Talk:Hypercube

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redirect to tesseract

Resolved

I changed the redirect from measure polytope to Tesseract. I agree a hypercube can imply HIGHER polytopes than a tesseract, but 99% of the time I expect people MEAN a 4-polytope, so I judge it better to link to tesseract specifically and let readers go to the more general (family) secondly from there. Tom Ruen 06:21, 14 October 2006 (UTC)[reply]

Since my redirect was reversed, I removed the redirect and put two links in here.

I still disagree - I NEVER mean "n-dimensional hypercubes" and say hypercube since that is ambiguous what I mean. I'd say n-cube or n-hypercube or n-measure polytope. If there's no dimension offer I assert 4 is implied. Tom Ruen 09:47, 12 November 2006 (UTC)[reply]

I disagree. I ALWAYS mean the general case when I say "hypercube", and would specify 4-dimensional if that's what I mean. I don't think 4d is the default, and I think hypercube should be the preferred name for what is now measure polytope. I see tesseract as a special case, that should be prominently linked from the hypercube page but not redirected from it. And according to WP:D disambiguation pages (such as the current state of this page) are for when there are several links to disambiguate, but here there are only two. —David Eppstein 18:18, 15 November 2006 (UTC)[reply]

Later: I made this a little more texty and added a link to hypercube graph. So now it looks a little more plausible as a dab page. I'd still prefer this to be the main name for measure polytope, though, and am still not convinced having a separate dab page is necessary since the tesseract and hypercube graph articles are linked from that page anyway. —David Eppstein 18:29, 15 November 2006 (UTC)[reply]

I agree with David Eppstein. The mathematical community does not mean 4-dimensional when it says "hypercube". I do believe the name "hypercube" has been used, e.g. in science fiction, for the 4-cube, and it would be good to give that as a secondary meaning with a link to tesseract.

Here's a suggestion: Keep "hypercube" as a disambiguation page, since there are several meanings, such as the graph. Create hypercube (geometry) to move measure polytope to. Put into measure polytope a short definition as a unit hypercube (named by Coxeter, not common). I throw these ideas out for reactions. Let's hear! Zaslav 01:41, 16 November 2006 (UTC)[reply]

Hypercube, Hypercube graph, n-cube, and measure polytope are too closely related to be considered disambiguous. While the A_(disambiguation) article could be referring to a number of different completely unrelated concepts from many different fields. Same with the error page, it has a separate disambiguous page, however, hypercube only has one unrelated concept mainly the movie. The hypercube article will now contain what was once the measure polytope page, have a single disambiguous link to the movie at the top, and the article will refer to an n-cube when referring to a specific dimension, measure polytope should only be used when applied to hypercubes of unit length (side length, wrong terminology) one. --ANONYMOUS COWARD0xC0DE 00:23, 25 December 2006 (UTC)[reply]

Good! That's what I meant all along, really, really! (Also, the principal meaning of "hypercube" is obviously the geometrical one.) Thanks to whoever carried out the merge/move. Zaslav 06:40, 30 January 2007 (UTC)[reply]

Merge Hypercube<--Hypercube graph

Resolved

To me it just seems that hypercube graph could be a section within Hypercube. Any dissenting view points? --ANONYMOUS COWARD0xC0DE 00:36, 10 January 2007 (UTC)[reply]

I don't see why a single merged article would improve the current situation, and I think the significant differences between hypercubes and hypercube graphs could make a merged article quite confusing. Hypercube graph lies in Category:Graphs, which may not be as relevant for hypercubes as geometric objects. There is also some important graph theory I hope to add but haven't yet concerning median graphs (retracts of hypercubes) and partial cubes (isometric subgraphs of hypercubes). A hypercube has faces of many dimensions while a hypercube graph has only vertices and edges. A hypercube is topologically equivalent to a ball while a hypercube graph has a large homology group. A hypercube has high dimension while the hypercube graph article talks about two-dimensional unit distance representations. We don't merge complete graph and simplex; why is this any different? —David Eppstein 07:41, 10 January 2007 (UTC)[reply]

Ok. I withdraw my request. --ANONYMOUS COWARD0xC0DE 04:51, 12 January 2007 (UTC)[reply]

"Measure polytope"

I changed the wording about and usage of this term, because a reading of Coxeter (1973) shows that he did not mean it to refer only to a unit hypercube. He meant it to be any "hyper-cube" (his spelling). Also, it seems clear to me, from his language, that he invented the name. Zaslav 17:41, 8 February 2007 (UTC)[reply]

n cube rotation

Can we get rid of this section? Maybe note that rotation involves 2 axes, so an N cube has N*(N-1)/2 axes of rotation (if we are counting only "square" rotations). —The preceding unsigned comment was added by Paul Murray (talk • contribs) 04:09, 30 April 2007 (UTC).[reply]

It's badly written in a way that makes it sound like OR, but actually the symmetries of hypercubes are well known, and their symmetry group is the same as the signed permutations. Probably this should be completely rewritten. I don't know why you would only consider square rotations, though. —David Eppstein 04:29, 30 April 2007 (UTC)[reply]

Significance

Could somebody explain what the significance of the Hypercube is? Why are we looking at it? What does it represent? ThePeg 18:49, 30 April 2007 (UTC)[reply]

I don't know but now I understand you make a "sweeping of the things" to make dimensions. Crikey I get it!!! WinterSpw 05:37, 24 May 2007 (UTC)[reply]

If you are going to require that mathematical concepts "represent" things or have concrete analogs, you are going to have to start discussions on a lot of pages. Much of the time, mathematical matters are allowed to exists in and of themselves, as "knowledge for knowledge's sake." Krychek 16:29, 8 June 2007 (UTC)[reply]

There are two important sets that hypercubes (or rather their vertices) can be be thought of as representing, though:

the set of all binary strings of a fixed length (the length of the string = the dimension of the hypercube)
the powerset of a finite set (the size of the set = the dimension of the hypercube).

In addition, the hypercube forms a fundamental building block for d-dimensional Euclidean space and forms the unit of measurement for volume in that space. It is much more than simply a shape that can be defined to exist in and of itself. —David Eppstein 23:48, 9 June 2007 (UTC)[reply]