Flatland flaw?

[Seldon]

I just reread Flatland...remember the part where the sphere takes A. Square out of Flatland and shows him the 3D world? Well, I don't think he could possibly understand it the way he does in the book. As we already know A. Square's vision is a line, meaning the sphere could not show him a cube the way the sphere sees it. All A. Square would see is one line of it. Even if he were moved past a cube or a sphere all he would see is lines. So really the only way he could understand it is if he was smart enough to visualize it and see it for what it really is.

So my question is this: is it possible for any being of n dimensions to truly visualize a figure of n+1 dimensions? Obviously A. Square does so for the purpose of the plot, but could he actually do it in reality? I'm not so sure.

[Keiji]

I think Sphere made Square somewhat 3D in the process of taking him out of Flatland. Just enough to be able to see in 3D. Flatland is already flawed enough - even if there were other dimensional universes intersecting with ours, no interaction would be physically possible between them, not even sight.

[Seldon]

I realize that the sphere wouldn't be able to see the square because he is infinitely thin, but wouldn't the square be able to see the sphere if he passed into his plane?

And what I was really trying to say is this: could any hypothetical being of n-1 dimensions fully imagine n dimensions, or is it truly impossible?

[Keiji]

I realize that the sphere wouldn't be able to see the square because he is infinitely thin, but wouldn't the square be able to see the sphere if he passed into his plane?

No. Between any pair of atoms there is a gap containing no matter, and this gap is on the order of 100,000 times bigger than an individual atom. Because of this, it's pretty much impossible for Sphere's atoms to happen to line up in just the right way for them to all intersect with Square's plane. Even if they did, Sphere would have to be at a temperature of absolute zero, otherwise vibration from heat would make them not line up again.

And what I was really trying to say is this: could any hypothetical being of n-1 dimensions fully imagine n dimensions, or is it truly impossible?

Yes. For example, you could have a dream which was in 4D. It's just highly unlikely, because our dreams are mainly built on what we know of real life.

[wendy]

It's like most movie flaws. It's a device to say that this happened, to allow the story to go on. In the 'planiverse', the device used is a nightly teletype conversation, but they did not have teletypes back when flatland was written, so they used a different device. Most movie devices is simply to let you know that something happened, and that the story goes on without expounding on this detail or that.

In any case, the device used is 'hyperspace', which means 'over-space'. The lessen is that three dimensions is hyperspace to two dimensions, as four dimensions is hyperspace to three dimensions. It's kind of like 'upstairs': it's relative, not absolute.