Book Review

The Annotated Flatland: A Romance of Many

Reviewed by A. K. Dewdney

The Annotated Flatland: A Romance of Many or society becomes Dimensions a challenge to be Edwin A. Abbott, with Introduction and Notes by surmounted, some- Ian Stewart times by a clever Perseus Publishing, 2001 twist of plot, some- ISBN 0-738-20541-9, 160 pages, $30.00 times by the creak- ing of heavy back- I must confess at the outset to never having been stage machinery a great fan of Flatland. Yet this new edition, deco- that barely saves the rated with insightful and helpful notes by Ian Stew- situation. Most art, has added a new depth (or is it thickness?) to works of science fic- the novel. Stewart is a fan of Flatland and has tion get over their recently published a fantasy sendup called premises in the first Flatterland. few pages, there- Imagine then a lesson from Euclid brought to life after launching into by a nineteenth-century Englishman, teacher, and the plot. But for the average Victorian reader, Flat- ordained minister. The characters are line seg- land had to be explained very carefully, and Abbott ments, triangles of various persuasions, and poly- was therefore led to write a natural history, rather gons, some of them multisided enough to mas- than a science (or pre-science) fiction novel. Nat- querade as circles. How do they move about in ural history was a popular topic in nineteenth- their plane world? No one, not Abbott, not Stew- century , including accounts of travels in art, nor I, knows. Yet move they do, with passions exotic lands, many of them imperial territories and and pleasures no less real, for all their lack of possessions. anatomy, than our own. But Abbott’s was a natural history with a dif- Writing a work of science fiction (Stewart iden- ference. Here was a land that everyone had visited tifies the work as pre-science fiction) that is based in school, being subjected to plane figures, theo- on certain rigid premises, such as dwelling entirely rems, and constructions from Euclid. But it was also, in the plane and being a polygon of one kind or an- under Abbott’s hand, a land of passion and violence, other, entails certain consequences which it be- not to mention extreme political incorrectness. comes the delight of the author to explore. That the Shortly after Flatland appeared, according to one premises lead sometimes to an awkward physics of Stewart’s notes, some readers objected to Ab- bott’s characterization of Flatland females. Males could be many-sided, social rank increasing with A. K. Dewdney is professor of mathematics at the University of Western Ontario. His email address is [email protected]. the number of sides, but females were digons, their

1260 NOTICES OF THE AMS VOLUME 49, NUMBER 10 sides coincident, mere segments. According to Thanks to a visit from a mysterious spherical being Stewart, Abbott was merely satirizing Victorian from three dimensions, Mr. Square has the math- views of women, not expressing his own. Although ematical equivalent of a religious experience and the class structure reminds me more strongly of ultimately gets locked up for proclaiming his Imperial India, satire or lack of it may be a red her- gospel. He witnesses a sphere manifesting in Flat- ring. Abbott had made his choices and pressed on, land as a circle that gradually widens until it reaches tongue in cheek, to see what would come out of it its fullest extent, thereafter diminishing. A perfect all. Thus women, who were lower in caste than the being able to change its size at will, what else could narrowest “irregular” triangle and (apparently) it be but semidivine? As Stewart explains, how- completely without brains, nevertheless possessed ever, Abbott himself took no spiritual message a deadly penetrating power in their infinitesimal from his own book. As a minister he was at pains heads or tails, being able to pierce the side of any to out (elsewhere) that seemingly miracu- triangle or other polygon with instantly fatal effect. lous powers, by themselves, do not warrant wor- To warn others of their dangerous presence, women ship. must utter “peace cries” and constantly wiggle Abbott lays his chapters out with methodical pre- their behinds. Hmmm. cision. In the first he sketches the two-dimensional The central metaphor of Flatland is of course the nature of Flatland, and in most of this he goes to ability to appreciate higher, as well as lower, di- great lengths to explain how Flatlanders see only mensions. As a teacher, Abbott had discovered a lines. In Chapter 2 he discusses the climate and marvelous door to this world. He first explains houses of Flatland. The houses are pentagonal be- how things in Flatland appear to Flatlanders and cause square (or worse, triangular) houses would then demonstrates their utter helplessness in imag- have sharp corners that might do injury to passers- ining something higher. The real target is you, the by. Instead of gravity created by large, nearby bod- reader. Can you put yourself in a Flatlander’s shoes? ies, there is a “constant attraction to the South” (felt (Excuse me, they have no feet.) If so, you will dis- keenly by many English people even today). Thus cover that you have no more knowledge of the rain must fall from the North. By this point in my fourth than a Flatlander does of the original reading of the book, I can recall feeling dis- third. Yet you may embrace the analogy far enough tinctly disoriented, not to say puzzled, by such a to know in which direction the Flatlander must peculiar way of arranging things. Yet, with no cen- point, even if his arm (if he had one) must leave the tral planet, what else can Abbott do if he wants to plane entirely. Aha! If you were to point in the di- have rain? There seems no reason why rain could rection of a fourth dimension, would your arm not not fall into Flatland from the third dimension, also disappear? for this is the origin of all light in Flatland! It is particularly helpful to be accompanied on Chapter 3 sketches the inhabitants of Flatland this voyage by a presence on the sidelines, some- as line segments (females), isosceles triangles (low- one who can explain the twists and turns of di- caste workers and soldiers), equilateral triangles mensionality as cleverly and completely as Stew- (“middle class”), squares and pentagons (“profes- art. And besides four-dimensional spacetime and sional class”), hexagons and higher (“nobility”). At space-filling curves, there is the cellular automa- the highest level of society are the near-circles ton (which demonstrates a substrate for thought, (“priestly order”). A curious progression is made thanks to von Neumann), Hilbert space (which has by the sons of regular polygons, who enjoy one infinitely many dimensions), geometrical con- more side than their fathers had. But the poor struction, and much else. Indeed, for those with a isosceles triangles are doomed to remain isosceles thirst for more dimensions and more about them, unless, by dint of perseverance and high achieve- there is an excellent essay at the end of the book ment, an isosceles father will have an equilateral entitled “The Fourth Dimension in Mathematics”. son (shades of Lamarck). “The birth of a True Equi- Our three-dimensional companion also provides lateral triangle from Isosceles parents [sic] is the ample historical material from his two-dimensional subject for rejoicing in our country for many fur- space beside the text. There are crucial notes on longs round.” There is social unrest among the Abbott’s personal and professional life, many notes lower classes, of course, but revolutions are nipped about historical figures, from British Prime Minis- in the bud by a curious law of compensation: “Thus, ters to noted scientists of the day, whimsical ob- in the most brutal and formidable of the soldier servations on calendars, verse in scientific papers, class—creatures almost on a level with women in sex in Flatland, and enough other entertaining ma- their lack of intelligence—it is found that, as they terial to divert the reader from drier stretches of wax in the mental ability necessary to employ their the work. tremendous penetrating power to advantage, so do The hero of the story is one A. Square who hap- they wane in the power of penetration itself.” pens to be a square and therefore of higher caste, Chapter 4 is about women. To what I have already somewhere between a merchant and a physician. said, I might only add that, although brainless, they

NOVEMBER 2002 NOTICES OF THE AMS 1261 suffer from being confined too much and are wont “Natural Law” that Abbott forgot to mention? In to rebel. “Hence, in their fits of fury, they remem- order to prevent the number x from increasing ber no claims and recognize no distinctions. I have forever, the relative distribution of individuals actually known a case where a woman has exter- within sidedness classes must remain the same minated her whole household and half an hour af- over time. For this to happen, the distribution terwards, when her rage was over and the fragments should be monotone decreasing in the direction of swept away, has asked what became of her husband higher sidedness, there always being the same ratio and children.” Abbott has the bit in his teeth by now, of priests to those in lower classes for example. The perhaps relishing the public reaction to this shock- number f (n) of n-gons may increase with every ing state of affairs. generation, but the ratio f (n +1)/f (n) must remain In Chapter 5, Abbott must get around the in- approximately constant at c, with c<1. ability of Flatlanders to recognize each other, es- Now the number of n-gons will increase to pecially to discern social class in encounters. They r(n)f (n) at the next generation, where r is the fe- “feel” each other (without hands), and Abbott goes cundity of that class. It follows that the fecundity r, on at great length about the social conventions as a function of n, must decline with increasing n, that arise from this practice: “Let me ask you to feel because the ratio of two populations at the next Mr. So-and-so.” And on it goes. Flatlanders have an generation will be the same: exquisite sense of touch that permits them to dis- r(n +1)f (n +1) tinguish between the rather small differences in the = c of a pentagon and a hexagon. Not content r(n)f (n) with his earlier treatment of vision, Abbott returns so that to the subject by invoking yet another Deus ex Machina, a pervasive fog that causes farther sur- r(n +1) f (n)

1262 NOTICES OF THE AMS VOLUME 49, NUMBER 10 to no improvement in one’s status. The doctrine has Technically, Square should be capable only of views drawbacks, of course, and when Abbott’s hero that are one-dimensional cross-sections of these states that some scalene fellows plead in court views, but by now the reader is swept up in the that their crimes were the result of their poor con- drama. If there is any place in the book where tech- figuration, Stewart leaps in with a discussion of the nical objections fall by the wayside, it is here. The “nature versus nurture” question in our own three- two float through the third dimension until they dimensional society, comparing the scalene’s plea can see the General Assembly Hall of the States of to that of the drunk driver: “I couldn’t help it, Your Flatland. As it happens, the administrators are in Honour, I was drunk.” the midst of passing a law that dictates death or Finally, in Chapter 13, Abbott settles to the main imprisonment for anyone preaching the “gospel of event, beginning with a visit to Lineland in a dream the third dimension.” Apparently the Sphere has of A. Square’s. Lineland is inhabited by line seg- been visiting other households. However, even a ments. Has he found the Planet of the Women? Not visit by the sphere to the council chamber itself so. Yet every man has two wives in a world that per- changes nothing. The presiding circle instructs mits no proximity whatever between male and fe- those present to take no notice. This happens every male. Courtship (and, presumably, fertilization) is millennium, it turns out. carried out entirely by sound, all inhabitants of There are further adventures in , in- Lineland being given to song. In this way Abbott cluding a visit to various three-dimensional solids gets around the difficulty that proximate courtship and an extended conversation with the Sphere in would create. Things get sillier than this in Lineland, which Square asks whether spaces higher than but Abbott hastens on to the next chapter, in which three might exist. The Sphere pooh-poohs this idea, his hero must explain to the King of Lineland the thereby granting Square a kind of moral and in- nature of Flatland. A. Square pleads with the King tellectual superiority by default. In the end, only to imagine an extra dimension, all to no avail. Square knows the truth, but he is imprisoned for I was wrong. Finally, in Chapter 15, Abbott set- it. He writes from prison. tles to the main event. It is the turn of the millen- Abbott also writes from a prison of sorts. He has nium (1999 in the Flatland era). In the midst of a followed his premises, adding more or less arbitrary lesson for his hexagonal grandson, ideas (further premises) that allow a humanlike A. Square demonstrates how an array of squares existence to his subjects. If the narrative creaks and three on a side has nine (the square of three) groans under these burdens, Abbott nevertheless squares in it. But then the bright little fellow recalls carries it off to the point of convincing the naive that Grandpa has been teaching him to cube num- reader that perhaps higher dimensions do exist, bers, as well. “I suppose three cubed must mean which is all he set out to do. In this edition, Stew- something in Geometry. What does it mean?” The art is the tour operator, clearly enamored of this child tries to construct a higher dimension and place and keen to give us the widest possible view. Grandpa must send him off to bed with, “If you He includes every conceivable scrap of history and would talk less nonsense, you would remember mathematics that bears on the strange adventures more sense.” Presently A. Square and his wife have of A. Square. a visitor from the third dimension, a sphere who With me, Stewart succeeds as well. How else did asks to be left alone with the hero. I become motivated to explore the population bi- The sphere appears as a circle and therefore ology of Flatland? How else to explain my renewed exercises great authority over our hero, who is nev- interest in the subject of two-dimensional exis- ertheless hard pressed to imagine what the circle tence? means by being able to see into Square’s house from a third dimension. The sphere describes the sleep- ing servants and other household contents of Square’s house without having visited any of the rooms. Failing to convince Square that he comes from a higher dimension, the alien visitor proceeds in Chapter 17 to a demonstration of the reality of the third dimension. He filches an accounting book from a closed cupboard without opening the door. Square is even more horrified when the sphere pokes him from the inside. The hero throws himself against the figure, judg- ing him to be an evil magician. In the next chapter the sphere reacts by plucking Square right out of Flatland. Giddily, Square witnesses the spherical surface, as well as the contents of his own house.

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