The fourth dimension as seen from Flatland

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Flatland is a beautiful book about imaginary people, an imaginary place, and world that is ... well, flat. Beginning with their homes, everything in Flatland is two-dimensional. However, their homes are like our homes: they have a big entrance hall, bedrooms; a big doorway entrance for men, and a tiny door for women. Windows were not needed.  All homes share a common architecture: all 'buildings' are five-sided houses.

This book is a milestone in the fiction literature and also in the scientific field. From the standpoint of fiction writing, the book is about "civilizations" that live in universes of two and one dimensions. Quite strange this approach because we tend to think of two-dimensional "beings" as mere shadows without any autonomy at all. However, shadows are just projections of solid masses, if the masses or bodies stand still the shadows must also stand still. If bodies move, their shadows must move accordingly.  Shadows do not have proper motion: they cannot move by themselves.

From the standpoint of scientific writing the impact of the story is not so presumptuous because the mathematical grounds of space dimensions was already laid by other authors. However, the story of people living on a plane, and the story of people living on a line is a unique form of presenting and simplifying an abstract topic of the domain of mathematics with lively characters with whom people can identify. The very fact that the main character was prosecuted and imprisoned for telling his people that he saw a three-dimensional being, make us remind that sometimes, even today, people can be prosecuted and punished for just thinking "outside the box", which in terms of the Flatlanders is the equivalent to say that one spoke with a being from the fourth dimension.

The book was written in 1884 by Edwin Abbott Abbot with the dual purpose of satirizing his contemporary society and at the same time give the reader a non mathematical view of the concept of dimensions and spaces with non conventional dimensions.

The importance of Flatland relies on the fact that it is the first book to deal with people from other dimensions. Before Flatland, other authors dealt with people or inhabitants in the Moon or in other planets, but none of them confronted the reader the possibility of life in other dimensions —not merely in other worlds.

Flatland is also the name of a place —the place where Flatlanders live. Flatland is a bi-dimensional world, a flat world like a tabletop. People from this land are somewhat like shadows on a plane, but not quite so, because shadows do have width and breadth but not height.

Right from the Preface, Abbott explains to his readers how can the inhabitants of Flatland can have some "height" and still be planar.

It is true that we have really in Flatland a Third Dimension called 'height' just as it is true that you have really in Spaceland a Fourth unrecognized Dimension called by no name at present, but which I will call 'extra height'.

No 'delicate micrometer' -as has been suggested by one hasty Spaceland critic -would in the least avail us: for we should not know what to measure, nor in what direction. When we see a Line, we see something that is long and bright; brightness, as well length, is necessary to the existence of a Line; if the brightness vanishes, the Line is extinguished.

Flatland is a plane universe so thin that no micrometer can measure it, but Flatlanders are not mere shadows that move on a plane. They are living people like you and me, they build houses, they have social rules for men and women, they have children, a highly stratified social class system, they have climate, courts, etc.

At the same time, those are some of the reasons why Flatland —the book— is also claimed as a critique of Abbot's contemporary society. A society with some rules for women and different rules for men.

Thus, the book is about people that live in a two-dimensional world but behave similarly to Abbot's contemporary society.

But the book is acclaimed for its lessons in the dimensions of space rather than the social critique that it conveys. For Abbot, it is perfectly possible to live and evolve in a plane world just like we do in our tridimensional space.

If Flatlanders are not like shadows on a plane, then, physically, how are the Flatlanders? We let Abbot describe this land via his main character whose name is A. Square (surely himself because Abbott Abbot is Abbott-squared).

Imagine a vast sheet of paper on which Lines, Triangles, Squares, Pentagons, Hexagons, and other figures, instead of remaining fixed in their places, move freely about, on or in the surface, but without power of rising above or sinking below it, very much the shadows –only hard and with luminous edges—and you will then have a pretty correct notion of my country and countrymen. Alas, a few years ago, I should have said “my universe”, but now my mind has been opened to higher views of things.

He uses the analogy of a coin placed on the middle of a tabletop; the coin looked from above will appear as a circle, but seen from an edge will appear as a line.

By what means do Flatlanders recognize among themselves? If women are lines “like needles” and men are triangles, soldiers are sharply pointed isosceles triangles, professionals and gentlemen are squares, etc, then how do they differentiate one from the others? Since all of them are simple lines when seen from an edge, then how do they distinguish one triangle from the others? How do they know when is soldier —and not a woman speaking?

A. Square tells us how they do it, but before going into explanations he remembers us how we unconsciously use the “gifts” of having two eyes, perceiving perspective, and see the colors around us.

You, who are blessed with shade as well as light, you who are gifted with two eyes, endowed with a knowledge of perspective, and charmed with the enjoyment of various colors, you, who can actually see an angle, and contemplate the complete circumference of a Circle in the happy region of the Three Dimensions—how shall I make clear to you the extreme difficulty which we in Flatland experience in recognizing one another’s configuration?

A. Square gives the following explanation:

The answer is threefold. The first means of recognition is the sense of hearing, which with us is far more highly developed than with you, and which enables us not only to distinguish by the voice our personal friends, but even to discriminate between different classes ...

This is understandable because this behavior is common among animals. We see it in bats, dogs, cats, etc.

The second means of distinguishing among themselves is by the sense of feeling. He describes it as:

Feeling is, among our Women and lower classes -about our upper classes I shall speak presently-- the principal test of recognition, at all events between strangers, and when the question is, not as to the individual, but as to the class. What therefore "introduction" is among the higher classes in Spaceland, that the process of "feeling" is with us. "Permit me to ask you to feel and be felt by my friend Mr. So-and-so" -is still, among the more old-fashioned of our country gentlemen in districts remote from towns, the customary formula for a Flatland introduction.

So far the Flatlanders arguments are acceptable, but how do they perceive angles, if the they cannot see the figures from above the planes because they are living in a flat-land?

I answer that though we cannot see angles, we can infer them, and this with great precision. Our sense of touch, stimulated by necessity, and developed by long training, enables us to distinguish angles far more accurately than your sense of sight, when unaided by a rule or measure of angles....

From there on, A. Square gives long explanations about how they sense and even measure angles.

However, each one of the highly developed senses that A. Square says they have are mere survival mechanisms, but none of them make of Flatlanders three dimensional people. That is, none of those mechanisms constitute an "extra dimension", therefore Flatlanders are good people, but pitifully, still flat 2-dimensional.

The next dimension for Flatlanders is just the "third" dimension of us. The fourth dimension for us is like the third dimension for them.

Near the end of the romance, A. Square asks The Sphere if he can show him the fourth dimension. This is analogous to a being of Lineland (1-dimension) asking A. Square (2-dimensions) to take him to see how the Spheres (3-dimensions) are. This is what A. Square asks:

But my Lord has shewn me the intestines of all my countrymen in the Land of Two Dimensions by taking me with him into the Land of Three. What therefore more easy than now to take his servant on a second journey into the blessed region of the Fourth Dimension, where I shall look down with him once more upon this land of Three Dimensions, and see the inside of every three-dimensioned house, the secrets of the solid earth, the treasures of the mines of Spaceland, and the intestines of every solid living creature, even the noble and adorable Spheres.

Sadly to say, but The Sphere is astonished to this demand, because it is easy for a being from any dimensional space to visualize lower dimensions, but not higher dimensions. The answer is simply the one we 3-dimensional beings would give to anybody that asks the same question:

But where is this land of Four Dimensions?

A. Square answers:

I know not: but doubtless my Teacher knows.

Poor Sphere, now he is mentally blocked, so his answer is plain and simple:

Not I. There is no such land. The very idea of it is utterly inconceivable.

In simpler words, it is very easy to accept lower dimensions, as it is very easy to deny the higher ones. We live comfortably in strong 3D houses; they also live comfortable in their planar homes. Everything is so similar: we cannot imagine the fourth dimension, and they cannot imagine the world we live in.

E. Pérez
10/10

Also in this Website:

Resources for the study of Flatland
About how Alfred Schofield found the fourth dimension in The Bible