Anthology of Ideas

“Flatland” by Edwin Abbot was written near the end of the romantic period and the beginning of the realism period, it combines elements from both and adds a healthy dose of satire, social injustice and science to create a masterpiece of short fiction that has been beloved by each succeeding generation. Although the book is not very long (less than 100 pages) and written in 1884 Abbot introduced some geometric and mathematical ideas that were far ahead of their time and also are still (for the most part) valid today. Abbot also takes a huge swipe at the the Victorian England class system and its thoughts of women in society.

Abbot describes a word of two dimensions through the eyes of “a square”, as the book progresses the square describes the world he lives in, a world of triangles, squares, pentagons and many other polygons. The square is ignorant of the possibility of a higher dimension than his two dimensional world, yet he dreams of going to a three dimensional world populated only by straight lines. The square one evening is contacted by a sphere who wishes to teach the square of the third spacial dimension. However although the square realizes the concept of lesser dimensions he is unable to grasp the idea of a dimension higher than his own. It is not until the sphere lifts him out of his two dimensional world that he is finally able to realize what the sphere attempted to explain. The square is awakened to the idea of multiple dimensions and after being returned to his home attempts to explain his experiences to others, but is unable to communicate the idea of “up”.

One of the major themes in the squares realization of higher dimensions is that if there is a third spacial dimension then there must be a fourth. The idea of higher dimensions is now just beginning to be proven through some of the “cutting edge” physics theories. The square realizes that since points can become a line and many lines a square, and many squares a cube that it must be possible for many cubes to become something more in a fourth dimension. Although the first real research papers on the fourth dimension was written in 1919 (and subsequently submitted to and rejected by Einstein), the idea of the fourth dimension was already circulating socially around England’s upper class in Abbot’s day. Abbot’s book was a an attempt to change the way the upper scientific establishment of the day thought of geometry and the ancient, strict and constricted form it was stuck in, just as it was a way to change the way class and women’s rights were thought of.

In the world of Abbots square there were five main classes (in no particular order). The first class was the women who were straight lines with little width and “devoid of brain-power, and have neither reflection, judgment nor forethought, and hardly any memory.” The second was isosceles triangles who had only two equal angles, and thus were considered less intelligent then other polygons with equal angles. These triangles could through subsequent generations become equilateral triangles The third was the equilateral triangles who had a chance of having square children. The isosceles and equilateral triangles were the serfs to the other polygons and were unable to advance beyond this status. The fourth class were the other polygons ranging from squares to hexagons and beyond, depending on the number of sides the polygon would have a different profession, for example all squares were lawyers. The fifth class was the priestly class who ruled over all others, these polygons had so many sides that they were nearly circles. Imperfection of any kind was abhorred in the two dimensional world, any shape having non-equal angles was destroyed.

Socially Abbot refers to two main issues, the first is the class system that was hard if not impossible to move out of, and the second was the Victorian view of a woman as a monastic, childcare provider, and housekeeper but nothing more. The class system in flatland was set up that it would be impossible for an polygon born to be able to become anything but what he was born too and that all his hopes and dreams to be pinned on the chance that his son would be better than him. Just as in Victorian England there was no room for people to change their social situation without miraculous means ala Oliver Twist. Woman in flatland lived under a special set of rules that if broken could mean execution. This is analogous with Victorian england double standard when it came to owning property and divorcing. The social ills of Victorian England were of stark contrast with the scientific advances of the same period. The Victorian age was one of change that was resisted by many but through his satire Abbot showed just how this idea was ridiculous at best.

It is rare indeed that a book so well accomplishes two very different purposes, not only does the book argue for class and women’s rights but also splendidly attempted to modernize mathematics. Through satire the book achieves both of its goals and leaves the reader not only with a greater sense of geometry and the realization that all scientific advances require a “leap of faith” but also can open the readers eyes to the ridiculousness of class systems based on preconceived notions and outward appearance.