[md224]

> I don’t think that works because the N-1 dimension is contained in the N dimension.

I'm not sure that's right. Each dimension is linearly independent [0] from the others, which means e.g. you can't add up a bunch of width and get height, or add up a bunch of length and get width. So in an important sense, they're not contained within each other.

You might be thinking of how a 2D plane contains the first dimension within it, but that's not the 2nd dimension... that's two-dimensional (a combination of two dimensions).

[0] https://en.wikipedia.org/wiki/Linear_independence

[wibagusto]

Hmm yeah that what I’m confusing. I suppose it’s misleading when folks say The Fourth Dimension since it depends on context! But in the case of the article the question becomes how do we visualize four or more dimensions? What the dimensions represent doesn’t necessarily matter unless there’s a specific problem being solved.