Talk:Hyperplane/Archive 1

Archive 1

"Not to be confused with Hypersonic aircraft." -- I didn't intend this as a joke, but if people think it's too dumb, take it out.

No, I think people could come here looking for hypersonic aircraft. It might even be a good idea to move it to the top as a "dab" to point people to the right article immediately. StuRat 03:36, 22 September 2005 (UTC)

I found that pretty funny, but it may nevertheless be useful to someone.. -Simon80 03:25, 31 July 2006 (UTC)

It's a good joke and useful to boot. Zaslav 06:19, 10 November 2006 (UTC)

k-hyperplane

No one calls a point or line as 0-1-dimmensional hyperplane, it does not have any sense, you have nice term k-dimensional affine subspace!!!

Even if some use this term it should be stoped, and no reason to mention it here.

So I remove it again

Tosha 04:55, 6 Mar 2004 (UTC)

BTW I would remove realm since again no one use it, and not clear why anybody would need this in principle.

Tosha 04:59, 6 Mar 2004 (UTC)

• we always include alternately used terms for something, regardless of whether anyone needs it or not
• if a term or definition is used, it should not be removed, but if it is agreed that the term is in bad usage, it should be explained why it is not good to use the definition or term, and not widescale removed

Dysprosia 05:04, 6 Mar 2004 (UTC)

k-hyperplane again

Ok, sure, but are you sure that this term is used anywhere except this article? (I do geometry and I did not see anyone who use it).

Even if it is used by few guys, it should be marked as extremally reare so no one will have an idea to use it again.

I will be indeed very surprised if it is used... (can you give me a name of book which use k-hyperplane?)

Tosha 05:22, 6 Mar 2004 (UTC)

Example:

• Jamison, R.E., 'Finding little hyperplanes in bigger ones'. In Linear algebra and its applications, Volume 35, (February 1981), pp: 11-19.

A k-hyperplane is just a logical abbreviation to refer to a k-dimensional hyperplane. Such an abbreviation is bound to be useful in some contexts.

A.N. Yzelman 10:39, 20 March 2007 (UTC)

Added too technical tag

I added the to technical tag to the article because I've read it several times and never came away with any kind of understanding of what a hyperplane is. I finally got a friend to explain it to me and its such a basic topic this article could do a much better job of being accessible to someone with only algebra under their belt. I'm not qualified to edit the article, unfortunately, but I know something's wrong when I see it. Triddle 05:10, 16 September 2005 (UTC)

I reworded it to put the simple part up front, and removed the technical tag. If you still think it's too technical, put the tag back on. StuRat 03:27, 22 September 2005 (UTC)

I think the article is missing a good example to drive the point home; I'm not 100% sure what this killer example would be but my friend got it into my head by using the RGB color space as an example. Plot out all the combinations of R, G, and B, then split the entire thing right along the middle of B. You formed a two dimensional hyperplane on the three dimensional color space. I'm not a mathwiz though, not even close. All that I know is that when I read that article I can't make heads or tails about what a hyperplane is but I get my friend's 3D color space explanation. Triddle 05:46, 22 September 2005 (UTC)

I don't think everyone should be able to understand all of the article, but should get the "basic concept", contained in the first paragraph. Do you get that part ? Would some pics help ? Maybe one to illustrate the 1-dimensional case, another for 2-dimensions, and another for 3-dimensions ? StuRat 14:50, 22 September 2005 (UTC)

Moving the paragraphs around sure did help a lot. The first paragraph is simple and I understand it but I think thats because I already understand it now. If I was still 'green' to this concept I'm not sure that would really do the trick. I think a picture would be great; maybe some concrete examples of what one can do with a hyperplane for us kinesthetic learners. Thanks for taking the time to address my concerns. =) Triddle 15:53, 22 September 2005 (UTC)

You're welcome. Does the diagram I added help ? StuRat 18:32, 22 September 2005 (UTC)

My edit, and kinds of hyperplane

I removed a lot of stuff. Part of it was repetition; saying once that a hyperplane divides the space in two half-spaces is enough. And note that just saying "a hyperplane divides a space in two" is incorrect, any surface divides a space in two. So, one has to be more careful with wording.

I removed the ascii art picture. It is clear enough I think what a line with a point on it is. And the picture is not pretty. A hyperplane cannot be visualized, so all one can do is carefully explain what it is by analogy; so that's done well in the article.

I think it is incorrect to say that a hyperplane is a projective subspace. It is only an affine subspace, which can be linear in certain cases.

And in general, an affine subspace is not linear as the article stated.

In conclussion, the article had a bunch of incorrect and confusing information. Now it is shorter, and I hope clearer. Oleg Alexandrov (talk) 07:39, 17 December 2005 (UTC)

You are mistaken to reject projective hyperplanes. A hyperplane can be a projective hyperplane, which is a hyperplane in a projective space, or a linear hyperplane, which is a hyperplane in a linear space (vector space), or an affine hyperplane, which is a hyperplane in an affine space. In all cases, it is a projective/linear/affine subspace whose dimension is 1 less than that of the whole space. All three kinds of hyperplane appear in the current mathematical research literature. The article should have all three. I will check when I have time, to make sure that is correct. Zaslav 05:53, 10 November 2006 (UTC)

Technical or nontechnical?

I think the beginning of an article should be accessible to as many readers as possible. Persons seeking more advanced or technical knowledge (though not highly specialized details) should also find it here, as they do in similar articles. Edits made since my last contributions more than a year ago have removed the advanced content. I ask contributors not to dumb down the article. Thank you. Zaslav 06:18, 10 November 2006 (UTC)

Examples of uses of hyperplanes

It may be a good idea to have examples of uses of hyperplanes. For instance hyperplanes are used in the perceptron neuron model, to seperate patterns. —The preceding unsigned comment was added by FrederikHertzum (talk • contribs) 14:41, 16 April 2007 (UTC).

Article Overhaul

I did an overhaul on this article. I describe 3 ways to define a hyperplane, and I added a section describing common operations on hyperplanes. My changes make this article more technical, rather than less technical as someone above would apparently prefer. I also added some examples, although my descriptions are quite brief. Frankly, if someone is reading about hyperplanes, there's a good chance he/she may want technical details. The correct solution is to improve the descriptions in non-technical sections, not destroy the information in technical sections. To put a few other debates to rest, the term k-hyperplane is valid, the general definition of hyperplane does include points as well as lines and planes, and hyperplanes do not need to have n-1 dimensions to be valid and useful. I also touched up the section about specific types of hyperplanes to make the article consistent. I am a computer scientist, not a mathematician, so it would be a good idea for a mathematician to verify that the terminology I use is consistent with the usage in mathematics.--Headlessplatter (talk) 16:30, 1 June 2009 (UTC)

Removed Example

Someone removed the following paragraph without explanation. I think it's a useful example, so I put it here for further critique, since destroying good info is so much easier than creating it:

A 3-dimensional polygon is bounded by 2-dimensional surfaces, which intersect at 1-dimensional lines, which intersect at points. Analogously, an m-dimensional polygon is bounded by (m-1)-dimensional hyperplanes, which intersect at (m-2)-dimensional hyperplanes, and so forth until they intersect at points, which are zero-dimensional hyperplanes. Such polygons are used, for example, in linear programming. —Preceding unsigned comment added by 128.187.80.2 (talk) 21:04, 4 June 2009 (UTC)

Merge to Flat (geometry)

Link to article: Flat (geometry)

These two terms are often used interchangeably. In my area of study, the term "hyperplane" refers specifically to a (d-1)-flat in a d-dimensional space. Does anyone have a differing opinion? Justin W Smith ^talk/_stalk 20:26, 5 May 2010 (UTC)

(Note, I was invited here, by a neutral invitation on my user talk page.) Almost the same here, a d-plane or d-flat are used almost interchangeably, with d-flat sometimes also referring to d-facet A hyperplane usually refers to an n−1-flat in an n-dimensional space, but sometimes is used interchangeably with flat. — Arthur Rubin (talk) 21:51, 5 May 2010 (UTC)

(Note, I was also invited here, by a neutral invitation on my user talk page. Many thanks to Justin for the invitation!) My experience is the same. Hyperplane usually refers to flats of codimension one, but is sometimes used to refer to any flat. I would vote for merging most of the contents of this page into "flat", but some content that refers specifically to the codimension one case should be kept here. ("Hyperplane" seems like too common a word to have no article at all.) Jim (talk) 22:10, 5 May 2010 (UTC)

@Jim: IMO, "Hyperplane" should be a section of the Flat (geometry) article. Then Hyperplane would simply redirect to that section. Justin W Smith ^talk/_stalk 22:58, 5 May 2010 (UTC)

BTW. I searched the history of the two articles for editors I knew to be active. (I invited David Eppstein b/c I know that geometry is one of his areas of expertise.) I apologize to anyone that I overlooked. Justin W Smith ^talk/_stalk 22:58, 5 May 2010 (UTC)

In "Arrangements of Hyperplanes" by Orlik & Terao, they explicitly define "hyperplane" as a (d-1) affine subspace in d-dimensions. And I know "hyperplane" is used analogously for projective spaces. The intro of Orlik & Terao's book appears to discuss "hyperplanes" as such as if there would be no confusion in its precise meaning. (I'm sure I have seen "hyperplane" used as the term for a "flat", but at the moment I don't recall where.) Justin W Smith ^talk/_stalk 22:58, 5 May 2010 (UTC)

I agree with the previous comments: a hyperplane is a codimension-1 flat. In principle I think hyperplanes and flats could be separate articles but when I look at the actual content of the hyperplane article as it is now, most of it seems to be more about flats in general rather than being specific to the codimension-1 case. —David Eppstein (talk) 23:09, 5 May 2010 (UTC)

The Guggenheimer reference, added today to the Flat article, has a hyperplane as an n−1 flat. That reference is 1977; the "flat" terminology is more recent that the classical term "hyperplane". After all, a line is a flat, and so is a plane; the term requires numeric prefix for precision. However, it is suggestive of multilinear algebra, an important aspect of linear algebra. In popular literature one notes indiscriminate use of the terms hyperspace and hyperplane, but we need not yet yield to such commonality. Merge not called for.Rgdboer (talk) 03:11, 6 May 2010 (UTC)

Definitely hyperplane should not be merged into flat. Both articles should be more clear about their setting, for instance one does not need a Euclidean space for these notions, an affine space (over any field) is the proper setting (they also exist in projective spaces, but that is not the most common use). Also one should not imply a vector space setting (that is, a fixed origin); by the way I strongly disagree with the Euclidean subspace article that implies that a Euclidean space is a vector space and therefore has an origin (if you want that, then say "Euclidean vector space"). Euclid never assumes a fixed origin, nor should modern geometry. I have no strong ideas about "flat", I think the traditional term is "affine subspace" which seems quite clear. However hyperplane is a specific kind of affine subspace, namely one of codimension 1. One should not pollute this term by saying that it could be of any dimension, even if some author can be found that uses it that way; anyone who does is just wrong. It is just as wrong as using "line" or "plane" to mean any subspace. The term "hyperplane" is there precisely to single out the codimension 1 case, because it has special properties. For instance in an affine space a hyperplane and a line not parallel to it (i.e., of which no translate is contained in the hyperplane) always have a unique point of intersection; symmetry with respect to a hyperplane inverses orientation; and there are many more properties that are not shared with arbitrary subspaces. So one needs a term to distinguish the case, and "hyperplane" is it. Marc van Leeuwen (talk) 05:50, 8 May 2010 (UTC)

The recent change to this article distinguishes "hyperplane" from the more general term, "flat". So, I think the merge won't be needed. Hyperplanes, like points, have unique characteristics among flats in d-space. (I doubt anyone would claim that the article on "points" should be merged to "flats".) Justin W Smith ^talk/_stalk 06:08, 8 May 2010 (UTC)

I agree with Justin W Smith.

Re Marc v. Leeuwen's remarks: A "hyperplane" is any codimension-1 flat, whether in an affine, projective, or linear (vector) space. I would not say that affine space is "the" proper setting, but it surely is one of the proper settings and anyone who assumes a hyperplane passes through the origin ought to make that explicit. Otherwise, I agree with his remarks, except that he's a little too authoritarian about "wrong" for my taste (though not actually wrong about it).

Finally, it may be worth stating that a "flat" is perhaps more often called a "subspace" (a total synonym in this context), and that in an older day (if my memory serves me) a "hyperplane" was indeed any "flat"; the assumption that its dimension is n−1 is from recent decades. Thus, one has to be careful in reading older literature. As for "plane", I believe that long ago it too was often used for a subspace/flat of any dimension, but now that usage is almost entirely gone. Zaslav (talk) 07:13, 25 October 2010 (UTC)

Hyperplanes must separate the space

Our encyclopedia has separating hyperplane theorem and supporting hyperplane theorem. For these articles to make sense we need the precise and classical notion of a hyperplane. Even a modern text like Jean Gallier Geometric Methods and Applications for Computer Science and Engineering (2001) has the classical meaning: an n−1 flat.Rgdboer (talk) 21:42, 7 May 2010 (UTC)

The article has been altered to conform to standard.Rgdboer (talk) 22:06, 7 May 2010 (UTC)

I think this change might be sufficient to obviate a merge. Some of the material removed from this article might be appropriately added to Flat (geometry). Justin W Smith ^talk/_stalk 22:23, 7 May 2010 (UTC)

Why don't we just specify that those two theorems only apply to hyperplanes of codimensionality 1? Pretending that all hyperplanes have codimensionality of 1, and throwing out everything related to hyperplanes that don't divide the space, was a terrible solution to a trivial problem--just be more specific. I understand that much of the literature makes the assumption that hyperplanes have codimensionality of 1, but that doesn't make the rest of it cease to exist. If you are decreeing that the term "hyperplane" shall henceforth refer only to hyperplanes that divide a space, then you also need to invent a new term to encompass non-dividing hyperplanes and include it with your decree.--128.187.80.2 (talk) 20:10, 7 June 2010 (UTC)

The "new term" is not "new" at all, and it's well established for this use. If a "hyperplane" (as you might call it) doesn't divide the space, then it's a "flat". See Flat (geometry). Justin W Smith ^talk/_stalk 20:34, 7 June 2010 (UTC)

Oh. Thanks. Sorry for getting worked up.--128.187.80.2 (talk) 17:35, 8 June 2010 (UTC)

I hadn't heard of flat (geometry) before, but intro definitions as given make sense, and hyperplane belong in separate article here, special since being (n-1)-dimensional, it's a bounding "surface" of n-space. Also a hyperplane is defined by a point and normal vector in any dimension. Mathworld actually defines flats as the intersection of a set of hyperplanes. [1] Tom Ruen (talk) 06:03, 8 May 2010 (UTC)

In complex geometry (finite-dimensional vector spaces of complex numbers) hyperplanes are still important but they do not separate the space. —David Eppstein (talk) 16:02, 14 May 2010 (UTC)

And furthermore, in finite geometry there is no notion of separating a space, but a hyperplane is still a "flat" of codimension 1.

I also note that in this context "flat" is synonymous with "subspace". Possibly "subspace" is more widely used. Zaslav (talk) 07:02, 25 October 2010 (UTC)

Archive 1