Synthesized answer
Flatlanders move freely on or in the surface of their world, which is described as a vast sheet of paper [1]. They can move about on this surface but do not have the power to rise above or sink below it [1]. This movement is compared to shadows, but they are solid with luminous edges [1].
In the context of Linelanders, a specific group within Flatland, their movement and eyesight are confined to a single straight line, which is their world [2]. Each Linelander occupies the entire width of this narrow path, and they cannot move left or right to allow others to pass [2]. This means that Linelanders cannot pass each other; once neighbors, they remain neighbors until death [2].
Synthesized from the book passages below. Chat with the book on Feynman for follow-up.
From the book
t because we call it so, but to make its nature clearer to you, my happy readers, who are privileged to live in Space. Imagine a vast sheet of paper on which straight Lines, Triangles, Squares, Pentagons, Hexagons, and other figures, instead of remaining fixed in their places, move freely about, on or in the surface, but without the power of rising above or sinking below it, very much like shadows—only hard with luminous edges—and you will then have a pretty correct notion of my country and countrymen. Alas, a few years ago, I should have said “my universe:” but now my mind has been…
hat he called his _inside_ or _stomach;_ nor had he even now the least conception of the region from which I had come. Outside his World, or Line, all was a blank to him; nay, not even a blank, for a blank implies Space; say, rather, all was non-existent. His subjects—of whom the small Lines were men and the Points Women—were all alike confined in motion and eyesight to that single Straight Line, which was their World. It need scarcely be added that the whole of their horizon was limited to a Point; nor could any one ever see anything but a Point. Man, woman, child, thing—each as a…
e water. Well, that is just what we see when one of our triangular or other acquaintances comes towards us in Flatland. As there is neither sun with us, nor any light of such a kind as to make shadows, we have none of the helps to the sight that you have in Spaceland. If our friend comes closer to us we see his line becomes larger; if he leaves us it becomes smaller; but still he looks like a straight line; be he a Triangle, Square, Pentagon, Hexagon, Circle, what you will—a straight Line he looks and nothing else. You may perhaps ask how under these disadvantagous circumstances we…
atement of facts followed by ocular demonstration ought to suffice. —Now, Sir; listen to me. You are living on a Plane. What you style Flatland is the vast level surface of what I may call a fluid, or in, the top of which you and your countrymen move about, without rising above or falling below it. I am not a plane Figure, but a Solid. You call me a Circle; but in reality I am not a Circle, but an infinite number of Circles, of size varying from a Point to a Circle of thirteen inches in diameter, one placed on the top of the other. When I cut through your plane as I am now doing, I…
moving parallel to itself upward. _I_. What? Northward? _Sphere_. No, not Northward; upward; out of Flatland altogether. If it moved Northward, the Southern points in the Square would have to move through the positions previously occupied by the Northern points. But that is not my meaning. I mean that every Point in you—for you are a Square and will serve the purpose of my illustration—every Point in you, that is to say in what you call your inside, is to pass upwards through Space in such a way that no Point shall pass through the position previously occupied by any other Point; but…