…the characters are assorted geometric shapes living in an exclusively Two-Dimensional world…the narrator, a middle-aged Square, has a disturbing dream in which he visits a One-Dimensional realm, Lineland, whose inhabitants can move only from point to point.
With mounting frustration he attempts to explain himself-that he is a Line of Lines, from a domain where you can move not only from point to point but also from side to side. The angry Linelanders are about to attack him when he awakens.
Later that same day he attempts to help his grandson, a Little Hexagon, with his studies. The grandson suggests the possibility of a Third Dimension-a realm with up and down as well as side to side. The grandfather Square proclaims this notion foolish and unimaginable.
That very night the Square has an extraordinary, life-changing encounter: a visit from an inhabitant of Spaceland, the realm of Three Dimensions.
At first the Square is merely puzzled by his visitor, a peculiar Circle who seems to change in size, even disappear. The visitor explains that he is a Sphere. He only seemed to change size and disappear because he was moving toward the Square in Space and descending at the same time.
Realizing that argument alone will not convince the Square of the Third Dimension, the exasperated Sphere creates for him an experience of Depth. The Square is badly shaken:
There was a dizzy, sickening sensation of sight that was not like seeing; I saw a Line that was no Line; Space that was not Space. I was myself and not myself. When I could find voice, I shrieked aloud in agony, “Either this is madness or it is Hell.”
“It is neither,” calmly replied the voice of the Sphere. “It is Knowledge; it is Three Dimensions. Open your eyes once again and try to look steadily.”
Having had an insight into another dimension, the Square becomes an evangelist, attempting to convince his fellow Flatlanders that Space is more than just a wild notion of mathematicians. Because of his insistence he is finally imprisoned, for the public good. Every year thereafter the high priest of Flatland, the Chief Circle, checks with him to see if he has regained his senses, but the stubborn Square continues to insist that there is a Third Dimension. He cannot forget it, he cannot explain it.
from the Victorian fantasy Flatland, as quoted in the Aquarian Conspiracy