Talk:Euler's theorem (rotation)

What is a rotation?

As I understand the first sentence, a rotation is defined to be "a motion of the rigid body that leaves at least one point of the body in place", but what is a rigid body motion? I think SE(3), i.e., all transformations of the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bold{x} \mapsto \bold{Rx} + \bold{b} }

with R in SO(3), however that does not seem to be what is meant in the article. -- Jitse Niesen 10:50, 14 May 2009 (UTC)

Yes, when b = 0 it is a rotation, provided R is an orthogonal matrix. When R = E it is a pure translation. I thought that "rigid body motion" would not have to be defined. See also Rotation matrix where I wrote the same (I'm still working on the latter). --Paul Wormer 11:23, 14 May 2009 (UTC)

But there are combinations of rotations and translations that leave points of the body in place. For instance, take

${\mathbf {x}}\mapsto {\begin{bmatrix}0&-1&0\\1&0&0\\0&0&1\end{bmatrix}}{\mathbf {x}}+{\begin{bmatrix}1\\0\\0\end{bmatrix}}.$

This transformation leaves the point (1/2, 1/2, 0) in place, but it's not a rotation. So I think it's wrong to define a rotation as "a motion of the rigid body that leaves at least one point of the body in place". -- Jitse Niesen 16:47, 14 May 2009 (UTC)

The vector (1/2, 1/2, 0) is an element of the difference space of the 3D real affine space. It represents the equivalence class consisting of pairs of ordered points P→Q related to each other by parallel translation (class of parallel oriented line segments). For instance O→P ∼ O′→P′ are represented by the same triplet with

${\begin{aligned}O\rightarrow &{\begin{bmatrix}0\\0\\0\end{bmatrix}}\quad P\rightarrow {\begin{bmatrix}1/2\\1/2\\0\end{bmatrix}}\\O'\rightarrow &{\begin{bmatrix}-1\\0\\0\end{bmatrix}}\quad P'\rightarrow {\begin{bmatrix}-1/2\\1/2\\0\end{bmatrix}}.\\\end{aligned}}$

Namely,

${\begin{bmatrix}1/2\\1/2\\0\end{bmatrix}}=\left({\begin{bmatrix}1/2\\1/2\\0\end{bmatrix}}-{\begin{bmatrix}0\\0\\0\end{bmatrix}}\right)=\left({\begin{bmatrix}1/2\\1/2\\0\end{bmatrix}}-{\begin{bmatrix}1\\0\\0\end{bmatrix}}\right)-\left({\begin{bmatrix}0\\0\\0\end{bmatrix}}-{\begin{bmatrix}1\\0\\0\end{bmatrix}}\right).$

A rotation is a motion of affine space that leaves invariant one point of affine space. To map affine space on the difference space (consisting of triplets of real numbers) we take a system of axes with origin in the invariant point. Hence an orthogonal map of difference space is a rotation of affine space if and only if it leaves the triplet (0, 0, 0) invariant, or, in other words, iff b = 0 and R orthogonal.

--Paul Wormer 08:23, 16 May 2009 (UTC)

To be precise: A rotation (at least, as used here) is not a motion in affine space (which has no metric), but in Euclidean affine space. Peter Schmitt 22:25, 7 June 2009 (UTC)

Slight change of title?

I think "Euler's theorem on rotation" (or similar) is a better title since "(rotation)" points to disambiguation. Peter Schmitt 22:29, 7 June 2009 (UTC)

Move matrix material to other page(s)?

I think, the matrix material would better fit into the general context of rigid motion, isometries of Euclidean spaces, orthogonal matrices, and linear operators.
Comments? Peter Schmitt 22:34, 7 June 2009 (UTC)

Introduction

I have rewritten the introduction in the attempt to make the statement of the theorem simpler (the fixed point need not be in the body), and to describe the modern mathematical view. Peter Schmitt 22:53, 7 June 2009 (UTC)

Reference

The reference is in the bibliography subpage. Why put it on the page? Peter Schmitt 08:17, 8 June 2009 (UTC)

Because I didn't see your reference. My fault, I didn't look at the bibliography page. Here we see a clear disadvantage of the subpage system: a subpage is easily overlooked. Further, I thought that I was unique in having checked the original, it did not occur to me that you had too, sorry. --Paul Wormer 08:38, 8 June 2009 (UTC)

No problem. I am rather new here, and it might well be that, in cases such like this, the reference should be given on both places. Peter Schmitt 09:49, 8 June 2009 (UTC)

Oh, I forgot: You think we should take the date of publication (1776) instead of 1775 (date of presentation)?

I changed the reference to the bibliography page. I'm in favor of having a reference in one spot, because if you change something you tend to forget to do it twice. About the year: I checked that reference about 20 years ago, long before I retired. Now I would have to bike to the university library to check it again, since I don't remember the details. You say it was first presented to the St. Petersburg Academy? Usually one would use the date that the paper appeared in the proceedings, I guess, i.e., 1776. --Paul Wormer 12:17, 8 June 2009 (UTC)

The year is not really important. Of course. references are cited by the year of publication. However, since this is intended as historical note, one might consider the "true" year instead. By the way, now you may check the references conveniently online (see the external links). It would have been much more time-consuming to look through the collected works in the library ;-) (I think, the basic idea of subpages is a good one. However, probably existing subpages should be shown more prominently on the page: maybe in the TOC, or in an extra box near it, and/or as subsections at the bottom of the page -- as a link, or simply included there.
Do you have an opinion on the other issues I mentioned earlier above?
Peter Schmitt 13:15, 8 June 2009 (UTC)

Amazing, that internet (again I overlooked a subpage, I didn't see your link before). It took me quite some time 20 years ago to find that reference. I had to go through Euler's collected works, which were in a messy state, many books and loose papers, no index, etc. and partly in French and partly in Latin (I read both with quite some difficulty).

You say that a rotation point does not have to be inside the body, but is it not a screw displacement then? (Translation plus rotation?) This is something I don't have sharp in my mind, I've seen references to Chasle's theorem, but I've never looked it up.

With regard to the title of the article: I don't mind what it is called, go ahead, call it what you want.

Is there anything else that you want me to answer about this article?

--Paul Wormer 14:27, 8 June 2009 (UTC)

A screw motion does not have a fixed point. Whether a fixed point (a centre of rotation) is inside or outside the body movement does not change the motion. I don't see what your problem is. Maybe I misunderstand you?

What I would like to know is your opinion on gathering matrix material in another article. You did some work in this area. I have not read it in detail, but I noticed that it is written from the point of view of a physicist (or a natural scientist). Nothing wrong with that! But the focus, and some notation and conventions, are different there than in mathematics. I wonder how this can be handled best: One article dealing with both varieties? But then you have to choose one "main view" and add remarks on the differences. That might be clumsy. Or two parallel articles labelled mathematics and physics? Even then there should be crossreferences, of course.
Peter Schmitt 22:42, 8 June 2009 (UTC)

About the point of view from which mathematical articles should be written: I wrote a little essay-like answer in the next section, generalizing the problem somewhat. --Paul Wormer 09:15, 9 June 2009 (UTC)

Applied v. pure mathematics

Peter Schmitt raises a question that plagues Wikipedia and has been discussed at our Forum too: at what level of sophistication should mathematical articles be written?

To start the discussion I like to point out that there is a quite a lot of mathematical knowledge in the physical sciences, computer science, and fields that use statistics. That knowledge is applied mathematics.

A large amount of mathematical knowledge exists that has no applications outside mathematics. That is pure mathematics. Pure mathematicians do not only develop new knowledge, but also new, more concise language that covers the new concepts. It is important to point out that this "new" language (sometimes called "Bourbakese" after Bourbaki) often is more concise in describing applied mathematical subjects than the "old" language.

As far as I can see, the problem we are facing is: do we apply the modern mathematical language and notation to the "old" applied mathematical topics? Many mathematicians say yes, because they know the meaning of the new concepts (for instance, they know what is meant by a graded A-module, a functorial morphism, or a hypercohomology) and see that the names are more precise, exact, and elegant, or, quoting from a WP edit war, "less clumsy". The other side of the medal is that most users of mathematics do not know or understand the modern mathematical terminology and they complain when mathematicians write articles in a language they don't understand.

I'm certain that all pure mathematicians understand perfectly well the "old" language of applied mathematics, whereas the converse is not true. So, my proposition is: write topics in applied mathematics in a language understood by the users (not the creators) of mathematics. Professional mathematicians will find this clumsy and will see all sorts of border cases that are not covered by the article (physicists call that "pathological" cases), but in my proposal they have to accept it. Articles about pure mathematics, written by and for pure mathematicians, can be written in Bourbakese.

An alternative could be two subpages, a technical and a more accessible page. But I doubt that we have the manpower to achieve this.

Finally, I realize that my proposition will cause difficulties to pure mathematicians writing about applied subjects. It forces them to regress into the language of their "mathematical childhood" and to leave untreated some border ("pathological") situations.

Note:

--Paul Wormer 09:09, 9 June 2009 (UTC)

What you write is, of course, an important, but difficult question. An encyclopedia that aims to be the best should address all possible readers, i.e., explain the material as simple as possible, but not exclude topics because they cannot be explained to everyone. Thus, both the needs of the pure mathematician and the applying one ("applied" mathematics can be rather theoretical nowadays) should be addressed, at least as the final goal. (You won't exclude modern theoretical physics just because most engineers do not use it!).

But what I meant above is something much easier: There are small, but possibly confusing differences in notation and terminology between physics and mathematics which already concern beginning students. For instance: How do you denote the conjugate complex number - by a dagger or by an asterisk. What does an asterisk mean: the conjugate or the adjoint matrix? And there are differences what should be stressed, e.g., in matrix theory, or in vector space theory. (There are also different needs for engineers and theoretical physicists.)

Peter Schmitt 09:46, 9 June 2009 (UTC)

Star versus bar, tilde versus T, first or second operand of inner product complex conjugate, that sort of things are minor nuisances, they have never bothered me much. I have my personal choices (which I learned in "Kindergarten"), but I don't mind at all when somebody makes different choices. I always try to remember to define what I mean when I use such notations.

I was thinking about choices in more conceptual things. As an example, the basis-free definition of a tensor product space is an order of magnitude more abstract and—for physicists—more difficult than in terms of bases. I say, choose the definition in terms of bases. More in general, the avoidance of coordinates and bases can be difficult for non-mathematicians. Group representations as matrices or in terms of KG-moduli, that's the sort of choice I was thinking of. (I say, use matrices first and perhaps generalize to automorphisms on vector space later. Group algebras and KG-moduli have the lowest priority and should go on a different page).

Again, I don't object at all when pure mathematicians write articles about pure maths that I cannot read. The same goes for theoretical physics, philosophy, biology, medicine, etc. We agree completely on that. CZ is not on paper, there is room for all levels of treatment. Only when a choice must be made, I go for the less-sophisticated level first.

--Paul Wormer 11:12, 9 June 2009 (UTC)

Sure, you and I (and others) don't mind differences in notations. However, those who are trying to learn about it are easily confused, if the notation is unfamiliar, even if the usage is clearly defined (and even more so, if a link leads from one form to another).

And on the level of concepts, as well. It is the same mathematics, but seen differently. There is a need for both views. A reader knowing the "practical" side may want to (or need to) know the formal approach (and how they are connected), for instance, because he has to read a "formalistic" book. Or a "pure" mathematician wants to learn about some application, and is confused because a tensor does not look like he is expecting it. "A tensor is something which transforms according to certain rules" and "tensor field" describe the same abstract object, but I think that you have to start differently to explain them. So, what I am looking for is a way to do both, explain both approaches, and to compare them. And what I meant (I may be wrong with it) is that two parallel pages are better for this than a single page.

Moreover, in most cases this will not be a question of level, but of approach. What looks easy for a physicist (using coordinates, ...) may look complicated and difficult for students of mathematics (and vice versa).

There may also be different needs: details vital for applications may hide the concepts a theoretician is looking for, and some exotic examples and counterexamples which help to clarify the theoretical situation are unnecessary "nonsense" for applications.

Peter Schmitt 14:58, 9 June 2009 (UTC)

Basically we agree, ideally the applied and formalistic approaches are presented side-by-side. My point is that CZ has so little manpower that it is better to aim for more, rather than for deeper/better/longer, articles. In other words, to get CZ off the ground we must choose, and then my opinion is that we should give priority to more applicable maths. With regard to notation: if CZ would impose a standard, I would gladly follow it (but that's up to the maths editors). --Paul Wormer 15:51, 9 June 2009 (UTC)

Well, as far as I can judge from what I have observed during the few weeks since I registered, the whole project is somewhat utopic. Why should one not discuss some equally utopic aims for some pages? (It can help when making some decisions.) Of course, for the moment any page from either side will be helpful. Peter Schmitt 19:59, 9 June 2009 (UTC)

You are right, write about whatever you enjoy most. Even when your writing is aimed at specialists it can be useful and may score high on Google. (For instance, I wrote solid harmonics and this article scores 2nd to 3rd on Google. The WP article scores 1st and is also mainly by me).--Paul Wormer 07:53, 11 June 2009 (UTC)