Flatland flaw?

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Flatland flaw?

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.
Seldon
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Re: Flatland flaw?

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.

Keiji

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Re: Flatland flaw?

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?
Seldon
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Re: Flatland flaw?

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.

Keiji

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Re: Flatland flaw?

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.
The dream you dream alone is only a dream
the dream we dream together is reality.

wendy
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Re: Flatland flaw?

In the movie of Flatland, Flatland was made to appear like a blue solid ground but A Sphere could just pass through it as though unobstructed
People may consider as God the beings of finite higher dimensions,
though in truth, God has infinite dimensions

Prashantkrishnan
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Re: Flatland flaw?

Seldon wrote: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.

Would it be possible for a 2D being to conceptualize a 3D object? The 2D being sees in 2D. It would have to construct a 3D object in its mind.

It would be much easier for us to do this than it would be for Mrs. 2D. But if you try it, you will find out how hard it is. Let's say you have a screen that shows you only one line of pixels at a time. It would be difficult to build up an image from this. A simple image like a cube, yes. Anything complicated, no, unless one of those unusual people who can memorize large amounts of arbitrary data.

If most of us can't do it, how can we expect a 2D creature to? 2D creatures have fewer neurons in their brains. One wonders whether they would have the raw capacity to accomplish this task.

So in general I think it isn't possible. There is too much data for a lower dimensional being to absorb. Essentially what you are doing is replacing every point with a set. But simple cases are doable. Cubes, spheres, things like that.

Through the study of perception one realizes that most of the time we observe a tiny fraction of what is going on around us. We get by. Our world has a great deal of order and continuity that we can take advantage of. The same can be done with higher dimensional spaces. The space may be huge, but the number of essential features could be quite small. I suspect that with practice one could develop enough facility to get around.
PatrickPowers
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Re: Flatland flaw?

Wrong, a 2D being doesn't see in 2D. It sees in 1D and infers the 2D by parallax, lighting and shade, prior experience, etc.. Though, of course, if you ask one, it would claim that it could visualize 2D quite well, since it would have honed its perception so much that the leap from 1D to 2D would pretty much be instinctual and unconscious.

Much like how we 3D beings imagine that we see 3D, but actually we don't. What we see is just 2D images from which our brains, having internalized the process after much exposure to such images from the eyes, unconsciously and instinctively construct 3D models based on things like parallax, light and shade, prior experience, etc.. (Which may or may not represent reality -- hence the existence of illusions like the Necker cube AKA the tumbling cube, the hollow mask illusion (that appears to be a face turning the other way), etc.).

Consider for a moment, how much our so-called 3D perception is really just manipulating 2D surfaces in disguise, so much so that we have come to identify many objects by their surface, regardless of their actual, internal, 3D contents. Even when we visualize containers, which arguably are more than just their outer 2D surface, we still think of them in terms of 2D -- as two sets of 2D surfaces: an outside set, and an inside set. We never think of, for instance, the wood grain structure of a wooden box, for example. It doesn't exist in our minds, because we are basically thinking in 2D, just with some clever imagination that pulls it just a tad closer to the 3D reality.

Yet, from the vantage point of hypothetical beings who can actually see 3D in its full glory, such as 4D beings with 3D retinas, the inside and outside surfaces of the box are laughably irrelevant -- they are no more than the outlines of the box, barely within notice -- while the 3D wood grain structure would be the primary feature of the box in their sight.

Nonetheless, for what it's worth, our pseudo-3D perception is pretty useful when it comes to visualizing higher dimensional space. It at least gives us a crutch upon which we can, just barely, peer into the world of 4D objects and their amazing geometries, given sufficient training along the right principles. It takes effort, certainly, but it is not insurmountable.
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