reader, by saying in the previous paragraph which cases we will discuss first, which next, and which last. I like this way less, but it also removes the problem, so I would accept it. Of course you can find other ways by yourself, so you won't have to agree with me. As long as they resolve the problem I will be happy with them.

If your intention is to make it harder for the reader, so she learns from the experience, as you have previously argumented here, then... man, you're wrong.

The blindness "joke" is not funny. Sorry.

Now about the wording.

I'm glad to see you read the dictionary link. It's a sign you're somehow trying to collaborate. Now we agree on "displacement" having more than one meaning. I don't see where's the irony on that, but it's good! At least it's a step forward. The meaning I considered when reading the proof is still the natural one. Using the word consistently and being correct doesn't change that. Now that you mention its use in the "compatible" paragraph, I recall reading "other displacements" was odd. "other displacement vectors" would do better, as it implies the intended meaning is the vector one. The illustration doesn't depict the meaning unambiguously at all. If we write displacement vector, arrows depict it. If we only write displacement, arrows depict the natural meaning. So maybe it's actually the picture what leads to that meaning being the one assumed.

Now please don't start with stuff like "displacement vector is redundant". That doesn't change my argument. Nor it does solve the problem.

Please make a faith effort and believe me when I tell you my corrections to the proof were not trying to make your proof clearer, but the proof I thougth it was meant. A different one. Which I like less, now that I've understood also the proof you really meant. Saying that my contribution is not related to the proof doesn't give any information to anyone, as it obviously is not. I've already said that before with these words: "If you had read more carefuly what I wrote you may have infered that I already grasp both proofs, yours and mine. Maybe the problem is you don't still understand mine (and that would be why you thought I was confused).". In any case, even if you still don't want to believe me, I prefer your original proof (to your second one, and to mine). This must be the third or fourth time I say it. The problem I have with it is the wording only.

Despite your immature insinuations about my several physical and mental disabilities, I'm more clever than average, and have background in Maths since I've studied Physics and Comp.Sci., so yes, as irritating as it is to you, the wording of the proof is to blame for my difficulties understanding it. Please make an effort and understand that you, knowing the proof beforehand, knowing the structure of the text beforehand, and knowing the intended meaning of each word beforehand (since you wrote them), cannot test by yourself the comprehensibility of the text. It's something so straightforward... --euyyn 01:38, 20 October 2007 (UTC)

CORRECTION: I wrote that way ago, and see that since then you have added "displacement vectors" to the first "incompatible" paragraph, so you're ok with that wording. I still think anyway that it is needed in the "compatible" paragraph, to remove the potential confusion as early as possible.

Ah, and thanks for the links. The "triangle musseum" story is very enjoyable. --euyyn 01:50, 20 October 2007 (UTC)

The reasoning behind my changes.

I undid KSmrq's revert of my "unhelpful edits"; here is why:

• 1: I changed the intro because it was misleading: it falsely implied that quasiperiodic structures have to be finite "in the mathematical formalism" (which is nonsense, because a crystal is not a "mathematical formalism"). A Penrose tiling, for example, is both infinite and quasiperiodic.
• 2: I removed the "compatible" part of the proof. It is not actually necessary, and as far as I can tell, it does not appear in Scherrer's paper. But I don't know German, so I could be mistaken in this.
• 3: I replaced the not-very-relevant example of a 4d double rotation with a better one, which actually shows an eightfold symmetry of the lattice.

Ben Standeven (talk) 19:25, 17 September 2008 (UTC)

A question about the second proof

Can anyone explain what is meant by "Due to periodicity of the crystal, the new vector r' which connects them must be equal to an integral multiple of r:"? Why should the vector A'B' be along any of the basis directions at all? Here is a case where I see a problem - take the hexagonal lattice, take A = (0,1), B = (0,2) and R = rotation by 60 degrees about (0,0) - then A' = (-1,-1), B' = (2,0), and A'B' is not a vector along the basis directions. Here the coordinates are w.r.t the primitive cell. Am I missing anything in the proof? —Preceding unsigned comment added by 210.212.55.3 (talk) 10:50, 8 October 2009 (UTC)

Trigonometry proof: rotations direction

The trigonometry proof used to let the point A be rotated in the opposite direction to the point B, which seemed right. Unless both rotations are in the same plane, and one is clockwise while the other is counter-clockwise (by $-\alpha$), A'B' doesn't end up parallel to AB for most rotation angles. The "opposite direction" phrase got removed in a recent edit, that also includes some other changes however. Wrwrwr (talk) 17:00, 5 January 2015 (UTC)