Talk:Hypercube

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Hypercube elements ${\displaystyle E_{m,n}\!}$ (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

The section Faces contains this fragment:

"The number of the ${\displaystyle m}$-dimensional hypercubes (just referred to as ${\displaystyle m}$-cubes from here on) contained in the boundary of an ${\displaystyle n}$-cube is

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

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 ${\displaystyle 2^{n-m}{n \choose m}}$ is then equal to 1, the correct count.

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 ${\displaystyle \mathbb {C} ^{n}}$. 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.

While thinking about 4D, I tried to visualize it, but unfortunately, I wasn't successful. I tried to put it on paper, but after thinking about it carefully, it just wasn't possible. I imagined myself transported to a 2D world and trying to show the characters from there what 3D was. My first idea was to draw it for them, but a problem arose. There's no depth, no matter what I tried, I couldn't draw it in their world. To draw 3D, you need 3D in 2D; you can only draw a 2D world. After thinking about it, I remembered the hypercube, which makes absolutely no sense because why would 3D be any different and have the ability to draw higher spaces?

Can someone prove me wrong? Azorororo (talk) 18:00, 10 January 2026 (UTC)Reply

Are you saying that a concept cannot exist unless we can portray it? —Antonissimo (talk) 06:02, 11 January 2026 (UTC)Reply