Talk:Measure polytope

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[edit] sources?

I mostly added the resource template because I'd like sources for this name. Tom Ruen 00:55, 29 July 2006 (UTC)

[edit] Names in higher dimensions

Where did all these -teron, -peton, etc. names come from? Who first called them that? I'm a little suspicious because dodecapeton, for example, gets zero Google hits. —Keenan Pepper 04:38, 4 September 2006 (UTC)

I agree. I never heard of these used by geometers. Maybe I don't read the right geometers. We need citations to the literature. Otherwise, I assume some overenthusiastic Wikipedian made these up, and they should be removed. But if they are (e.g.) known from the 19th century, then keep them. Zaslav 01:29, 16 November 2006 (UTC)

[edit] merge with n-cube? or hypercube?

I've heard n-cube used, but the existing article at n-cube seems to be something different. Tom Ruen 00:45, 4 November 2006 (UTC)

I am a mathematician and I often refer to the n-cube or hypercube. I never heard of a "measure polytope". The common name for this geometrical object is hypercube or n-cube. If anything is merged, this article should be merged into n-cube, not the other way round. However, I recommend the name "hypercube" as an article title, since that is the most correct general name. (n-cube means a specific dimension, n. Hypercube does not specify the dimension, so it's the most general.) Zaslav 17:51, 15 November 2006 (UTC)

Measure polytope comes from Coxeter, quoting selectively from "regular polytopes", p122-123:
  • "The n-dimensional generalization [of a parallelogram] is known as a parallelotope. It has 2^n vertices, [...] 2^(n-k)*(n choose k) [k-dimensional parallelotope subelements]."
  • "If the n-vectors are mutually perpendicular, the parallelotope is an orthotope, the generalization of a rectangle and the 'box'.
  • "If the n perpendicular vectors all have the same magnitude, the orthotope is a hyper-cube of measure polytope, γn and the corresponding lattices determine the cubic honeycomb δn+1.
  • "The name measure polytope is suggested by the use of the hyper-cube of edge 1 as the unit of content (i.g. the square as the unit of area, and the cube as the unit of volume)."

My own resistance against using hypercube as naming the family is that the name implies "beyond the cube", so it would seem to exclude the lower elements.

There's also the dual family, cross-polytope to consider. Do you have a shorter name for that? I've heard people use n-diamond (in the "baseball" diagonal sense) perhaps?

Tom Ruen 00:08, 16 November 2006 (UTC)

Thanks for these valuable comments. Let me reply: The name "hypercube" in mathematics is completely standard for all dimensions from 0 on up. There is no exclusion of small dimensions. This is well-established terminology. The name "measure polytope" has no record of general use in math that I know of (but I don't know everything!). Note that, according to your quote, even Coxeter has it only as a secondary name (with a justification, which suggests that possibly he was inventing a new name). In fact, a close reading shows that Coxeter means by "measure polytope" only what we usually call the "unit hypercube", not any hypercube of arbitrary size. I think the case is very strong for writing the article under the name "hypercube".
I'm not sure what the problem is with "cross-polytope". That name is standard. An alternative that is also found (less commonly) is "hyperoctahedron". These are the names used by geometers, so as I understand it, they should be the ones in Wikipedia, on the grounds that Wikipedia is not the place for new inventions (research or terminology). Does this sound right? Zaslav 01:26, 16 November 2006 (UTC)

Matches in MathSciNet for "hypercube": 2014. Matches in MathScinet for "measure polytope": 5. Matches in Google Scholar for "hypercube": 77600. Matches in Google scholar for "measure polytope": 36. I am quite confident that the majority of matches for hypercube in both cases do not restrict themselves to exactly four dimensions. —David Eppstein 01:50, 16 November 2006 (UTC)

I surrender but not on search results implying dimensions of meaning. It still seems wierd that "cube is a geometric object" and "hypercube is a class of geometric objects". Simplex has the same problem, but we don't talk about hypertriangles to imply simplices of any dimension! At least Coexter is consistent with his notation - adding "tope" suffix to objects of general dimension.

I questioned "cross-polytope" because I considered it in the same category as "measure polytope" - a slightly clumsy term for a class of objects.

I am most comfortable with n-cube since it is a clear set rather than an object. There's still the n-cube article of little discussion and minimal edits, added anonymously ONCE, touched a couple times.

How about:

  1. Clear n-cube
  2. Move this to n-cube, and redirect here to n-cube?

Tom Ruen 04:17, 16 November 2006 (UTC)

In general, I prefer "n-object" to "hyperobject" (object = cube, sphere, torus, simplex, etc.) You don't really hear the hyper terminology much (actually people usually just say cube, sphere, torus, simplex, etc., with the dimensionality clear from context). I don't really like having to specify n in the title, but you have to disambiguate somehow. One could also point out that n-plane and hyperplane (codim-1 plane) are different concepts. -- Fropuff 04:26, 16 November 2006 (UTC)
We should remember that we are writing for a general audience for whom "hypercube" is more likely to be encountered and is a more English-like and less technical name than "n-cube." I think the solution is clearer exposition. We should also be consistant across the encyclopedia and hypersphere is already the primary name used rather than n-sphere. Here is some possible language:
In mathematics, a hypercube is the generalization of the ordinary cube to spaces of any dimension, particularly dimensions greater than three. Mathematicians also use the term n-cube, where n is the dimension, and often just say cube when the dimensionality clear from context. The hypercube where the length of each edge is 1 is called the unit hypercube or unit n-cube. H.S.M. Coxeter introduced the term measure polytope for this object. The hypercube in four dimensions, the 4-cube, is also called the tessaract.
--agr 11:18, 16 November 2006 (UTC)
I guess my biggest complaint is that I don't think I've ever heard a mathematician call a n-sphere a hypersphere or an n-cube a hypercube. I know the terminology exists, but it just isn't used in mathematics. I know we are writing for more than just mathematicians here, so I'm not 100% opposed. I do agree we should be consistent. I would fully support moving hypersphere to n-sphere. In fact, I've wanted to do that for a long time. Perhaps we should bring this discussion to WT:WPM and see what others think. -- Fropuff 15:50, 16 November 2006 (UTC)
For sphere I tend to agree with you, but hypercube is certainly used. See the numbers from my search results above. In comparison to 77600 hits for hypercube in Google Scholar, I got only 11200 hits for n-cube, so while I consider it a valid synonym it doesn't seem to be winning the popularity contest. —David Eppstein 15:59, 16 November 2006 (UTC)
A similar search in Google Scholar produces 7650 hits for hypersphere and 4470 for n-sphere. In general Google the number of hit are 322000 for hypersphere vs 84500 for n-sphere. Also hypersphere is in my Random House Dictionary of the English Language, n-sphere is not. I think the reason you don't hear the terms hypersphere and hypercube used much by mathematicians is that they are comfortable with higher dimensions and just say sphere and cube. Indeed they will say 2-sphere to be clear when talking about the ordinary sphere. I'd also point out that that three of the headings in the hypersphere article use the term "hyperspherical." Substituting "n-spherical" would not work.--agr 20:27, 16 November 2006 (UTC)
Well if you want to argue by search counts: a Google scholar search for "sphere" returns 375,000 while "cube" returns 345,000. Granted many of these are for the low-dimensional concepts, but certainly not all. As for the titles in the hypersphere article, they could be replaced with "spherical"; it is clear from context that multiple dimensions are being discussed. I guess what I'd really like to see is sphere (something) and cube (something). I'm just not sure what the something should be. Maybe "n-dimensional"? -- Fropuff 23:28, 16 November 2006 (UTC)
From WP:NAME "Generally, article naming should give priority to what the majority of English speakers would most easily recognize, with a reasonable minimum of ambiguity, while at the same time making linking to those articles easy and second nature. There is no reason to create disambiguated versions of cube and sphere when there are perfectly good English words for the concepts in question, hypercube and hypersphere.--agr 02:12, 17 November 2006 (UTC)
I think the reason you don't hear "hypersphere" and "hypercube" used by mathematicians is who you hang out with. I hear "hypercube" used often enough, but not "hypersphere" because I work with flat objects, not round ones, in my mathematics. I think it's widely understood that "sphere" and "cube" used for arbitrary dimensions are abbreviations.
The term "n-cube" or "hypercube" or "n-sphere" or "hypersphere" does not mean a family, it means an object of a certain kind. All objects of that kind do constitute a family.
The difference between "hypercube" and "n-cube" and the reason I prefer the former as an article name is that the latter specifies a particular dimension. It is often used loosely to mean "for any n", but that's sloppiness. The term "hypercube" is a better article title because it doesn't specifically imply a particular dimension (despite having sometimes, but now rarely, been used for a 4-cube).
Is it time to carry out the move? Zaslav 03:46, 20 November 2006 (UTC)
Yes the kicking point for me was the Google scholar results. --ANONYMOUS COWARD0xC0DE 23:30, 24 December 2006 (UTC)

[edit] Can Someone please explain why the hypercube looks the way it does

I can understand taking the cube, and sweeping it such that we get this picture Hypercube. What I don't understand, is what this picture is supposed to represent Confusing Hypercube

The two images are just different ways of representing a hypercube in two dimensions. The first way projects the hypercube onto 2-D space using isometric projection, which keeps parallel lines parallel to one another. The second way is a perspective projection. Gandalf61 16:56, 12 November 2006 (UTC)
See Schlegel diagram Tom Ruen 00:16, 13 November 2006 (UTC)


I kind of understand it now, although isometric projections are easier to understand that perspective projections. Thanx Paskari 19:57, 30 November 2006 (UTC)