Talk:Hypercube

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)

Text
"Increasing axis vectors" means exactly what it means, increasing the amount of axes in any dimension. Examples include x axis, y axis, z axis, w axis, v axis, etc... Adding a coordinate axis does increases the vertices of a hypercube by a multiplication of two. Moreover the text is useful because it links to 4-cube, 5-cube and so on and makes clear the relationship between the number of vertices and the hypercube. Lebesgue measure is not irrelevant, as the text is in specific relationship to a hyper-volume, it's not WP:SUBMARINE because it's directly linked to the text. 21:04, 17 May 2024 (UTC) Des Vallee (talk) 21:04, 17 May 2024 (UTC)


 * You do know that squares and cubes do not need to be aligned to the axes of any particular coordinate system, right? Perhaps you should also know that providing formulas for the number of faces of different dimensions, including vertices, is already done in the "faces" section, and that your attempt to add a very partial piece of that information counting only the vertices is misplaced in a section about coordinates. —David Eppstein (talk) 21:14, 17 May 2024 (UTC)

More natural viewpoint
The section Faces contains this fragment:

"''The number of the $$m$$-dimensional hypercubes (just referred to as $$m$$-cubes from here on) contained in 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 $$n$$.''"

But there is no good reason to limit the counted faces to the boundary.

The n-cube is a perfectly fine polytope, and it has exactly one additional face beyond those on the boundary: its single n-dimensional face.

What's more, this corresponds to the case above where m = n, and it is easy to see that the very same formula $$ 2^{n-m}{n \choose m} $$ is then equal to 1, the correct count.