ALTHOUGH I had less than a minute for reflection, I felt, by a kind of instinct, that I must conceal my experiences from my Wife. Not that I apprehended, at the moment, any danger from her divulging my secret, but I knew that to any Woman in Flatland the narrative of my adventures must needs be unintelligible. So I endeavoured to reassure her by some story, invented for the occasion, that I had accidentally fallen through the trapdoor of the cellar, and had there lain stunned.
The Southward attraction in our country is so slight that even to a Woman my tale necessarily appeared extraordinary and well-nigh incredible; but my Wife, whose good sense far exceeds that of the average of her Sex, and who perceived that I was unusually excited, did not argue with me on the subject, but insisted that I was ill and required repose. I was glad of an excuse for retiring to my chamber to think quietly over what had happened. When I was at last by myself, a drowsy sensation fell on me; but before my eyes closed I endeavoured to reproduce the Third Dimension, and especially the process by which a Cube is constructed through the motion of a Square. It was not so clear as I could have wished; but I remembered that it must be “Upward, and yet not Northward,” and I determined steadfastly to retain these words as the clue which, if firmly grasped, could not fail to guide me to the solution. So mechanically repeating, like a charm, the words, “Upward, yet not Northward,” I fell into a sound refreshing sleep.
During my slumber I had a dream.1 I thought I was once more by the side of the Sphere, whose lustrous hue betokened that he had exchanged his wrath against me for perfect placability. We were moving together towards a bright but infinitesimally small Point, to which my Master directed my attention. As we approached, methought there issued from it a slight humming noise as from one of your Spaceland blue-bottles, only less resonant by far, so slight indeed that even in the perfect stillness of the Vacuum through which we soared, the sound reached not our ears till we checked our flight at a distance from it of something under twenty human diagonals.
“Look yonder,” said my Guide, “in Flatland thou hast lived; of Lineland thou hast received a vision; thou hast soared with me to the heights of Spaceland; now, in order to complete the range of thy experience, I conduct thee downward to the lowest depth of existence, even to the realm of Pointland, the Abyss of No dimensions.2
“Behold yon miserable creature. That Point is a Being like ourselves, but confined to the non-dimensional Gulf. He is himself his own World, his own Universe; of any other than himself he can form no conception; he knows not Length, nor Breadth, nor Height, for he has had no experience of them; he has no cognizance even of the number Two; nor has he a thought of Plurality; for he is himself his One and All, being really Nothing. Yet mark his perfect self-contentment, and hence learn this lesson, that to be self-contented is to be vile and ignorant, and that to aspire is better than to be blindly and impotently happy. Now listen.”
He ceased; and there arose from the little buzzing creature a tiny, low, monotonous, but distinct tinkling, as from one of your Spaceland phonographs, from which I caught these words, “Infinite beatitude of existence! It is; and there is none else beside It.”
“What,” said I, “does the puny creature mean by ‘it’?” “He means himself,” said the Sphere: “have you not noticed before now, that babies and babyish people who cannot distinguish themselves from the world, speak of themselves in the Third Person? But hush!”
“It fills all Space,” continued the little soliloquizing Creature, “and what It fills, It is. What It thinks, that It utters; and what It utters, that It hears; and It itself is Thinker, Utterer, Hearer, Thought, Word, Audition; it is the One, and yet the All in All. Ah, the happiness, ah, the happiness of Being!”
“Can you not startle the little thing out of its complacency?” said I. “Tell it what it really is, as you told me; reveal to it the narrow limitations of Pointland, and lead it up to something higher.” “That is no easy task,” said my Master; “try you.”
Hereon, raising my voice to the uttermost, I addressed the Point as follows:
“Silence, silence, contemptible Creature. You call yourself the All in All, but you are the Nothing: your so-called Universe is a mere speck in a Line, and a Line is a mere shadow as compared with—” “Hush, hush, you have said enough,” interrupted the Sphere, “now listen, and mark the effect of your harangue on the King of Pointland.”
The lustre of the Monarch, who beamed more brightly than ever upon hearing my words, shewed clearly that he retained his complacency; and I had hardly ceased when he took up his strain again. “Ah, the joy, ah, the joy of Thought! What can It not achieve by thinking! Its own Thought coming to Itself, suggestive of Its disparagement, thereby to enhance Its happiness! Sweet rebellion stirred up to result in triumph! Ah, the divine creative power of the All in One! Ah, the joy, the joy of Being!”
“You see,” said my Teacher, “how little your words have done. So far as the Monarch understands them at all, he accepts them as his own— for he cannot conceive of any other except himself— and plumes himself upon the variety of ‘Its Thought’ as an instance of creative Power. Let us leave this God of Pointland to the ignorant fruition of his omnipresence and omniscience: nothing that you or I can do can rescue him from his self-satisfaction.”
After this, as we floated gently back to Flatland, I could hear the mild voice of my Companion pointing the moral of my vision, and stimulating me to aspire,3 and to teach others to aspire. He had been angered at first—he confessed—by my ambition to soar to Dimensions above the Third; but, since then, he had received fresh insight, and he was not too proud to acknowledge his error to a Pupil. Then he proceeded to initiate me into mysteries yet higher than those I had witnessed, shewing me how to construct Extra-Solids4 by the motion of Solids, and Double Extra-Solids by the motion of Extra-Solids, and all “strictly according to Analogy,” all by methods so simple, so easy, as to be patent even to the Female Sex.
1 Abbott hedges his bets by making the Sphere acknowledge the correctness of A. Square’s arguments (on the existence of arbitrarily many dimensions) by setting it within a dream. The dream occurs in a fictitious setting and is thus doubly fictitious. In Victorian times, casting a story as a dream was a literary cliché, and more than one otherwise compelling story ends with the reader being informed that “it was all a dream.” This is the case, for instance, with Alice’s Adventures in Wonderland:
At this the whole pack [of playing cards] rose up in the air, and came flying down upon her; she gave a little scream, half of fright and half of anger, and tried to beat them off, and found herself lying on the bank, with her head in the lap of her sister, who was gently brushing away some dead leaves that had fluttered down from the trees upon her face.
“Wake up, Alice dear!” said her sister. “Why, what a long sleep you’ve had!”
“Oh, I’ve had such a curious dream!” said Alice.
Here, the statement that the story was really just a dream does not spoil the illusion—perhaps because a significant feature of Alice is its remarkably dream-like quality—but modern readers usually find such revelations anticlimactic and irritating.
2 Abbott rounds out his analogy by reducing the dimensions of space even below the limited realm of Lineland. (Recall that Poincaré and Menger went one stage further and assigned dimension minus one to the empty set, but it is not surprising that Abbott did not go so far.) The Being that inhabits Pointland is the ultimate solipsist, and Abbott can make good use of him to castigate complacency and lack of imagination. The wonderful thing about solipsists is that you can insult them as much as you want and they can only blame it on themselves.
3 Abbott was a great teacher. Many tributes from former pupils are reported in City of London School, and they are made credible by Abbott’s general reputation— that is, they are not just polite remarks made at Old Boys’ dinners with the benefit of rosy memories and a surfeit of wine. For instance, the Rt. Hon. H. H. Asquith, who became prime minister in 1908, said that Abbott “taught his pupils as well as any schoolmaster of his time.” According to William Bram-well Booth (1856–1929), second general of the Salvation Army, “The chief service rendered me . . . came through the encouragement of Dr. Abbott, the brilliant Head. . . . He kindled a little flame of intellectual aspiration.” And Oscar Browning, a prominent university academic, made his views crystal clear: “When I have been asked who was the most distinguished headmaster in England I have never had any hesitation in replying ‘Dr. Abbott’.” Characteristically, Abbott draws a more positive lesson from the self-satisfied Being of Pointland than merely berating the unimaginative for their complacency.
4 Today’s mathematics has taken Abbott’s message to heart, although it followed its own route to get there. An important example of the use of high-dimensional geometry occurs in celestial mechanics, the motion of stars and planets. In this subject, the bodies concerned are usually reduced to point masses, and the state of each such body is represented by six numbers: three coordinates of position and a further three coordinates of velocity. (One of the features of Newton’s Laws of Motion is that the future trajectory of a particle is determined not by its position alone but also by its velocity. Think of throwing a ball, and you’ll see why the velocity matters.) A typical problem in celestial mechanics is the motion of a three-body system, such as the sun, moon, and Earth. This problem involves eighteen independent “coordinates”—six for the sun, six more for the moon, and another six for the Earth. The “phase space” of all possible states of the three-body system therefore has eighteen dimensions.
The nine planets of the solar system, plus the sun, form a sixty-dimensional mathematical system. At a conservative estimate, the human body, with its numerous flexible joints, is 101-dimensional. And mathematicians have found that it really pays to set up the problem that way and to use geometric concepts (all originally derived by analogy with two and three dimensions) to help solve it.
Multidimensional spaces are important in other areas of human activity as well; a good example is economics. A national economy depends on the sales and purchases of, say, a million different goods. The price of each good is in principle an independent quantity, so the state of the economy can be represented by a “price vector” in a milliondimensional space. Again, mathematicians have found that it pays to set up the problem that way and to use geometric concepts to help solve it. An example is the simplex method invented by American mathematician George Bernard Dantzig (1914– ) to answer such questions as “How should the economic activity be divided among various goods?” The simplex method is based on an analogy with triangular faces of a three-dimensional polyhedron. (See chapter 4 of Flatterland, “A Hundred and One Dimensions.”)