Talk:Crystallographic restriction theorem

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[edit] 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)

[edit] 4D examples

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

[edit] 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)