Square to Cube, Cube to Tesseract

[Yoshi]

I read an explanation of a cube being an infinite squares on top of eachother.

If you took an infinite cubes and put them one on top of eachother, it would make this really tall rectangle.

Sometimes camparing 1 dimension to another isn't really the best way to go.

[PWrong]

That's a good joke, but it's not really accurate. You wouldn't stack the cubes on top of each other, you'd stack them through the fourth dimension to make a hypercube.

I'm not sure about that explanation of a cube though. Infinity is a tricky number. If a square has length and width, but infinitesimal depth, and you had infinity of them stacked together, their total height would be equal to one. But one of what? And what if the length and width of the squares is 2?

[RQ]

That's true, but the thing is that stacking an infinite number of squares on top of each other means that all 3D object with respect to a finite and defined 2D shape are the same height, since I think we can all agree that infinity=infinity and that an infinite number of squares would make a length of x and x only for all 3D objects, so the third dimension would only have one polyhedron. Or perhaps one isn't the best word to describe it since it will probably have an infinite height. However maybe there would be more than one different shapes in 3D, since if we have the two dimensions of the 2D shape of which we are stacking infinitely, we would have an object with infinite height, yet different length and width. This is possible, where you might argue that all of the volumes would be equal to infinity, but just like different parabolas, and hyperbolas they wouldn't be the same. However this brings us to another point. The transition from 1D to 2D, which would be the stacking of an infinite number of segments parallel to each other, which would make an object in the 2nd dimension with an infinite length, but finite width. This would make the 3D dimension have only finite width. Yet if we even go back to a point, its transition to the 1st dimension would have to give it infinite width, so basically, it would all be a big jumble of infinite dimensions, if infinite stacking is all we do.