Monday, February 15, 2016

Flatland: A Romance of Many Dimensions

I CALL our world Flatland, not because we call it so, but to make its nature clearer to you, my happy readers, who are privileged to live in Space.
Having recently re-read Edwin A. Abbott's Flatland: A Romance of Many Dimensions (1884), a late Victorian novella which I assign as reading in the geometry course I teach every spring, I am struck with the fact that here is a bona fide fantasy that rarely makes the canons of fantasy fiction. British fantasists like H. Rider Haggard, William Morris, and George MacDonald are always named, though rarely (one suspects) read. Earlier satires like Gulliver's Travels and Utopia receive honorable mention. But Flatland, which is enjoyable as both a fantasy and a satire, is sadly excluded, and its author, a clergyman, unknown to fantasy-lovers.

It is easy to see why self-appointed historiographers like L. Sprague de Camp (Literary Swordsmen and Sorcerers) and Lin Carter (Imaginary Worlds) left it by the wayside, if they were aware of it at all, and why the latter excluded it even from very eclectic collections like Golden Cities Far and Dragons, Elves, and Heroes. A novel that takes place in a two-dimensional universe wouldn't have been in their line, fixated on material elements as they were, despite the large role world-building plays in the work. There really is nothing quite like Flatland.

The narrator, A. Square, begins by helping the reader imagine what life in Flatland is like.
Imagine a vast sheet of paper on which straight 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 the power of rising above or sinking below it, very much like shadows – only hard and with luminous edges – and you will then have a pretty correct notion of my country and countrymen.
A busy thoroughfare in Flatland
Alas, a few years ago, I should have said "my universe": but now my mind has been opened to higher views of things.
With perceptions limited entirely to the plane, the world appears to its denizens as a line, much as our three-dimensional world is perceived by us through a two-dimensional field of vision (as in a television screen, which is flat).
As there is neither sun with us, nor any light of such a kind as to make shadows, we have none of the helps to the sight that you have in Spaceland. If our friend comes closer to us we see his line becomes larger; if he leaves us it becomes smaller: but still he looks like a straight line; be he a Triangle, Square, Pentagon, Hexagon, Circle, what you will – a straight Line he looks and nothing else.
This exposition of Flatland society and history continues through half the book, touching on the stratification of social classes according to number of sides, the operation of schools and prisons, domestic arrangements, relations between the sexes, the rise of Chromatistes and the art of painting, the Universal Colour Bill, the machinations of the Chief Circle Pantocyclus, the violent suppression of the chromatic sedition, and so forth.

Before the Sanitary and Social Board.
The satire of late Victorian society is heavy but not (to me, at least) altogether transparent. For instance, women in Flatland are both despised and feared – despised, for they are regarded as irrational and foolishly sentimental, and feared, for their bodies are extremely sharp line segments, and they are capable of unthinkingly slaughtering their own families if provoked. A. Square belabors the point in several passages, but it seems plain from the forward that this is to be taken ironically. What precisely Abbott was driving at escapes me, unless it was to ridicule Victorian mores by showing a society in which the strait confinement of women really was a cogent necessity, though even this is questioned within the narrative itself.

A well-bred Hexagon yielding to a Lady.
The second half of the book presents a sequence of visions and visitations. In the first, A. Square descends upon Lineland in a dream, coming to revile its King for his narrow-minded inability to conceive of more than one dimension
"Besotted Being! You think yourself the perfection of existence, while you are in reality the most imperfect and imbecile. You profess to see, whereas you can see nothing but a Point! You plume yourself on inferring the existence of a Straight Line; but I can see Straight Lines, and infer the existence of Angles, Triangles, Squares, Pentagons, Hexagons, and even Circles. Why waste more words? Suffice it that I am the completion of your incomplete self. You are a Line, but I am a Line of Lines, called in my country a Square: and even I, infinitely superior though I am to you, am of little account among the great nobles of Flatland, whence I have come to visit you, in the hope of enlightening your ignorance."
Hearing these words the King advanced towards me with a menacing cry as if to pierce me through the diagonal; and in that same moment there arose from myriads of his subjects a multitudinous war-cry, increasing in vehemence till at last methought it rivaled the roar of an army of a hundred thousand Isosceles, and the artillery of a thousand Pentagons. Spell-bound and motionless, I could neither speak nor move to avert the impending destruction; and still the noise grew louder, and the King came closer, when I awoke to find the breakfast-bell recalling me to the realities of Flatland.
Our narrator is then visited in his turn by a Sphere from Spaceland, who appears to him as a circle (or priest) who can change sizes at will, and before whom A. Square is no better off than the denizens of Lineland were before him. When arguments fail, the visitant resorts to deeds:
"The higher I mount, and the further I go from your Plane, the more I can see, though of course I see it on a smaller scale. For example, I am ascending; now I can see your neighbour the Hexagon and his family in their several apartments; now I see the inside of the Theatre, ten doors off, from which the audience is only just departing; and on the other side a Circle in his study, sitting at his books. Now I shall come back to you. And, as a crowning proof, what do you say to my giving you a touch, just the least touch, in your stomach? It will not seriously injure you, and the slight pain you may suffer cannot be compared with the mental benefit you will receive."
Before I could utter a word of remonstrance, I felt a shooting pain in my inside, and a demoniacal laugh seemed to issue from within me. A moment afterwards the sharp agony had ceased, leaving nothing but a dull ache behind, and the Stranger began to reappear, saying, as he gradually increased in size, "There, I have not hurt you much, have I? If you are not convinced now, I don't know what will convince you. What say you?"
Though at first bewildered by his subsequent elevation above Flatland, which permits him to see "through" walls, A. Square is eventually led to posit the existence of more than three spacial dimensions.
[T]ake me to that blessed Region where I in Thought shall see the insides of all solid things. There, before my ravished eye, a Cube, moving in some altogether new direction, but strictly according to Analogy, so as to make every particle of his interior pass through a new kind of Space, with a wake of its own – shall create a still more perfect perfection than himself, with sixteen terminal Extrasolid angles, and Eight solid Cubes for his Perimeter. And once there, shall we stay our upward course? In that blessed region of Four Dimensions, shall we linger on the threshold of the Fifth, and not enter therein? Ah, no! Let us rather resolve that our ambition shall soar with our corporal ascent. Then, yielding to our intellectual onset, the gates of the Sixth Dimension shall fly open; after that a Seventh, and then an Eighth –
The Sphere, overcome with ire at this impertinence, returns A. Square to Flatland. There the narrator inevitably shares the fate of all enthusiastic visionaries out of step with their ruling classes when he tries to spread the "Gospel of Three Dimensions."

The "still more perfect perfection" of the cube referred to above is the regular polytope now known as a hypercube or tesseract. The latter term was coined by Charles Howard Hinton in 1888. Incidentally, this figure (or, rather, its five-dimensional analogue) plays a large role in A Wrinkle in Time by Madeleine L'Engle, in which the dimensional analogy is pursued considerably less competently; that novel also glances upon a two-dimensional world, and would seem to be partly inspired by Flatland. However, in treating time as a fourth "spacial" dimension, it adopts the erroneous conception of space-time expounded upon by H. G. Wells in The Time Machine (1895).

Two recalcitrant revolutionaries executed after trial.
Abbott's lucid approach to the fourth dimension by way of analogy is almost astonishing, considering that it comes from a clergyman (presumably) untutored in such matters, and well before the theories of Ludwig Schläfli were well known. Schläfli, a Swiss mathematician, originated the idea of regular polytopes in the 1850s, but his work did not receive recognition until much later. In Regular Polytopes (1947), H. S. M. Coxeter notes that regular polytopes were independently rediscovered by nine different mathematicians between the years 1881 and 1900, even as Flatland was being written. The time, evidently, was ripe.

Though relatively obscure at its publication, Flatland went on to receive widespread acclaim among mathematicians and physicists in the twentieth century. In our own time, theoretical physics has for a number of years been feeling its way toward the possibility that more than three spacial dimensions play a role in the structure of the universe. My doctoral work focused on higher-dimensional geometry and applications to particle physics, so this is something I know a bit about.

But the book as a whole is quite enjoyable purely as a work of speculative fiction. At every point it compels the reader to ponder what life would be like in a two-dimensional world.
There being no sun nor other heavenly bodies, it is impossible for us to determine the North in the usual way; but we have a method of our own. By a Law of Nature with us, there is a constant attraction to the South; and, although in temperate climates this is very slight – so that even a Woman in reasonable health can journey several furlongs northward without much difficulty – yet the hampering effect of the southward attraction is quite sufficient to serve as a compass in most parts of our earth. Moreover, the rain (which falls at stated intervals) coming always from the North, is an additional assistance; and in the towns we have the guidance of the houses, which of course have their side-walls running for the most part North and South, so that the roofs may keep off the rain from the North. In the country, where there are no houses, the trunks of the trees serve as some sort of guide.
Stephen Hawking suggests in The Universe in a Nutshell that such a creature would be unable to digest food, since a "tube" through the body would separate the unfortunate polygon into two halves. But, perhaps, like flatworms, the Flatlanders expel waste material through the mouth, which, Abbott tells us, also serves as the eye, indicating a physiology markedly different from our three-dimensional preconceptions.

Two small country houses with an antiquated square outbuilding.
Other questions arise. Writing is mentioned, for instance. Flatland writing must needs be one-dimensional, however; of what does this writing consist? Something like printed Morse code, perhaps? What are books like? Does Flatland geometry predominantly consist of the study of magnitudes on a line, much as ours consists of shapes in a plane? What would they think of the Cantor set, I wonder?

Whatever could the hills and mines mentioned by the narrator be like? And what of trees, which are referred to as growing from south to north? And so on.

Alas, the book is much too short to begin answering such questions. Not that most other readers would be as interested in them as I am. I happen to know the ins and outs of classical Euclidean geometry and its modern extensions and generalizations pretty well, and I can imagine any number of brave new worlds for A. Square to explore. Others have tried their hands at sequels before now; perhaps I shall join their number some day.

Lately, though, I've been piecing together digital collages of Flatland life in spare moments here and there, ostensibly to put an illustrated version of Flatland on my faculty website for my students to peruse. (These are the color pictures on this post; the drawings are Abbott's.) I have to say, I rather like the results, not that that means much.

When I took art as a teenager, I quickly found myself at the front of the class; however, every year, my teacher would begin by tasking us with forming a composition out of geometrical shapes, and I inevitably received F's on these assignments without ever knowing why. It was quite maddening.

In recent years, I've spent a good bit of time pondering the relation between artistic and mathematical abstraction, as a perusal of my artsy posts will show. (See here and here, for instance.) These collages are, like my fractals, a step toward abstraction in art from the far side. In making them I'm reminded of my old composition assignments. What I'm trying to say is, I hope I wouldn't still get F's on them.

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