by Edwin A. Abbott
Reviewed by Colin C. Adams
In 1884, the English minister, headmaster, and biblical and Shakespearean scholar Edwin Abbott Abbott produced a thin volume titled Flatland: A Romance of Many Dimensions. It was both an introduction to the notion of higher dimensions and a satire of Victorian society and norms. At that time, there was substantial interest in the idea of higher dimensions, both within the scientific community and also in the more general population. Abbott's work provided a simple story that allowed lay audiences to grasp the idea of dimensions beyond the familiar three. Flatland helped to set the stage for many of the scientific advances to come.
In the pantheon of popular books about mathematics, one would be hard-pressed to name another that has lasted so long in popularity or had such a dramatic impact. Generations of students have gained their first true appreciation of higher dimensions by reading this slight story written by a schoolmaster more than a century and a quarter ago. Of the more than 50 books that Abbott wrote, this is the one for which he is remembered.
The book's appearance in England was followed a year later by its publication in the United States, where it has yet to go out of print. Just since 2007, more than 20 different publishers have produced editions of the book -- a testament to its popularity, profitability and expired copyright. The cheapest version is available from Dover Thrift Editions for just two dollars. At that price, members of my department hand them out to students as prizes, and the students don't have to impress us all that much to merit a copy. A new edition jointly published this year by Cambridge University Press and the Mathematical Association of America contains enough notes and commentary by William Lindgren and Thomas Banchoff to more than double the length of the book.
Flatland is the story of a two-dimensional creature, A Square, who is actually a square and is one of the inhabitants of Flatland, a world consisting of a single plane. The name A Square is essentially a spoken version of A2, which is a play on the author's double name. The book is littered with playful references such as these, many of which are easily missed on a first reading.
Other residents of Flatland include isosceles triangles, who make up the lower classes and aspire to be equilateral, and regular polygons with greater numbers of edges. The larger one's number of edges, the larger one's angles, and larger angles provide greater intelligence and higher social status. All polygons aspire for themselves and their progeny to increase their number of edges, hoping to approach the circular ideal. Subsequent generations do inherit greater numbers of edges, so one has hope for the social advancement of one's children. However, all of this holds true only for the males. Women are relegated to being line segments, with zero angle and hence little intelligence. Moreover, they are extremely dangerous because of their sharp endpoints. They are therefore required, "under penalty of death," to warn others of their presence by "moving their backs constantly from right to left" and "continually keeping up [a] Peace-cry." (The reference to women "moving their backs constantly" alludes to the fashion of wearing bustles, which, as Banchoff and Lindgren explain, "focused attention on a woman's backside and emphasized the movement of that body part.")
For modern-day audiences, this may be painful to read, but Abbott's portrayal of women in the book was intended as parody of Victorian customs that he himself deplored. In fact, he was a fierce supporter of women's rights: As documented in an appendix on his life and work, he was active in the women's suffrage movement and worked tirelessly in support of the rights of women to an education.
Roughly half the book (part 1, titled "This World") is devoted to explicating life in Flatland -- its society and physical environment and the various means that Flatlanders use to understand and interpret their surroundings. Part 2 ("Other Worlds") begins on the eve of a new millennium. That night, A Square dreams of Lineland, a world that consists only of a straight line populated by points and line segments, the longest of which is the king. A Square does his best to convince the king of the existence of dimensions beyond the one he knows, all to no avail.
The next day A Square is visited by a creature from the higher dimensions. This being is a sphere, which impinges on Flatland, appearing as a circle that can change its diameter continuously at will. The Sphere visits Flatland once every millennium, when he is allowed to try to convince Flatlanders of the existence of the third dimension. Ultimately, in order to convince A Square of the reality of the three-dimensional world (Spaceland), the Sphere is forced to pull him out of Flatland and allow him to experience Spaceland's full effects.
A Square, having recognized that the line world is a one-dimensional slice of Flatland and that Flatland is a two-dimensional slice of three-dimensional space, reasons by analogy that three-dimensional space is a slice of four-dimensional space and so on up the ladder of dimensions. The Sphere is taken aback by this suggestion and refuses to consider the possibility. Even those who are supposedly enlightened have their own blinders on.
Eventually A Square is returned to his planar world. Having seen his brother imprisoned just for being present when the Sphere entered the council chambers, he knows to keep his mouth shut and successfully does so for 11 months. But finally, hearing some Flatlander spout nonsense about the two-dimensionality of the universe, A Square can contain himself no longer and proceeds to espouse the gospel of higher dimensions. For this he is imprisoned for life, because the leaders of Flatland do their best to suppress knowledge of any world outside their own. And so A Square forlornly relates his story to us seven years later, still imprisoned, having made not a single convert, questioning the reality of the higher and lower dimensions he has experienced, and by analogy, questioning the reality of his own two-dimensional existence as well.
Although Flatland does an excellent job of opening up the world of higher dimensions, it is at least as interesting for its social commentary. Abbott manages to pack a surprising number of the hot-button issues of his day into just a bit over 100 pages. In addition to raising questions about women's rights, he lampoons the class system that was dominant in England at the time. He satirizes the economic power structure and the chicanery employed by the aristocracy to restrain the impoverished masses. That one's shape determines one's intellect and character is an allusion to the pseudoscientific claim of phrenologists that a person's intelligence and personality can be determined by the shape of his or her skull. In addition, Abbott certainly intends comparisons with the cave posited by Plato, within which people are chained so that they are only capable of looking at manipulated shadows cast on a blank wall. The liberation of an inmate from the cave is an experience very similar to A Square's emancipation from Flatland into space.
With their informative notes, Banchoff and Lindgren add immeasurably to the text. In addition to explanations of arcane terminology and of the mathematics involved, they provide the background necessary to understand the book in the context of Victorian England. It is a period during which entrenched ideas, both social and scientific, were undergoing dramatic metamorphosis. Banchoff and Lindgren's comments on Abbott and his milieu allow the reader to comprehend this fascinating turning point in history.
In fact, this is not the first edition of Flatland with substantial notes. Ian Stewart, a well-known mathematical expositor and author of a follow-up book to Flatland called Flatterland, is the editor of Flatland: The Annotated Edition, which was published by Perseus in 2001. In the introduction to that book, Stewart quotes Martin Gardner: "I see no reason why annotators should not use their notes for saying anything they please if they think it will be of interest, or at least amusing." Stewart certainly agrees: His notes, which run from a paragraph to several pages long, often veer substantially away from the text; he is using Abbott's book as a platform from which to launch a variety of excursions into math and elsewhere. For example, the mention in Flatland of the beginning of a new millennium provides Stewart with an opportunity to discuss when the millennium should actually begin and how we arrived at the calendar we use today. Throughout, he goes off on extended tangents, but they are always interesting.
If you have never read Flatland, should you start with one of these annotated editions? That's not what I would recommend. It is too tempting to read the notes on the facing pages as you go, and that would cause you to lose the flow of the story, which is critical to an appreciation of the original work. But once you have read the two-dollar Dover edition, then you really should reread Flatland in either or both of the annotated editions. The plethora of fascinating background information and detail will make you appreciate the book at a much deeper level.
If you would like to be culturally literate in mathematics, you should read Flatland. It's not great literature. The plot is creaky, and -- dare I say it -- the characterizations lack three-dimensionality. They have no depth. But the book is great mathematical literature. And perhaps it is unfair to judge Abbott's depiction of the Flatlanders so harshly. It's hard to give depth to such thin creatures. And Flatland does leave one thinking. Might there be higher-dimensional creatures looking down on us, wondering why we are so dimensionally impoverished?
Colin C. Adams is Thomas T. Read Professor of Mathematics at Williams College in Williamstown, Massachusetts. He is the author of several books, including Riot at the Calc Exam and Other Mathematically Bent Stories (American Mathematical Society, 2009), Why Knot? (Key Curriculum Press, 2004) and The Knot Book (W. H. Freeman and Co., 1994).