The story of “Flatland, A Romance of Many Dimensions”, was first published 100 years ago and is considered by some as “mathematical fiction”.
The author presents us first with a 2-dimensional world made up of regular Polygons which have at least three straight sides and angles but can have five or more. This world exists as if it were a flat piece of paper with lines and shapes on it. The residents of this world are like the lines on the flat paper. Shape determines rank and if deformities in shape are found it can be a criminal issue.
This society has very strict standards and if a deviation in shape are beyond the established standard then the solution is euthanasia of the shape. If it isn’t beyond the standard, but is still a deviation, then the shape may be allowed to grow and develop to see if it can be cured. If it can’t then the shape will be “painlessly and mercifully consumed”. A shape that is off-standard, but leaning in a different direction than where the more fatal standard exists, will be assigned to live their lives as civil servants or in lower level jobs. Class structure is very rigid.
The “Square” shape is a favored member of a special group of professionals and gentlemen in this two-dimensional world. The books narrator’s name is A-Square’s and he comes from this group.
Men are polygons with various numbers of sides, but women are simple line-segments. Women are required to make a noise if they move so they will not bump into the shapes. Classes are distinguished by the "Art of Hearing", the "Art of Feeling", and the "Art of Sight Recognition" or even just the sound of one’s voice.
The first seven chapters of the book considers existing in a two-dimensional universe, but then A- Square dreams about a visit to a one-dimensional world called “Lineland”. The inhabitants of this world can’t see A-Square as anything but a group of points on a line and they themselves exist only as just points.
After this dream A-Square is visited by a three-dimensional sphere named “A-Sphere” who has come from a 3-dimensional world. A-Sphere appears to A-Square but he still only looks like a circle to him. A- Sphere levitates both of them, up and down, and shows him what 3 dimensions look like. The two of them discuss the possible existence of a fourth, fifth and sixth dimension.
A-Square returns to his dimension, anxious to tell other about the other dimensions, but makes no progress in convincing the others of what might be possible, and he winds up in prison for what he is saying. Seven years after being imprisoned, A-Square writes out the book Flatland for a future generation so that they can see beyond their two-dimensional existence.
“Attend to your Configuration.”
― Edwin A. Abbott