> the Empire State Building can fit entirely inside of an n-cube with sides of 1 cm, for sufficiently large n. The length of the diagonal of such a cube grows without bound, and so does any constant-dimensional cross-section.
I still find this analogy confusing, because the units don’t match. The Empire State Building has some volume in cm^3, but the hypercube volume is cm^n. 1 cm^5 is not 100 times more volume than 1 cm^3, right?
Am I thinking about it wrong? I guess you can think about filling up a 3D cube with quasi-2D slices, is it like that?
Ah, thinking about the other comments a bit, you mean the diagonal of the hypercube can be larger than the biggest dimension of the building?
It’s not about area, but about actually physically placing an object in a higher dimensional object, like placing a long stick (1D) in a room (3D): the stick can be longer than the longest side of the room, but not by that much. In higher dimensions, this effect is more extreme.
Thanks, I see what you mean now. I think I was keying off the word “entirely” and trying to make it work for all three dimensions of the building at once. It seems like the building will fit along each coordinate inside the diagonals, but the edges of the hypercube would still cut through the building, right? That’s where I was getting confused
> the building will fit along each coordinate inside the diagonals, but the edges of the hypercube would still cut through the building, right?
No, you can fit the whole thing inside the 1mm hypercube, as long as you have enough dimensions. Once you have fit the height of the building along one diagonal, you can add new dimensions and rotate the building along that diagonal so that the width and the height will fit along the diagonals of the other dimensions.
I still find this analogy confusing, because the units don’t match. The Empire State Building has some volume in cm^3, but the hypercube volume is cm^n. 1 cm^5 is not 100 times more volume than 1 cm^3, right?
Am I thinking about it wrong? I guess you can think about filling up a 3D cube with quasi-2D slices, is it like that?
Ah, thinking about the other comments a bit, you mean the diagonal of the hypercube can be larger than the biggest dimension of the building?