Talk:Hypercube

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[edit] redirect to tesseract

Resolved 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)

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)

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)
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)
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)
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)
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)

Every geometry textbook I've looked at uses the term hypercube to mean tesseract specifically, and it is clearly and patently illogical and nonsensical to use it to name figures with fewer than four dimensions. What "David Eppstein" "always [means]" by hypercube is utterly irrelevant. —Preceding unsigned comment added by 24.183.105.240 (talk) 12:53, 8 December 2007 (UTC)

I hesitate to pull rank, but given that some of my professional research papers concern hypercubes I think it is relevant. —David Eppstein (talk) 17:14, 8 December 2007 (UTC)
I'm curious what text books you have? Sommerville calls a tesseract a hexahedral 8-cell. Coxeter says tesseract. Coxeter does use hypercube in a table of regular polytopes {p,q,r}, but that implies which hypercube. He also says regular simplex in the context of 4-d figures, which would otherwise be ambiguous.
I might support calling this article the Hypercube family since it's not a single object all. Others support n-cube which parametrizes the family member. I agree saying a cube is a hypercube sounds silly. I also tend to think 4-cube/tesseract when someone says hypercube, but again, it's a family. People call me by my family name, and I can understand, but its still ambiguous. Tom Ruen (talk) 20:45, 8 December 2007 (UTC)
Re the silliness of low-dimensional hypercubes: Google scholar has 30 hits for one-dimensional hypercube, 112 hits for two-dimensional hypercube, and 277 hits for three-dimensional hypercube. —David Eppstein (talk) 21:33, 8 December 2007 (UTC)
The fewer dimensions meant, of course, the sillier it is to use the term hypercube, and you'll have noted that, correspondingly, according to your little (and inconsequential, I'm afraid) survey, the fewer dimensions the fewer "hits". —Preceding unsigned comment added by 24.183.105.240 (talk) 07:49, 11 January 2008 (UTC)
Re: "I also tend to think 4-cube/tesseract when someone says hypercube, but again, it's a family.":
Logically the term hypercube can mean an n-dimensional figure (with "family traits") such that n > 3, although it is commonly used to mean tesseract in particular. It cannot logically mean one such that n <= 3. —Preceding unsigned comment added by 24.183.105.240 (talk) 08:01, 11 January 2008 (UTC)

[edit] Merge Hypercube<--Hypercube graph

Resolved 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)

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

[edit] "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)

[edit] 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).

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)

It states "a 2-dimensional hypercube can only be rotated by vertex" In a 2 dimensional plane, a square can only be rotated by its center or its edge. If you rotate it by its vertex, it will lift of the plane, directly contradicting what a 2d object is. —Preceding unsigned comment added by 71.99.30.19 (talk) 02:52, 1 February 2008 (UTC)

[edit] 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)

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)
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)

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

  1. the set of all binary strings of a fixed length (the length of the string = the dimension of the hypercube)
  2. 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)

Furthermore, understanding the properties of hyper-dimensional objects is crucial for work in modern M-Theory which states that 7 of the 11 dimensions in our universe are curled up. —Preceding unsigned comment added by 71.99.30.19 (talk) 02:48, 1 February 2008 (UTC)

[edit] Fatal inconsistency!

We say that a hypercube is convex, but immediately proceed to describe it as a union of line segments. A few lines later we revert to describing each hypercube as the result of sweeping its predecessor in a direction orthogonal to itself. Do we wish hypercube to denote the n-simplex or its 1-skeleton? Or do we wish to retain the flexibility to use it in both ways? Whichever choice is made, the current lack of rigor scarcely supports aspirations to an encyclopedic treatment of the topic.----PaulTanenbaum 21:27, 17 August 2007 (UTC)

I don't understand this criticism enough to answer anything in a helpful way. I see nothing so dramatic. What does a simplex has to do with this? With all polytopes there's interpretations of interior vs surface. Feel free to improve it or make suggestions. Tom Ruen 22:34, 17 August 2007 (UTC)
Tom, here's the problem. The 1-skeleton (i.e. the union of line segments) is not convex for any n > 1, in other words for the square or beyond. But I'm happy to say that David Eppstein has fixed things up nicely.

PaulTanenbaum 13:21, 18 August 2007 (UTC)

[edit] Cellular Automaton

The recurrence relation that generates the number of elements in an n dimensionalc cube struck me as similar to the cellular automaton of Wolfram. It is a repetitive iterative process based on a very simple rule, albeit not producing black or white squares, but still very similar. Is this not worth a mention? In addition I believe Wolfram has suggested automata could be doing something fundamental in physics. It might be worth finding out if he has reported automata could have some connection with dimensions/hypercube. --- Plexos | Talk 05:16, 27 October 2007 (UTC)

Recurrence relations and cellular automata are two quite different things, and neither one was invented by Wolfram. The recurrence relation is easy to explain directly, by the way: if you view an n-cube as the Cartesian product of an interval and an (n−1)-cube, each m-dimensional face of the n-cube is either the product of an m-dimensional face of the (n−1)-cube with an endpoint of the interval (there are 2Em,n−1 of these) or the product of an (m−1)-dimensional face of the (n−1)-cube with the whole interval (thera are Em−1,n−1 of these). Add these two sets of faces and you get the recurrence. But I'd hesitate to put an explanation like that into the article without a proper citation for fear of violating the prohibition against original research. For the same reason, you should probably avoid putting things in because they "strike you as similar" and wait until you can find a reliable source for them. —David Eppstein 05:23, 27 October 2007 (UTC)
All Em,n for all hypercubes can be generated from the linear recurrence relation of the form e = 2a + b, there's nothing more to it. This is Occam's razor at it's sharpest and you might say that the relation "is" the hypercubes. Can the cartesian product system derive this relation or a simpler one for all m and n? If it can then I would be interested to see the maths. You only mention faces (cases En − 1,n) above but what about others like lines and points for a cube? As for original research, Jimmy Wales is wrong about this, what better place than Wikipedia with all it's credibility, software tools, discussion system, history control, etc on tap for creators. In addition the increased presence of such experts would raise standards throughout. --- Plexos | Talk 17:49, 27 October 2007 (UTC)
The objects counted by En − 1,n are "facets". "Faces" are elements of any dimension. As for whether Wales is right or wrong: if you disagree with him you're welcome to start your own encyclopedia but we have to play by the rules here. —David Eppstein 18:14, 27 October 2007 (UTC)
I had a look at the NOR page and it has a point, you can't have all sorts of cranks posting wrong information. It's a pity genuine creators go down with the ship though. They could allow special original research pages like say Original Research:Hypercube, though with their own ruleset and moderators. --- Plexos | Talk 07:48, 30 October 2007 (UTC)

[edit] Visualization

It's possible to see a 3-cube on a 2D image.

Is it possible to see a 4-cube if we made a 3D model of it? —Preceding unsigned comment added by 125.238.140.173 (talk) 23:18, 20 November 2007 (UTC)

What do you mean by "possible", "see", and "model"? It's certainly possible to make 2d drawings and 3d objects that help visualize some aspects of 4-cubes; there are several in tesseract. —David Eppstein (talk) 23:32, 20 November 2007 (UTC)

The picture of the tesseract projection in the article is actually of a transparent tesseract (otherwise it would look just like a regular transparent 3d cube). I think it would be better to either change the second picture to a simple transparent cube and leave the first one unchanged, or change only the first picture to a transparent cube. Chronometrier (talk) 23:05, 11 January 2008 (UTC)

[edit] The Values of n

All the given examples talk of n-cubes in which the value of n is an integer, I think it should be mentioned whether or not it is always the case. Knowing how pure mathematics work, I see no reason why hypercubes would only be integer-cubes and never (-3)-cubes, 1,5-cubes, pi-cubes or i-cubes, so it would be more complete if it were mentioned no?. 24.200.59.201 03:26, 1 December 2007 (UTC)

I think you mean whole numbers (o-oo), not integers. This does not happen simply because n tells us how many lines can be put in at a right angle at a corner. Look at a 3n cube. At each corner there are 3 right angles. So, how can you have 1/2 of a line? Any length of line is one whole line. Same thing with negative values. How can you have -3 lines? —Preceding unsigned comment added by 71.99.30.19 (talk) 02:39, 1 February 2008 (UTC)

[edit] The formulas of hypercubes elements

It is said that the number of vertices of a n-cube is 2^n and the number of "(n-1)-cubes sides" is 2n, but maybe some formulas for other elements could also be given: the number of edges, I calculate, is n2^(n-1) and the number of faces is 2^(n-2) times the summation of "a" from 1 to (n-1)[or the (n-1)th triangle number] and all elements are probably so calculable. I simply do not want to waste my time on calculating all of them myself if the formulas are already known (which they probably are) or, worse, if there is some mean of calculating any number of m-dimentional elements in a n-cube with a single formula. 24.200.59.201 03:26, 1 December 2007 (UTC)

The formula is given as a combination:

The number of m-dimensional hypercubes (just referred to as m-cube from here on) on the boundary of an n-cube is

E_{m,n} = 2^{n-m}{n \choose m} ,     where {n \choose m}=\frac{n!}{m!\,(n-m)!} and n! denotes the factorial of n.

Tom Ruen 03:53, 1 December 2007 (UTC)

Oh, and that is what is written in the article, I had misunderstood what it meant. thanks. 24.200.59.201 (talk) 23:39, 5 December 2007 (UTC)

[edit] Maple program

The following text was removed from Hepteract. Tom Ruen 19:07, 2 December 2007 (UTC)

for i from 0 to 13 do seq(binomial(i, j)*2^(i-j), j = 0 .. i-1) od;##

                                  2
                                 4, 4([[square]])
                               8, 12, 6  ([[Cube]])
                            16, 32, 24, 8 ([[Tesseract]])
                          32, 80, 80, 40, 10 ([[Penteract]])
                      64, 192, 240, 160, 60, 12 ([[Hexeract ]])
                   128, 448, 672, 560, 280, 84, 14 ([[Hepteract]])
              256, 1024, 1792, 1792, 1120, 448, 112, 16  ([[Octeract]])
           512, 2304, 4608, 5376, 4032, 2016, 672, 144, 18
      1024, 5120, 11520, 15360, 13440, 8064, 3360, 960, 180, 20
 2048, 11264, 28160, 42240, 42240, 29568, 14784, 5280, 1320, 220, 22
4096, 24576, 67584, 112640, 126720, 101376, 59136, 25344, 7920,1760, 264,  24
8192, 53248, 159744, 292864, 366080, 329472, 219648, 109824, 41184, 11440, 2288, 312, 26

Other:http://www.research.att.com/~njas/sequences/A038207

[edit] Hypercube names

George Olshevsky calls a 10-cube, a dekeract [1], but only case I know. Tom Ruen (talk) 01:59, 15 December 2007 (UTC)

George Olshevsky can continue calling it that on his web site. We should use more reliable sources here. —David Eppstein (talk) 02:14, 15 December 2007 (UTC)
you could continue the pattern. a 5-cube could be a penteract, sexeract (lol), septeract, octeract, noveract, ect. —Preceding unsigned comment added by 71.99.30.19 (talk) 02:43, 1 February 2008 (UTC)
It's not septeract; it's a hepteract. Weatherlover819 (talk) 03:44, 15 March 2008 (UTC)

[edit] Consideration

I think the Cartan's triangle is the appropriate arithmetic triangle for hypercubes. I think the Cartan's triangle (sandbox) should have its own article. I have written a sandbox, so put this into consideration. Weatherlover819 (talk) 03:44, 15 March 2008 (UTC)