# Talk:Reflection (mathematics)

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## Untitled

reflection (also spelt(also spelled 'spelled') reflexion)? :)

This article does not contain information for the novice (me). Anyone want to translate into non-technical english?Hyacinth 23:37, 22 Dec 2003 (UTC)

a reflection is a mirror image of something. put a triangle on a graph. then imagine moving the origin to the triangle or away from it. that is translation. now draw the same triangle upside down. then, without flipping or redrwaing the triangle, try to move the origin to it so that they are the same triangle. in fact, you cannot do this, since it requires flipping the triangle over, or imagining the mirror image to do so. that is reflection. it is the mathematical concept representing the everyday concept of a mirror. Oemb1905 19:24, 6 March 2007 (UTC)

## Curious

Reflection is everywhere in the universe but a simple logical discussion is necessary for me to work on P vs. NP.

Would anybody know of a book I can buy or look at in a library?

My theory:

Look at your hands. They both have fingers. They both have the ability to pick up objects. They both have the ability to snap their fingers. They DO NOT have the ability to interchange. That would mean your left hand and your right hand could be exchanged but for all *practical* purposes they would be backwards.

Now you have the concept for my theorom:

That P != NP for all cases But for NP Complete cases you coulds see true interchangeability.

Whew! I have a theory so please be kind and email me if you have any thoughts or ideas..

Wha??? Melchoir 03:30, 1 December 2005 (UTC)

I always thought that I knew what reflection is about, but after reading this article I am more confused than before. I think it needs a rewrite. In particular, looking at this diff, it looks to me that the old version is better than what we have in here. It is not as rigurous, but does a better job of explaining the essense of the thing without going into too many specifics. Comments? Oleg Alexandrov (talk) 00:21, 20 October 2005 (UTC)

What's wrong with going into specifics? Giving the formulas for reflections makes precise what is stated in the introduction. -- Fropuff 00:28, 20 October 2005 (UTC)
I don't refer to the formulas section, rather to the section right before. Sorry for not being specific. Oleg Alexandrov (talk) 00:35, 20 October 2005 (UTC)
Ah. In that case, I agree. That section could well do with a rewrite. -- Fropuff 00:52, 20 October 2005 (UTC)
The old section seems confused. It claims that you can't reflect through a point; in fact, I reflect through points all the time in both math and physics, in different dimsensions, and I don't need a Euclidean structure to do it. Melchoir 20:53, 1 December 2005 (UTC)
I totally agree that this needs a re-write. And many Math Articles Do!! Even though I understand this article, as I am a Math Teacher, I think this article reminds me of an issue of mine. That is, in all the math articles, we should divide the articles into two parts: a plain English and non-mathematical summary, and a Mathematical Model. Thanks and good night. Oemb1905 19:28, 6 March 2007 (UTC)

I noticed that the two articles Reflection (linear algebra) were on largely the same thing, with similar algebra for the general reflection and at a similar level, and also that this article needed some improvement, so I've done the two things at once. Merge that article into this (mostly the reflection in a line section, but also some matrix related points and the link to Householder transformation for the more algebraic approach, then tidy up this article: splitting it into sections with a better introduction, then copyediting the result so it was clearer.--JohnBlackburnewordsdeeds 12:44, 27 January 2010 (UTC)

## Reflections other than in a hyperplane

The old section seems confused. It claims that you can't reflect through a point; in fact, I reflect through points all the time in both math and physics, in different dimsensions, and I don't need a Euclidean structure to do it. Melchoir 20:53, 1 December 2005 (UTC)

A persistent point of disagreement is over whether a "reflection" must be only in a hyperplane or may be in a subspace of any dimension, such as a point. There has just been a well-meant rewrite of the introduction that changes the principal definition from the latter to the former, after I rewrote it to change from the former to the latter. We should not be going in circles like this.
To me it seems indisputable, from reading a range of mathematics, that the term reflection in mathematics has as its fundamental meaning reflection in a point, line, plane, or any affine subspace of a Euclidean (and possibly more general) space. Therefore, I think it is necessary for Wikipedia to state the principal meaning to include reflection in any subspace.
In particular situations the subspace may be required to be a linear (homogeneous) subspace. The term "reflection" may be restricted to reflection in a hyperplane, as in the theory of groups generated by reflections. However, there is plenty fo geometry where that is not true, and I think one cannot dispute that the historical and general meaning of the term in math (and physics, chemistry, etc.) does not have either of these restrictions.
I ask for discussion of this point before again revising the definition. I'm sure we don't want an edit war! Zaslav (talk) 10:59, 11 April 2010 (UTC)
It is not so clear to me what the issue is. Often the term "reflection" refers (exclusively) to reflection in a hyperplane, as in "reflection of a figure produces its mirror image" (actually the mirror image article could be clearer, but I'm pretty sure that d is not considered a mirror image of p). Sometimes the word reflection is used in a broader sense of an orthogonal symmetry with respect to an affine subspace of an Euclidean space. The particular instance of symmetry with respect to a point does not involve any Euclidean structure at all, so in some settings is the only type of "reflection" readily available. And Coxeter prefers to call this a "central inversion", and I think he has some reason to want to avoid the word "reflection".
The current lead gives priority to the narrow definition, but spends a whole paragraph to discuss the alternative use, so it doesn't really push any particular point of view. Is there any particular reason to prefer the narrow definition? Yes there is, there are lot's of things that can be done with reflections in hyperplanes but do not necessarily extend to more general orthogonal symmetries. I've just given a course on Euclidean geometry, and somewhat to my surprise found out just how much can be done with hyperplane reflections (I was tempted to subtitle the course "it's all done with mirrors"); it would have been very tiring to have to carry the "hyperplane" qualification around everywhere (I didn't have to, because here in France definitions are respected (for instance nobody contests 0∈N), and the definition of "reflexion" does mention "hyperplan"). In the setting of reflection groups the fact of fixing a hyperplane is more crucial than being of order 2 (which complex reflections do no have to). Also almost everything in this article requires or considers the narrow definition of reflection (only the short section "Construction" is valid more generally). Is there on the other hand any particular reason to prefer the wider definition? Well of course there are some properties these notions have in common, but I think most of these follow from the fact that these symmetries are products of commuting hyperplane reflections (in fact the are precisely that). I know of no substantial theorem that applies precisely to reflections in the wider sense. Examples would be welcome though.
OK, so I think I've shown that my personal point of view is not precisely neutral in this issue (and in fact I am more generally opposed to gratuitously widening notions without a mathematical justification, as in the case of monomial), but I think I've not gone overboard in the current lead, have I? Marc van Leeuwen (talk) 14:13, 11 April 2010 (UTC)

Thank you for your full explanation of your position, which makes it much easier for me to explain mine. It sounds to me that perhaps you don't have wide experience in geometry (but maybe I'm wrong). I don't have access to any reference books now, so I can't examine them to find citations (I promise to do that as soon as I can), but let me make a partial answer. I base my remarks on usage in English; I don't know the French literature, but also, this is the English WP!

Quick remark: I agree that d and p are not mirror images.

Basically, I disagree with putting the narrow definition first in the introduction because the right general definition for mathematics as a whole (including applications, but not only for that reason) is the broader one. Therefore, the broader definition is the "right" one to put first. When there is a divergence of usage, it is right to mention in the introduction other established definitions, even at some length (as in this case), especially if they are commonly used. But the first definition should be the basic one. (Many WP articles do this.)

This dispute is not a matter of "respecting definitions" or "gratuitously widening notions". It is a matter of choosing the definitions that are in actual use across mathematics. Not the definitions that are in use in one part of mathematics. Let me state what I believe (subject to checking in reference books) are some facts, and some errors in your statements.

1. Yes: A reflection in point is not a reflection in a mirror. By definition, a mirror is a plane in 3-space, generalized to a hyperplane in n-space. So, mirror reflections are the same as reflections if and only if you choose the narrow definition of reflection. This leaves the question open.
2. I think it's clear that there is inconsistent usage across math. The question (to me) is then: which usage is primary? I claim the general definition is the basic math one, accepted for centuries, and it has been narrowed in certain branches. (I may be wrong. This is a factual question.)
3. I would not like to base the definition on "orthogonal symmetry". That involves advanced math concepts rather than the fundamental idea. Here is a my idea of a definition which states the fundamental idea of a reflection in math (and which has, it's true, been refined and restated in different ways as math advances): The reflection of a point P in a (linear or affine) subspace S of Euclidean space is obtained by dropping the perpendicular from P to S and continuing it an equal distance beyond S; the resulting point is the reflection of P in S. (Reflection of a set follows obviously.) Even in linear algebra and Coxeter groups you'll find this definition used to justify the linear-algebraic definition (when the writer bothers to justify it).
4. You are apparently basing your conclusions on the theory of groups generated by reflections. In that particular area, reflections are always reflections in hyperplanes. But this is a small (if important) part of mathematics. It is not a good reason for choosing a definition across all of mathematics.
5. It is historically incorrect to say a "reflection" need not have period 2. The proper term for a "complex reflection" is "pseudo-reflection". I suggest a search in Math Rev or Zbl for that term (with or without the -). The name is shortened by some people. "Complex reflection" is another name, probably a shortening: in full it would be "complex pseudo-reflection", but the term "complex" suffices to disambiguate it.
You must have noticed that people like to shorten names, often by adding restrictions to the ordinary definition. That's okay, as long as no one is likely to be confused, but does not change the basic definition. In a whole area of math, this may happen, as people (often, newcomers) forget or don't realize what the original name was. (I've seen this happen.) That is usually not a disaster in the narrow area, because the participants know what they mean, but it can be troublesome for people outside the area. That, I claim, is what has happened in the subject of "groups generated by reflections". In that subject there is no use for reflections not in a hyperplane, and there is use for pseudoreflections, so the names have been simplified in common usage. That applies only to the one area, though.
In the other direction, people often generalize names. So, as you say, the most general definition isn't necessarily the basic one.
1. The fact that certain kinds of reflection have more theorems than other kinds does not imply a definition. It implies a good subject. What implies a definition is usage and tradition. I guess you give weight to tradition (you spoke of "respecting definitions"). I'm sure that when I learned geometry, it was clearly stated that (regarding the plane) there are reflections in a line and reflections in a point. This seemed to be an old usage, not a mistake by the authors.
2. In linear algebra we learn that there are two kinds of isometry: rotations and reflections. The difference is in what happens to orientation, which we detect by the sign of the determinant. Thus, this distinction is only valid over the reals, not C or finite fields. There are several theorems about this, which is a fine thing for linear algebra over R. But, the definition doesn't necessarily apply to other areas of math. (The linear algebra usage is obviously related to usage in Coxeter groups.)
3. Coxeter, although a great geometer, was not the single authority on terminology. He had particular reasons for defining "reflection" as reflection in a hyperplane. We must consult more references. Let me suggest M. Berger's Geometry (French or English), and maybe Grunbaum and Shephard's book on Tilings and Patterns. Also, Altschuler-Court's College Geometry.

I hope I haven't been boring, and I thank you again for explaining your position carefully.

(BTW, the question of whether 0 is "natural" does not seem to have one "right" answer. At least, my limited research into the question last autumn suggests there was disagreement right from the beginning. Can you justify the statement that one definition is the correct one? Of course, it is unfortunate to have contradictory definitions in different parts of math, but it is very common. The use of ⊂ to mean ⊆, which is illogical but increasing, is a troublesome example.)

Re "monomial", I guess from other articles -- see semidirect product -- that some contributors think the "right" definition is the most general one they've ever seen, and that you oppose this. Let me express my total agreement with you. However, this case and that one are different. The issue is what the facts are in a particular subject. We need more facts about usage of "reflection" across math. Zaslav (talk) 11:39, 12 April 2010 (UTC)

I think that in a general-purpose encyclopedia such as this one, it is probably best to start out with the more specific and easily understood definition (that does not require the reader to know all about affine subspaces), and then have the general definition further down (see WP:MTAA). This is the opposite of how it would usually be done in a mathematical text, where general definitions are usually given first, followed then by special cases as examples. But to a certain extent, this is like arguing over the color of the bikeshed. Obviously the article needs to have more substantial content added (introduction, history, applications, more general formulations, etc.) Then perhaps the natural focus of the lead will be clearer. Oh, also I note that the Springer EOM article Hazewinkel, Michiel, ed. (2001), "Reflection", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 gives us a perspective that is somewhat different from either of the two currently emphasized in the article. Sławomir Biały (talk) 16:09, 12 April 2010 (UTC)
Thank you for the reference. The Springer EOM article is definitely more advanced. I suggest it should appear as an advanced definition, not in the intro. It may be that article was obliged to be very condensed, but I don't at all like its approach. It makes the simple very hard.
I see the issue exactly the opposite to how you do. The simplest definition is by dropping a perpendicular. One can state this as an introductory definition without the words "affine", "linear", and possibly even "subspace". Later, one can give the many different precise definitions, as in many other WP articles. However, maybe the best solution is to shorten the intro by taking out technical details, moving them to the next section, and only giving the intuitive content of the definitions with subspaces, with hyperplanes, and in more abstract spaces. Zaslav (talk) 17:48, 12 April 2010 (UTC)

## Recent edits

The phrase "reflection in a hyperplane" is thoroughly standard; "reflection about a hyperplane" and "reflection through a hyperplane" are not. It is used, for instance, in Coxeter's "Introduction to geometry". Coxeter is widely regarded as authoritative. It is in Dan Pedoe's "Geometry: a comprehensive course". It is also used in every textbook on linear algebra that I have consulted with. For instance, Apostol's "Calculus and linear algebra". Even if other usages were found, these references alone show that "reflection in" is quite standard and should be kept in the spirit of WP:RETAIN. (Also I should add that, contrary to the IP's edit summary, "reflection in" is a standard English idiom, in addition to being a standard mathematical one—one says "reflection in a mirror", not "reflection about a mirror" or "reflection through a mirror".) Sławomir Biały (talk) 11:45, 4 August 2013 (UTC)

A search of Google books shows that "reflection about" and "reflection through" are commonly used in geometry books. --50.53.46.137 (talk) 08:09, 3 October 2014 (UTC)
Your point? The same search for "reflection in" returns an order of magnitude more hits, not to mention the authoritative books on the subject to which I referred. Sławomir Biały (talk) 14:51, 3 October 2014 (UTC)