In this classic masterwork of perspective, Abbott examines the science of multiple spatial dimensions while satirizing the absurdity of truth by consensus and extending a subtle invitation to consider how what we take as our givens limits our grasp of reality, presenting us with a false view of the world warped by our way of looking at it.
The story is narrated by a protagonist named A. Square, a native of Flatland — a world whose geometric denizens only live and see in two dimensions. But the square has a transformative experience that renders him “the sole possessor of the truths of Space.” On the eve of a new year, he has a hallucinatory vision of journeying to a faraway place called Lineland, populated by “lustrous points” who see him not as a shape but merely as a scattering of points along a line. Frustrated, he tries to demonstrate his squareness to their king by moving from left to right. The king, ignorant of directions, fails to perceive the motion and clings to his view of the square as points on a line.
But then the square himself is visited by a creature from another world — a sphere from the three-dimensional Spaceland. The very notion of three dimensions is at first utterly unimaginable to our hero — he sees the visitor merely as a circle. And yet when the sphere floats up and down, thus contracting and expanding the radius of the perceived circle based on its distance from our grounded observer, the square begins to suspect that he, like the inhabitants of Lineland, might be congenitally blind to the existence of another dimension.
When he returns to Flatland and tries to awaken his compatriots to the revelatory existence of a third dimension, he is met only with obtuse denial and declared mad. Decrees are passed to make illegal any suggestion of a third dimension and all who make such claims are to be imprisoned or executed.
The square himself is eventually thrown in jail, where he spends seven years and composes Flatland as a cautionary memoir he hopes will inspire posterity to see beyond the limit of two dimensions.