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

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.

Generalized hypercubes?

The first paragraph of the section Generalized hypercubes is as follows:

"In complex Hilbert space, regular complex polytopes can be defined and are called generalized hypercubes, γ^p
_n = _p{4}₂{3}...₂{3}₂, or ... Real solutions exist with p = 2, i.e. γ²
_n = γ_n = ₂{4}₂{3}...₂{3}₂ = {4,3,..,3}. For p > 2, they exist in $\mathbb {C} ^{n}$. The facets are generalized (n−1)-cubes and the vertex figure are regular simplexes."

But the meaning of all this notation is unclear and needs to be explained, if any reader is going to understand this section.

In particular, what does this mean:

  γp
n = p{4}2{3}...2{3}2

???

I hope someone familiar with this subject will make this section comprehensible.

offset

The hypercube (offset) family is one of three regular polytope families, labeled by Coxeter as γ_n.

Why is that word (offset) here? Is it a synonym of hypercube?? Is it leftover from a syntax error? —Tamfang (talk) 16:25, 24 June 2025 (UTC)

I don't know and searching Google Scholar didn't find anything relevant. I think it should be removed. —David Eppstein (talk) 23:29, 24 June 2025 (UTC)