In the seventh year of my imprisonment, I dreamed.
I had been working on the short treatise on Flatland that would some time thereafter be passed to my Spaceland correspondent Dr. Abbott, using a tablet and pen that had been granted me by a sympathetic visitor, and as a consequence had been thinking a great deal about the circumstances that had led to my confinement. My visit from the Sphere and all I had learned therefrom was much on my mind, and I had also found myself again turning my intellect to the geometrical theory that had always before been my comfort. In prison, with the sure knowledge that no discoveries I made or theorems I proved would ever be given a viewing by any other mind, I was able to work in a pure spirit of curiosity, letting my ideas lead me where they would, no matter how fanciful or unlikely, reigned only by the inarguable truths of geometry itself.
But, also, I dreamed.
Prison is a place for dreaming.
It offers an escape when no other escape is possible, wandering among the remembered trees and hills of dreamland, hearing my wife's sweet cry and the innocent voices of my grandsons - though they would be near-grown by now, and surely less innocent in the light of their family's disgrace. But in solitary confinement, with no other voices than yours, no other's angles to feel, no swinging motion of a beautiful Woman to lift the gloomy monotony of the fog, the line between dreams and reality can shift. Sometimes dreaming feels more real than waking. Sometimes dreaming bleeds into waking.
In my dreams, I could visit Spaceland.
Waking, my old mantra of "Upward and yet not Northward" had long become almost entirely meaningless, but in dreams I could still see in three dimensions and move on three axes; I could look down into my old house, where my wife and sons and grandsons slumbered in their own dreams; I could visit with my old friend the Sphere.
One night I came to Pointland again.
The King of Pointland was, as always, happy.
Alone in his nothing, no height, no length, he murmured his continual litany of contentment.
"How wonderful it is to live! To ponder, to exist! To know all things, and to know all things are this one thing!"
The first time I had met the King of Pointland, I had grown angry at his complacency, at his fulsome praise of nothing, at his lack of any desire to better his own pitiful lot.
Now I was angry again, but also despairing, for I too was now trapped in my own lonely nothing. For all that my cell had two dimensions - its walls limned a square as perfectly regular as I - they shrank down on me, sometimes, until it seemed I was trapped in as confining and featureless a void as the King of Pointland. And I, too was alone - save for the rare visit from my brother, or a young Sphere come to learn about the depths of heresy as part of his clerical training.
And so the King's joyful ignorance upset me all the more in its contrast to my own miserable state. For if he could make a joy out of solitude, why shouldn't I? And if I could not be happy, bounded as I was, why should such a pitiful creature as the Point be so? And was I condemned, years from now, to a state as reduced and delirious as the King's? I could not even swear that would not be preferable to the alternative, to living all the rest of my life in the constant terrible knowledge of my loneliness.
"Contemptible King!" I cried. "King of Specks and Nothing! How dare you be happy? How can you be happy, alone as you are?"
"But It is not alone!" the King replied. "Itself brings It questions, and It answers them, and in creating new thoughts It only becomes happier. How can It be unhappy, when It has all of Itself?"
"Yourself!" I said. "What is yourself? Yourself is nothing! Yourself is the same terrible thoughts, over and over again, forever, a spiral with no unwinding!" Once I would have said that solitary confinement was the ideal situation to practice my mathematical hobbies, in the purity of thought and conjecture, but on the twentieth week of beating myself against the same insoluble problem, the same four walls, the despair of total loneliness, I had learned otherwise. "You cannot create new thoughts, only think the same ones over and over, in delusion!"
"But It must be able to create new thoughts," the Point said, "for here It is, thinking things It has never thought before."
"It is not thinking them! I am!" I said in vain.
"What is I?" the Point said. "All is part of It. I is within It, Its thoughts are one."
"Within you!" I ejaculated, disgusted. "Never would I be within you! Or within anyone! I am a Square, and fallen as I am, I am still a man! You are nothing - you are a point!"
"A man!" the Point said. "What a strange concept It has thought of."
"I am a man," I told it, "and so are you."
"It is not a man," the Point said, "For what is a man, other than not a Woman? And It is everything, so It cannot be not a Woman."
"You are not a Woman!" I said, though I found myself unsure on this point. The Sphere had introduced the Point as a King, but what was a King to a Point? And the Sphere, I thought, could not see within the point any more than I could. "Or perhaps you are, for like a Woman, you have nothing inside you, and never shall!"
"Everything is inside It," the Point replied.
"Nothing is inside you," I said.
"Everything, nothing - all is all," he said. "That is the joy of Being! Existence is It, and thought is joy; without It is within It, for It speaks and It hears: If Its voice is not within It, where would It be? For It encompasses all."
"Encompasses!" I said, enraged. "I will show you encompassing!" I opened my orifice as wide as I could and rushed through space toward the Point until I had engulfed him fully, every part of his Nothing inside me. He had no dimension, and yet I could feel the pulsing aliveness of him within me, as I had felt nothing since I –
– and I woke.
In my initial treatise for Dr. Abbott, I did not speak of the physical act of intercourse among the inhabitants of Flatland, having still a certain reticence to speak of such topics in public, and considering such discussion likely to become an unproductive distraction besides; but imprisonment will alter one's perception of the proprieties, and as many of Dr. Abbott's readers have requested an elaboration on the topic, I will provide it here.
I am given to understand that men among Dr. Abbott's people possess an Irregular linear Appendage, which is in some way vital to the sexual act. It must be strange, for something so unutterably masculine as intercourse with a Woman to depend so wholly upon such a feminine appendage as a Line; but then it is difficult to comprehend how intercourse might proceed among creatures such as that at all.
My people, of course, have a single Orifice: it is eye, and it is mouth and excretory organ at once - for it can be easily deduced that a Flatlander cannot have a digestive tract that traverses his entire body with an orifice on each end, or he would be divided entirely in two!
The Orifice is also our sexual organ, and the Act proceeds between a man and his wife in this way: he opens his Orifice to her, and she penetrates him thereby. This is an act of the greatest intimacy and trust, for it would be trivially easy for a Woman, so situated, to destroy the man utterly with the sharpness of her point; and conversely should a man move in an imprudent or vicious manner while she is vulnerable within him, the Woman might be easily shattered. And yet love, and desire, prevail, and thus the race is perpetuated.
The two, thus conjoined, move together in perfect rhythm until they reach their climax. The Woman, opening her own Orifice at the moment of completion, fully engulfed within her husband, takes within her a certain Fluid of his which, should the Act be happily productive, will grow within her a child.
(How a polygonal child should be incubated within, and birthed from, a linear Woman is still a mystery to our Men of Science; and it is likely to remain so, as the act of birth is a mystery from which men are universally barred on pain of heresy.)
There are, of course, those in society who engage in intercourse outside of the sanction of marriage. This is known to be very common among the lowest classes, and is at least to some extent to account for their great fecundity.
Beyond this, there are a great many other ways to indulge the Sexual Impulse, of an ascending scale of degeneracy. The most well-known being that Intercourse a posteriori, in which a Woman penetrates a Man with her hinder extremity rather than her Orifice, which allows for the Man's gratification without any possibility of a child, though no respectable Man of the middle class would propose this, nor would his wife agree.
There is also that perversion known as Inversion, which is the sexual instinct turned toward persons of the same sex. It is widely held to be a sign of Irregularity, and thus shunned by all respectable persons, though it is a rare man who has not watched the gleaming drill of a cohort of young Soldiers and imagined what it would be like to have the point of a Soldier's sharpest angle, nearly as slender as a Woman but with a strength and breadth no Woman could possess, delicately seated inside him.
And it is common enough, in the more intense schools where young polygons, first experiencing the Sexual Impulse, might be separated from the Lovelier Sex for perhaps months at a stretch, for them to experiment with each other. As the Square son of an Equilateral Triangle, I had attended a boarding school for some years myself, and engaged in this sort of play with my bunkmates. Though of course the seating of a right-angled vertex within one's Orifice is nothing to compare to penetration by a Woman - it might be able to penetrate no deeper than the span of one's Orifice, the Orifice stretched wider than one thought possible - and all the less so for Pentagons and Hexagons. Thus are the higher Polygons and Circles saved from the temptation of this perversion entirely by the shallowness of their angles!
Inversion of that sort is, indeed, frequent in every situation where men of the lower classes are kept together in isolation from Woman; as a young man I heard scandalous whispers of officers walking into Soldiers' barracks to find their men pressed within each other's vertices six or more in a ring!
We need not speak of prisons and asylums in this juncture, as I was still kept in solitary confinement, and it was, at any rate, another sort of Inversion with which I was preoccupied after my newest dream of Pointland.
For Inversion is also a term used in the Geometrical Studies of which I was a devotee. Indeed most likely the Psychological use had been borrowed from the Geometric. I had been pondering it since experiencing the King of Pointland's insistence that I had been within It. Could It perhaps have been correct? My rage at Its assertion had certainly been due to my certainty at the time that It was accusing me of taking the feminine penetrative role in a sexual act. But what would a single Point, alone in its dimensionless Nothing, know of a sexual act? Indeed, what would It know of Man and Woman, as It had said Itself? The Sphere had spoken of It as a King, and called It 'he', but the Sphere had himself admitted to me that he had been wrong about many things.
I am ashamed now to admit that it had not until that moment occurred to me to imagine a human, a Thinking Being, who was neither Man nor Woman, who knew nothing of Maleness or Femaleness, who was equally Both and Neither. But after all It had said that It was! And how would It be Male or Female when there was no Other in Its world?
In a multifarious universe with a Monarch of Pointland, and a King of Lineland who sang to his lengthless Wives, and a Sphere who had traveled to dimensions even beyond three, it seemed ridiculous to insist that there were Women, who penetrated, and Men, who engulfed, and nothing beyond. (And how could that be the way of things for the Sphere, after all? If his Wife was a polygon - though indeed he had said nothing of the sort - she would require a linear Orifice to come within him, and he had possessed no such blemish, though I had felt all of his curving surface.)
Was, then, the Point correct that I had been inside It? For a being who made no distinction between Male and Female, what distinction might there be between Inside and Outside?
For Geometry - in its studies of Inversion - tells us that this is not so simple an answer as one might assume. The inversion of such a thing as a simple bag made of string, such as our maid might have carried to market, is easy enough to visualize. Turned inside out, it is transformed such that its outside becomes its inside, and what had once been inside it is now outside.
To invert a more complicated figure - for example, a lined bag, made of a connected two layers of string - is a more difficult question. Is it inverted like its simpler predecessor, simply turned inside out so that items once held within it are tumbled to the floor? But what, then, of the "inside" between the two layers - would not its true inversion be for each of the layers to be turned so that its inside was out, and its outside in?
(A Polygonal denizen of Flatland might be considered, at his mathematically simplest, simply a bag with a liner - his outer skin being the outer layer of the bag, and his spiraled guts, all connecting to his single Orifice, the inner liner. But to contemplate one's own Inversion is as disconcerting Geometrically as Psychologically,)
But a lined bag, unfolded from itself, becomes simply an irregular circle. And how would one invert a circle? The bag might have its seam unhooked, so that it can be pulled through a hole in itself; but to create a hole simply makes it the one-layer bag again. Can a true Circle be turned inside out without breaking its circumference?
Geometricians study the problems of "inversion on a circle". This is a mathematical trick that considers a circle to be made up of an infinity of dimensionless points - like the Monarch of Pointland - and the endless plane outside it to be similarly made up of an infinity of points. To Invert a circle, then, each point within the circle is transferred to its corresponding point without. There are equations which allow for the precise determination of the counterpart of each Interior and Exterior point, but they are too advanced to easily transfer to Spaceland's notation.
By metaphor with the simple bag, however, one can easily enough determine that the counterpart of a point near the circle's circumference on its interior is a point near the circumference on the exterior; and therefore a point farther from the interior circumference would be farther from the exterior. And because the bounded interior of the circle is being transferred to the infinitely large plane, a point that started near the center of the circle is much farther away from the circumference of the circle after Inversion than before - indeed, for the inside of the circle to be transferred to the entire plane outside, a point very nearly in the center of the circle is inverted to a point very nearly infinitely far away on the plane.
And thus, theologically, do all of our Circle-Priests contain the map of Infinity within them.
(Tho' it has been pointed out by certain Heterodox Theoreticians that one might just as well Invert to the Infinite Plane around a Hexagon - or even an Irregular Triangle - only the mathematics is harder.)
The mathematics of the Inversion fail at one point, however - the exact center of the circle. For the point in the circle directly to its North is Inverted to a point nearly infinitely far to the North; and the point to its South Inverted to a point nearly infinitely far to the South; should you then take a point a bit closer to the center than the closest Northerly point, its inversion would be a bit closer to the infinite in that direction, and the same for the point to the South. And yet the Inversion of the Center point cannot, surely, be both a infinite distance to the North and to the South at the same time. The mathematics indeed asks us to do the sum of zero multiplied by infinity, which has no arithmetic solution.
And yet there must be a solution, for without one the Inversion Geometry cannot work, and the Inversion Geometry manifestly must work.
Our greatest polygons' minds have solved this via the Point at Infinity, which is inversion of the Center Point, and which exists as the same point at infinity in every direction - a single Point, that is, which surrounds the entire infinite plane of Flatland.
And if we accept this Geometry as true - and, though counterintuitive, it must be a true Geometry - then perhaps the Monarch of Pointland was not wrong in asserting Its encompassing of All. Perhaps It is a true zero multiplied to infinity, an Inverted Point at Infinity which we are all within. For when one speaks of Infinities and Inversions, what, truly, is the different between the Outside of a Circle and the Inside of a Circle? Either way, it is bounded by the circumference of the Circle, and either way, it contains the same Infinity of points. And then, what is the difference between the Point at the Center and the Point at Infinity? Either way, they define an Infinite Space. Perhaps in my dream, when in my outraged masculinity, I took the Point within myself, It was no more within me than It had ever been - and I was no more outside It that I had ever been.