Flatland: A Romance of Many Dimensions

  Need I say that I was at once arrested and taken before the Council?

  Next morning, standing in the very place where but a very few months ago the Sphere had stood in my company, I was allowed to begin and to continue my narration unquestioned and uninterrupted. But from the first I foresaw my fate; for the President, noting that a guard of the better sort of Policemen was in attendance, of angularity little, if at all, under 55°, ordered them to be relieved before I began my defence, by an inferior class of 2° or 3°. I knew only too well what that meant. I was to be executed or imprisoned, and my story was to be kept secret from the world by the simultaneous destruction of the officials who had heard it; and, this being the case, the President desired to substitute the cheaper for the more expensive victims.

  After I had concluded my defence, the President, perhaps perceiving that some of the junior Circles had been moved by my evident earnestness, asked me two questions:—

  Whether I could indicate the direction which I meant when I used the words “Upward, not Northward”?

  Whether I could by any diagrams or descriptions (other than the enumeration of imaginary sides and angles) indicate the Figure I was pleased to call a Cube?

  I declared that I could say nothing more, and that I must commit myself to the Truth, whose cause would surely prevail in the end.

  The President replied that he quite concurred in my sentiment, and that I could not do better. I must be sentenced to perpetual imprisonment; but if the Truth intended that I should emerge from prison and evangelize the world, the Truth might be trusted to bring that result to pass. Meanwhile I should be subjected to no discomfort that was not necessary to preclude escape, and, unless I forfeited the privilege by misconduct, I should be occasionally permitted to see my brother who had preceded me to my prison.

  Seven years have elapsed and I am still a prisoner, and—if I except the occasional visits of my brother—debarred from all companionship save that of my jailers. My brother is one of the best of Squares, just, sensible, cheerful, and not without fraternal affection; yet I confess that my weekly interviews, at least in one respect, cause me the bitterest pain. He was present when the Sphere manifested himself in the Council Chamber; he saw the Sphere’s changing sections; he heard the explanation of the phenomena then given to the Circles. Since that time, scarcely a week has passed during seven whole years, without his hearing from me a repetition of the part I played in that manifestation, together with ample descriptions of all the phenomena in Spaceland, and the arguments for the existence of Solid things derivable from Analogy. Yet—I take shame to be forced to confess it—my brother has not yet grasped the nature of the Third Dimension, and frankly avows his disbelief in the existence of a Sphere.

  Hence I am absolutely destitute of converts, and, for aught that I can see, the millennial Revelation has been made to me for nothing. Prometheus up in Spaceland was bound for bringing down fire for mortals, but I—poor Flatland Prometheus—lie here in prison for bringing down nothing to my countrymen. Yet I exist in the hope that these memoirs, in some manner, I know not how, may find their way to the minds of humanity in Some Dimension, and may stir up a race of rebels who shall refuse to be confined to limited Dimensionality.

  That is the hope of my brightest moments. Alas, it is not always so. Heavily weighs on me at times the burdensome reflection that I cannot honestly say I am confident as to the exact shape of the once-seen, oft-regretted Cube; and in my nightly visions the mysterious precept, “Upward, not Northward,” haunts me like a soul-devouring Sphinx. It is part of the martyrdom which I endure for the cause of the Truth that there are seasons of mental weakness, when Cubes and Spheres flit away into the background of scarce-possible existences; when the Land of Three Dimensions seems almost as visionary as the Land of One or None; nay, when even this hard wall that bars me from my freedom, these very tablets on which I am writing, and all the substantial realities of Flatland itself, appear no better than the offspring of a diseased imagination, or the baseless fabric of a dream.

  AFTERWORD

  It’s been a hundred and thirty years since Edwin Abbott Abbott wrote Flatland, the now classic story of A. Square (a play on the author’s double name), a most thoughtful citizen of a fanciful two-dimensional world. The geometry of higher dimensions and the social structure and practices of his planar universe constitute the two primary topics of A. Square’s musings and wanderings.

  The interweaving of these disparate topics gives the book a creative coherence that helps account for its continued popularity. I’ve always believed that if there were an algorithm for creativity, it would involve the extended rubbing together of two unrelated realms and a full exploration of their connections. This rubbing together is certainly what transpires in Flatland, and it is what attracted me and countless others to its odd but seamless amalgam of geometry and social commentary and their reciprocal illumination.

  Reading it for the first time in junior high school, I was more fascinated by the discussions of dimensionality than the social commentary, which I only dimly glimpsed. Around that time I’d also read a short story about a locked-room murder in which the perpetrator escaped through the fourth dimension the same way a criminal in Flatland might step out into the third dimension to escape a two-dimensional enclosure. Martin Gardner, who also stoked my nascent mathematical interest, wrote about a similar phenomenon. In fact, Gardner, Abbott, Gamow, and Asimov were acronymically instrumental in my going GAGA over mathematics and science.

  Abbott’s very slightly veiled criticism of Victorian social hierarchies resonates even more insistently today. So do his snide references to phrenology, the discredited belief that one’s intelligence is determined by the shape of one’s head, just as polygonal Flatlanders with more sides are believed to be smarter than those with fewer sides. Circles are the Flatland ideal, but there is a path upward: male children have one more side than their fathers. Women in Flatland, however, aren’t even accorded polygonal status and are merely line segments. In real life Abbott was a supporter of women’s suffrage, and he intended Flatland’s humiliating requirement that women constantly wriggle their backsides to be a parody of then contemporary attitudes to sex and gender. The description of the Chromatic Innovation—the introduction of color into this austere world—and its suppression also has fairly obvious real-world analogues.

  As important as these societal issues are, I believe that even more significant is A. Square’s mental liberation brought about by a sphere’s visitation to Flatland from the third dimension. The sphere appears as a circle that can change the size of its intersection (i.e., its diameter) with Flatland at will. Temporarily removed from the confines of Flatland by the sphere, A. Square returns home and eventually becomes a higher-dimensional evangelist and pays the price for his apostasy. A. Square’s brief view of the third dimension has great consequences for him—he is imprisoned physically—and for the reader it brings to mind Plato’s allegory of the cave in which mankind is depicted as seeing not real things, but only their shadows on the cave walls. It also suggests Galileo’s muttering under his breath that, contrary to Church doctrine, the Earth revolves around the sun and not the other way around, “Still, it moves.” Abbott could just as well have had A. aver, “Still, there are more dimensions.”

  We shouldn’t be too surprised that Abbott, who was educated at Cambridge, took highest honors not only in mathematics but also in classics and theology. Some of his other works also demonstrate his liberal, pedagogical and theological leanings, but none was anywhere near as popular as Flatland.

  This timeless (or does this word subtract a dimension?) tale of higher dimensions has many descendants—stories, speculations, and scientific theories (science fiction novels, popularizations of the many-worlds interpretation of quantum mechanics, worlds with bizarre topologies)—that in one way or another occupy the same conceptual area of our common culture. How about Schizoland, in which there are two li
nked time dimensions, each of which is sometimes the past of the other? I’ll leave that for some T. Square to write.

  I myself have sketched an attempt along these lines that utilizes the rich notion of dimension, and there is probably no better place than here to sketch it. Specifically, I’ve developed a mathematical metaphor for the old chestnut “There are two kinds of people in the world: those who are very strange and those whom you don’t know well.” In other words, I’m suggesting that each of us is actually very strange and not a completely integrated personality, and there is a way in which higher dimensions can illustrate this.

  If by “very strange” people, you mean people who, along some measurable dimensions, are statistically way off the charts, then this is almost certainly true. “Dimension” can be something geometrical, as in Flatland, but needn’t be. Think, for example, of dating services that advertise that they check prospective couples for compatibility along dozens of possible dimensions such as aspects of personality, obsessions, fears, hobbies, family backgrounds, political views, et cetera.

  Or consider people as consumers whose tastes differ along many more dimensions. One can ask whether they prefer one kind of product to another. We can inquire as well about the intensity of their preferences. We can also include dimensions of kinkiness, which I’ll leave to readers’ imaginations. Having partially listed this large indeterminate number of dimensions, we define people solely in terms of numbers along them, i.e., reductively as a collection of atomic traits and as nothing more coherent—people as sequences of numbers, as points in Big Data, as consumers.

  Employing these dimensions and the related notion of a physical dimension, we can see more geometrically why we are all a bit strange. Toward this end, imagine a straight line ten units long, along which people can be measured on some dimension of interest. Universality is preferable, but I’ll here be somewhat parochial and take the units to be inches. Let’s consider the parts of the line within a half inch of either end and call it the extreme parts of the line. The normal part is the middle section, which constitutes 90 percent of the line.

  Now consider a square ten inches on a side, along both of whose dimensions people can be measured. Consider the part within a half inch of a side of this square and call this border area the extreme part of the square. The normal part of the square is the middle section, which constitutes 81.0 percent of the square. This can be seen by noting that the whole square is 102 or 100 square inches, and the normal part constitutes only 92 or 81 square inches. (Note that a half inch off each side shortens every dimension by a whole inch; i.e., ten inches - ½ inch - ½ inch = nine inches.)

  (If you like circles better than squares, realize that the interior of a ten-inch pizza with a half-inch crust is only 81 percent of the area of the whole pizza. A one-inch crust leaves only 64 percent for the interior. Why?)

  Next up, let’s picture a cube ten inches on a side, along all three of whose dimensions people can be measured. Consider the part within a half inch of an outside face of this cube and call it the extreme part of the cube. The normal part of the cube is the middle section, which constitutes 72.9 percent of the cube. Once again, this can be seen by noting that the whole cube is 103 or 1,000 cubic inches, and the normal part constitutes only 93 or 729 cubic inches. (Remember that a half inch off of each side shortens every dimension by a whole inch, to 9 inches.) Returning to Italian food, I note that a similar analysis applies to spherical meatballs (which, like the circular pizza, in Flatland might be worshipped).

  Although it can’t be pictured as easily, the same idea makes sense with hypercubes, a hyperdimensional notion that A. Square gradually comes to glimpse. He understands that line-world is a one-dimensional section of Flatland in the same way that Flatland is a two-dimensional section of three-dimensional space. He suggests that three-dimensional space too is a section of four-dimensional space, and so on and on. When he mentions this to the sphere, the latter is shocked and refuses to even consider such speculations. Abbott no doubt means to point out that even those with much sophistication can be unwilling to expand their horizons further.

  We needn’t be as constrained as the sphere, however, and to that end let’s imagine a four-dimensional hypercube ten inches on a side, along all four of whose dimensions people can be measured. Consider the part within a half inch of the outside of this hypercube and call it the extreme part of the hypercube. The normal part of the hypercube is the middle section, which constitutes 65.6 percent of the hypercube. This also can be seen—please forgive the repetition—by noting that the whole hypercube is 104 or 10,000 cubic inches, and the normal part constitutes only 94 or 6,561 cubic inches.

  Further note that as the number of dimensions increases, the normal interior part of the hypercube constitutes a smaller and smaller part of the volume of the hypercube in question. We can continue this game and consider not four but fifty dimensions along which people can be measured and perform the same sort of calculation. If we do, we’ll find that that the interior normal part of the resulting hypercube constitutes only about one half of 1 percent of the volume of the hypercube. For one hundred dimensions, the interior or normal part shrinks to only .0027 percent of the total volume.

  The vast majority of points in the hypercube will be extreme along some dimensions and hence be at the “extreme” edge of the hypercube. In something like this same sense, most people are very strange and live beyond the edge of “normality.” Furthermore, there are surely many more than fifty or a hundred dimensions of various sorts along which people differ. We can also define an extreme score along any given dimension to be one that’s even more extreme—as, say, the top and bottom .5 percent rather than the top and bottom 5 percent of the given dimension. (The same sort of argument can be made in probabilistic terms rather than geometric ones and can also employ the so-called normal distribution rather than the uniform distribution I assumed above.)

  Given this suggestive little model of multidimensional but unintegrated human beings, it’s literally true and not very surprising that there are two kinds of people in the world: those who are very strange and those whom you don’t know very well. Almost all of us live in the extreme outer shell of a multidimensional hypercube (or hypersphere or hypermeatball) whose interior is largely devoid of other humans.

  This vignette may seem to have drifted from Abbott’s Flatland, but, as mentioned, it and many others are descendants of the book’s geometry and social commentary. It suggests that rather than being polygonal figures floating about a two-dimensional world, we are isolated points floating on or very near the surfaces of hyperdimensional cubes or spheres. The extra-mathematical “lesson” of the above is a bit more postmodern than Abbott’s criticism of Victorian hierarchies and hypocrisies. We’re merely alienated, somewhat incoherent collections of traits; sequences of numbers, each of us strange in our own way, and almost all of us residing on the surface of a vast hyperdimensional world.

  Obviously, more social commentary and character development are needed to come up with even a feeble analogue of Flatland’s story, although it must be admitted that the characters in Flatland are—excuse me for saying—somewhat flat. Nevertheless, Strangerland, if you’ll excuse the term, with perhaps an introspective protagonist named Nowhere Point, does help to illustrate the continuing appeal of Abbott’s classic book. Employing another geometric metaphor, I think that Flatland and A. Square appeal in the sense of a chaotic strange attractor, always pulling us to them, always along a slightly different trajectory.

  Whatever else it is and does, Flatland certainly suggests that there is more to the world than what is directly in front of our noses (or, should I say, our most pointed polygonal vertices).

  —John Allen Paulos

  * The Author desires me to add, that the misconception of some of his critics on this matter has induced him to insert in his dialogue with the Sphere, certain remarks which have a bearing on the point in question, an
d which he had previously omitted as being tedious and unnecessary.

  *When I say “sitting,” of course I do not mean any change of attitude such as you in Spaceland signify by that word; for as we have no feet, we can no more “sit” nor “stand” (in your sense of the word) than one of your soles or flounders.

  Nevertheless, we perfectly well recognize the different mental states of volition implied in “lying,” “sitting,” and “standing,” which are to some extent indicated to a beholder by a slight increase of lustre corresponding to the increase of volition.

  But on this, and a thousand other kindred subjects, time forbids me to dwell.

  *“What need of a certificate?” a Spaceland critic may ask: “Is not the procreation of a Square Son a certificate from Nature herself, proving the Equal-sidedness of the Father?” I reply that no Lady of any position will marry an uncertified Triangle. Square offspring has sometimes resulted from a slightly Irregular Triangle; but in almost every such case the Irregularity of the first generation is visited on the third; which either fails to attain the Pentagonal rank, or relapses to the Triangular.

  *When I was in Spaceland I understood that some of your Priestly circles have in the same way a separate entrance for Villagers, Farmers and Teachers of Board Schools (Spectator, Sept. 1884, p. 1255) that they may “approach in a becoming and respectful manner.”