How can a 2-D Flatlander actually see anything, and how does this relate to 3-D?

Meatbot

It seems to me that in order for him to actually see objects around him, the objects and his eyes would need to have some non-zero thickness in a third dimension. In Flatland, the inhabitants see each other as line segments, but isn't this really impossible? In order to see a line segment it must actually be other than a line segment. It must actually have a nonzero thickness in a perpendicular dimension. A true line segment would be invisible.

Also, it seems a 1-D being would have to be other than 1-D in order to see. It seems he would have to be 3-D and his object would have to be 3-D.

Ok, so what does this have to say about 3-D people viewing 3-D objects? Can the 2-D example be applied here or is this where it stops? If it stops, why is 3-D special? Do 3-D eyes and 3-D objects actually have to be 4-D in order for the eyes to see the objects?

Math Jeans

Well, you could think of it this way. In a 3-D world, people actually see things in 2-D. You could say that our vision could be captured on a 2-D frame. For a 4-D entity, the 4-D entity would wonder how we are able to see 2-D things. The reality of it lies in the plane that the flatlanders live on. If one of use were to decend to flatland and view the world from the same point that the flatlanders do, of course we wouldn't see anything, as what happens is the plane of flatland would occupy 0% of our vision. However, for a flatlander, who knows nothing of the 3-D world, they would view the world as 1-D, however, for them, their movement would for them make them think that they are looking in 2-D, the same misconception that we have in the third dimension. For a flatlander, however, the plane of flatland occupies 100% of their vision, or mainly, their eyes view things as 1-D, and it is all they see, however, they can see a distance on the 2-D plane.

I hope that clears things up.

mgb_phys

I think you are reading slightly too much into the analogy.
The 2d world is imaginary, you don't have to worry about the number of photons emitted from an infinetly thin line.

Math Jeans is obviously a mathematician - so doesn't see the problem, you just assume that photons are infinetly thin as well 1

jim mcnamara

Besides Flatland is a flop in modern "math"* classes.

My sons denounced it thoroughly, as did their teachers who were compelled to use it by the local school bored of education. Probably backed by the Flat Earth Society.

"math"* = whatever the school bored in our district says it is.

PS: I liked Flatland when I read it. But I'm old.

mgb_phys

There is a very good sequel 'Flatterland' by Ian Stewart following Victoria's visit to a weird spherical land with three dimensions.

Meatbot

Could you explain that in more detail? Seems to me like a 3-D person seeing a 2-d plane perpendicular to his point of view is like a 2-d person seeing a 1-d line head-on (a point). I don't think 2-d guy sees the point since it has no area, but the 3-d guy should still see the plane.

But why wouldn't the flatlanders experience the same thing we would and see nothing? I mean, yeah it's 100% of their field of view but their field of view is zero because a line segment has no thickness. The same reason we wouldn't see anything applies to them too I would think.

Office_Shredder

To a 4-D person, a 3-D object would take up no 'volume' in their field of vision, and hence be invisible

Meatbot

Cool. Yeah I see that now. But it would only be invisible if the 4-d guy looks at it from 3-d guy's perspective (like a 3-d guy putting his eye in the 2-d plane). If he moved and looked at it from any other viewpoint (in 4-d) he'd see it, right?

Office_Shredder

Exactly. Similarly, if you put yourself into the flatlander's point of view, you think you can't see anything, but that's only because you're accustomed to seeing things with width

Meatbot

Thanks! But why would it just appear that I can't see it? I would think I would actually not see it. How can a 1-d line ever actually appear to someone in any dimension including a 2-d guy? Why isn't 2-d guy's field of vision exactly zero? All objects would appear to be of size zero. How could you see something with zero size?

Office_Shredder

Because it doesn't have size zero. It appears that it would have size zero to you because you're used to having eyes that see into 3 dimensional space. This means you see a 2-d image, and hence to see something it needs to have area. For a 2-dimensional sight, it sees a 1-d image and hence things only need to have length in his field of vision for him to see. If you're willing to accept the existence of a 2 dimensional person, you should be willing to accept this too

Math Jeans

Actually, I'm not a mathematician.

Yes. That is a great book.

Meatbot

That just doesn't seem right to me intuitively, but that doesn't mean it isn't.

I guess I don't understand why it would appear to have a nonzero size for him. I see the analogy that 3-d guy sees a 2-d plane, so 2-d guy sees a 1-d line. But it seems that there is a qualitative difference between a 2-d plane and a 1-d line in this case. I don't think it follows that because 3-d guy CAN see a 2-d plane, that 2-d guy can also see a 1-d line. A plane has area but a line doesn't. Seems to me like you need area in order to see something. I don't know....

Math Jeans

Yes, our vision covers the 2D plane, however, our vision only covers the 2D plane. A 2D flatlander DOES only see a 1D line, but you are thinking that a 2D object would be able to see into the 3D plane, in which the 1D line would be invisible, however, like the 2D plane covers 100% of vision for us, the 1D plane covers 100% of vision for a flatlander, and is all they see simply because they are 2D, and not 3D.

Meatbot

But why isn't it 100% of zero? It seems their field of view has zero area so they would necessarily be blind.

Math Jeans

because their eyes only see with the vision of a line. Think of it this way. Our eyes see things with the vision of a plane, while theirs see the vision of a line. Once again, a 4D entity would not be able to see a 2D image, as their eyes capture images in 3D, their vision holds a 2D image a 0% of their field of vision.

Basically, a 4D object sees in 3D, and cannot see 2D, a 3D object sees 2D, and cannot see 1D, a 2D object sees 1D, but cannot see 0D, and a 1D object sees 0D. This all happens due to the restrictions to the respective dimensions. I wish I could give visual evidence.

Read a few books on this and it might become clearer.

Jimmy Snyder

Flatland works better as a mathematical world than it does as a physical one. For instance, even though there is no third dimension to Flatland, the sphere can see Flatland. That means that photons go off in the third direction. So a physicist in Flatland wouldn't have conservation of energy to work with. Conservation of matter would be hard to justify too with spheres popping in and out at will.

There is a book called "The Planiverse" that discusses the science of a two dimensional world.