Talk:Hypercube

Does anybody know how to transpose this table for better display when page is enlarged?

Hypercube elements $E_{m,n}\!$ (sequence A038207 in the OEIS)

			m	0	1	2	3	4	5	6	7	8	9	10
n	n-cube	Names	Schläfli Coxeter	Vertex 0-face	Edge 1-face	Face 2-face	Cell 3-face	4-face	5-face	6-face	7-face	8-face	9-face	10-face
0	0-cube	Point Monon	( )	1
1	1-cube	Line segment Ditel	{}	2	1
2	2-cube	Square Tetragon	{4}	4	4	1
3	3-cube	Cube Hexahedron	{4,3}	8	12	6	1
4	4-cube	Tesseract Octachoron	{4,3,3}	16	32	24	8	1
5	5-cube	Penteract Deca-5-tope	{4,3,3,3}	32	80	80	40	10	1
6	6-cube	Hexeract Dodeca-6-tope	{4,3,3,3,3}	64	192	240	160	60	12	1
7	7-cube	Hepteract Tetradeca-7-tope	{4,3,3,3,3,3}	128	448	672	560	280	84	14	1
8	8-cube	Octeract Hexadeca-8-tope	{4,3,3,3,3,3,3}	256	1024	1792	1792	1120	448	112	16	1
9	9-cube	Enneract Octadeca-9-tope	{4,3,3,3,3,3,3,3}	512	2304	4608	5376	4032	2016	672	144	18	1
10	10-cube	Dekeract Icosa-10-tope	{4,3,3,3,3,3,3,3,3}	1024	5120	11520	15360	13440	8064	3360	960	180	20	1

Most of the volume is near the corners

I've see a few places where it has said most of the volume of a n-cube is near the corners. This is by comparison to a n-ball. It might be worth putting in though I had a look and it is more nuanced than that. You might like thislittle proof that most of the volume is not at the corners, though it's not published so you can't put it in!

Consider an n-cube of side 2. Then each corner is an n-cube of side 1 with the centre of the original n-cube at the opposide corner. Then how big an n-cube round the corner would be needed to have proportion P say 9/10 of the whole n-cube? That would have to be of side nth root of P. As n grows P^1/n tends to 1 so practically none of the volume of an n-cube is in its corners! For instance for n-cube shaped corners of a tesseract to contain 9/10 of the total volume they would have to take (9/10)^(1/4)=0.974 of each side and just leav 0.026 wide strips round the centre. Considering the volume isn't in an n-ball round the centre either it seems to just be slipping away in the cracks ;-) NadVolum (talk) 20:57, 16 October 2021 (UTC)

Adding a Parallel projection

I noticed that we have a perspective projection of an n-cube, so I was thinking of adding a parallel (graphical) projection of a 4-cube shown below:

It may be nice to add it under the perspective projection image to help readers have a better visual and mathematical aid of the structure. If no one opposes this I may add it.

Mike1291 22:46, 22 November 2021 (UTC)

15 Jan 2022, Lem Dear Mike (and others?), I am not a polytopist, but have a question. In 4-D, are projections (i.e. 'images' in 3-D ever discussed; the algebra, if not a supposed visualization. (That would be similar to the 2-D projections we can draw of 3-D objects.) If so, has the following version of an all-integer 'Laplace four Square' Equation come up? & For edge E, and the four projections A, B, C, and D: 3*E^2 = A^2 + B^2 + C^2 + D^2. I've obtained several, and used these to construct equations for those five variables using four parameters. Each contains the same square root of the sum of six pairwise products or the four parameters. & For N dimensions, this generalizes to: (N-1)*E^2 = A1^2 + A2^2 + A3^2 + ,,, +[Asub(N-1)]^2 + [Asub(N)]^2. N = n parameters are needed. The n parametric equations each contain n-1 simple terms, plus the SQRT containing nC2 pairwise products. I could add more details if any of this rings a bell for any polytopist out there. Lemchastain (talk) 04:49, 16 January 2022 (UTC)

No idea re the integer equations, but we would need a source to add it to the article.

Incidentally, I undid the addition of the image above, with the edit summary: "This article is already overloaded with images, including at least two parallel projections. Also, I'm unconvinced that this is a correct parallel projection: what is the projection vector and how does it produce that particular combination of edge lengths and angles?" —David Eppstein (talk) 05:39, 16 January 2022 (UTC)

Tesseract

Why is this article different from that of the Tesseract? It feels like roughly the same thing, at least close enough to be merged. Language Boi (talk) 20:33, 22 December 2023 (UTC)

That one is only about four-dimensional things. This one is about the generalization to any number of dimensions. Four dimensions is still low enough that the regular polytopes there are special — in five or more dimensions there are always exactly three regular polytopes (the hypercube being one of them) but just as three dimensions has the five Platonic solids, four dimensions has six regular polytopes. So I think that as part of this set it has enough independent notability as a mathematical topic to justify a separate article from the general hypercube article. And certainly the same is true for the section on cultural uses. —David Eppstein (talk) 20:37, 22 December 2023 (UTC)

Ah, I see. That should probably be clarified at the top of the article though--because I would probably use the two words interchangeably. Whether that's correct or not I don't know, but I know it's common. Language Boi (talk) 21:21, 23 December 2023 (UTC)

You mean like the clarification that already exists in the hatnote at the top of the article? Or the one already in the caption of the figure at the top of the article? —David Eppstein (talk) 22:39, 23 December 2023 (UTC)