Book Review The Annotated Flatland: A Romance of Many Dimensions 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 England, 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 point 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 dimension 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.