I would call Flatland: A Romance of Many Dimensions a mathematical satire fiction (satirical novella). It was first published in 1884, written by the English schoolmaster Edwin A. Abbott.

Flatland is a fictional world that is two-dimensional and all citizens are geometric figures. Men are polygons with different numbers of sides in Flatland, and women mere line-segments.

I love this book because it humanized very simple yet elegant concepts in geometry and linear algebra, at the same time projecting back to the human world how rigid the artificially constructed rules and custom are.

## Social Hierarchy

To understand Flatland, the first thing to know is that not all figures are the same.

Our Soldiers and Lowest Class of Workmen are Triangles with two equal sides, each about eleven inches long, and a base or third side so short (often not exceeding half an inch) that they form at their vertices a very sharp and formidable angle.

Our Middle Class consists of Equilateral or Equal-Sided Triangles.

Our Professional Men and Gentlemen are Squares and Five-Sided Figures or Pentagons.

Next above these come the Nobility, of whom there are several degrees, beginning at Six-Sided Figures, or Hexagons, and from thence rising in the number of their sides of Polygonal, or many-Sided.

Finally when the number of the sides becomes so numerous, and the sides themselves so small, that the figure cannot be distinguished from a circle, he is included in the Circular or Priestly order; and this is the highest class of all.

The secret to keeping this hierarchy stable is limited mobility:

It is a Law of Nature with us that a male child shall have one more side than his father, so that each generation shall rise (as a rule) one step in the scale of development and nobility.

The catch is that this rule does not apply to the lowest class in the hierarchy, i.e., the Soldiers and Lowest Class of Workman.

## Irregularity

In Flatland, all figures must “have their sides equal”. This is because “the whole of the social life in Flatland rests upon” this fundamental fact. Because there is hierarchy, lower class citizens need to pay respect to higher class ones. But the only way to tell different figures apart in Flatland is to “feel” the angle of other figures. If figures have unequal sides, then it would be impossible to tell different figures apart in the two-dimensional world.

Hence, the “irregular”, “is from his birth scouted by his own parents, derided by his brothers and sisters, neglected by the domestics, scorned and suspected by society, and excluded from all posts of responsibility, trust and useful activity”.

Isn’t this funny to look at in parallel with the 1997 Think Different Apple Commercial? “The round pegs in the square holes. The ones who see things differently. They’re not fond of rules. And they have no respect for the status quo.”

## Women

They are lines because women are shallow and they have no depth. Bah.

## Color

One obvious solution that came to mind when reading about the rules of the Flatland is color coding different shapes. Chapter 9 of the book talks about “the Universal Colour Bill” – why color is banned in flatland. Because colors can be used in such a way to make figures of different classes appear the same in two-dimensional world. This is obviously a danger to social hierarchy.

## Limitation of Dimensions of Existence

The story of Flatland is told from the point of view of a Square, who was chosen from a Sphere from an outer-universe, called Spaceland, to enlighten the “truth” about the existence of a higher dimension world.

The irony lies in that the inevitable incomprehensibility of a higher dimension.

Before meeting the Sphere from Spaceland, the Square had a dream to the lower dimensional world, Lineland, where all citizens are lines. The Square tried to reason with the king of the Lineland and try to explain to him in vain the concept of “left” and “right”. The king of the Lineland could not possibly believe how the Square could move in and out of the Line.

In the same fashion, the Square himself could not possibility comprehend the concept of “Upward, and yet not Northward” that was explained to him by the Sphere. The Square was also appalled to see the Sphere entering and leaving the Flatland.

Finally, and most interestingly, when the Square was taken to Spaceland and finally believed in a higher dimension, he reasoned with Sphere:

In One Dimension, did not a moving Point produce a Line with two terminal points?

In Two Dimensions, did not a moving Line produce a Square with four terminal points?

In Three Dimensions, did not a moving Square produce - did not this eye of mine behold it - that blessed Being, a Cube, with eight terminal points?

And in Four Dimensions shall not a moving Cube - alas, for Analogy, and alas for the Progress of Truth, if it be not so - shall not, I say, the motion of a divine Cube result in a still more divine Organization with sixteen terminal points?

The Sphere rejected this absurd idea, and “in his voice of thunder, reiterate his command of silence, and threaten me with the direst penalties if I persisted.”

In linear algebra class in college I had a hard time trying to imagine what a four-dimensional space would look like. Then I learned about vector spaces and subspace, basis, orthogonal bases, projection, etc. Abstractly I could understand four-dimensional space, but I cannot see it. I cannot see what the world look like in four dimensions. Maybe time is the fourth dimension that we actually can go forward and back to, although it appears so impossible, just as the idea of going left appears to be impossible in one-dimensional space?

In a way, it’s deeply sad to see that we can not overcome our own limitations of existence.

## Language

The language of the book is very Shakespeare like and kind of fun to read. It has rhythm in it but a bit hard to read for the first few pages.

## The End

The Square has gone mad because of his vision of a higher-dimension world. The final irony is that the Square’s grandson was, at one point, wondering about higher-dimensional space but the Square and the rest of the Flatland scolded him for being delusional. After Square returned from the Spaceland, he thought his grandson would have been the perfect person to preach this vision to.

But the kid has already given up the belief and turned into believing what everyone else believes in, or claims to believe in.