> 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).