|Ivars Peterson's MathTrek|
December 20, 1999
Written in 1884 by Edwin A. Abbott (1838-1926), this slim volume has long served as a doorway to the fourth dimension and beyond for many explorers of geometry. The book's central figure and narrator, "A Square," takes visitors into a two-dimensional world where a race of rigid geometric forms live and love, work and play.
Like shadows, the denizens of Flatland freely flit about on the surface of their world but lack the power to rise above or sink into it. All Flatland's inhabitants--straight lines, triangles, squares, pentagons, and other figures--are trapped in their planar geometry.
On the surface, Abbott's narrative appears to be simply an entertaining tale and a clever mathematics lesson. From Flatland's beguiling text and quaint drawings, readers can begin to imagine the strictly limited vistas open to those trapped in a low-dimensional realm.
Flatland is also a sharply delineated satire that reflects widely debated social issues in Victorian Britain. Abbott was a strong advocate of women's rights, and he couldn't resist taking a satirical swipe at his class-conscious society's attitudes toward women. Flatland women are merely Straight Lines. Lower-class men are Isosceles Triangles; Squares make up the professional class; Nobles are regular polygons with six or more sides; and Priests, the highest-ranking members, are Circles.
"[A] Woman is a needle; being, so to speak, all point, at least at the two extremities," A Square comments. "Add to this the power of making herself invisible at will, and you will perceive that a Female, in Flatland, is a creature by no means to be trifled with."
Nonetheless, Flatland women also are judged "devoid of brain-power, and have neither reflection, judgment nor forethought, and hardly any memory." In this planar world, men believe that educating women is wasted effort and that communication with women must occur in a separate language that contains "irrational and emotional conceptions" not otherwise found in male vocabulary.
When he wrote Flatland, Abbott was headmaster at the City of London School, an institution that prepared middle-class boys for places at unversities such as Cambridge. He produced dozens of books, including school textbooks, historical and biblical studies, theological novels, and a well-regarded Shakespearean grammar that strongly influenced the study of the Bard's plays.
At first glance, Flatland appears out of place within this collection, but a closer look shows that it combines elements of Abbott's broad range of interests, from the reform of mathematics education to the nature of miracles.
Abbott was a member of a group of progressive educators who sought changes in the mathematics requirements for university entrance, which at that time included memorization of lengthy proofs in Euclidean geometry. Abbott's group considered such exercises a waste of time and felt that they narrowed the study of geometry unnecessarily.
Abbott's interest in higher dimensions was also anomalous. Despite evident public curiosity at the time about the concept of a fourth spatial dimension, the mathematics establishment in Great Britain generally refused to admit that higher-dimensional geometries were even conceivable. Conservative mathematicians maintained that such concepts would call into question the very existence and permanence of mathematical truth, as so nobly represented by Euclidean geometry.
Abbott challenged such a narrow viewpoint and deliberately called Flatland's university "Wentbridge"--a sly dig at Cambridge.
Flatland also represented one of Abbott's attempts to reconcile scientific and religious ideas and to illuminate the relationship between material proof and religious faith.
In the New Year's Eve, 1999, episode, A Square receives a visit from a ghostly sphere, who tries to demonstrate to the bewildered Flatlander the existence of Spaceland and a higher dimension.
The visiting sphere argues that he is a "Solid" made up of an infinite number of circles, varying in size from a point to a circle 13 inches across, stacked one on top of the other. In Flatland only one of these circles is visible at any given moment. Rising out of Flatland, the sphere looks like a circle that gets smaller and smaller until it finally dwindles to a point, then vanishes altogether.
To a Flatlander, a sphere passing through Flatland appears as a circle of changing diameter.
When this vanishing trick fails to persuade A Square that the sphere is truly three-dimensional, the visiting sphere tries a more mathematical argument. A single point, being just a point, he insists, has only one terminal point. A moving point produces a straight line, which has two terminal points. A straight line moving at right angles to itself sweeps out a square with four terminal points.
Those are all conceivable operations to a Flatlander. Inexorable mathematical logic forces the next step. If the numbers 1, 2, and 4, are in a geometric progression, then 8 follows. Lifting a square out of the plane of Flatland ought to produce something with eight terminal points. Spacelanders call it a cube. The argument opens a path to even higher dimensions.
After a harrowing but eye-opening adventure in Spaceland, A Square awakes on New Year's Day, 2000, refreshed and filled with an evangelical fervor to proclaim and propagate the Gospel of Three Dimensions. Sadly, no one takes him seriously, and he ends up in prison for his beliefs.
Through mathematical analogy, Abbott sought to show that establishing scientific truth requires a leap of faith and that, conversely, miracles can be explained in terms that dont necessarily violate physical laws. Miracles could be shadows of phenomena beyond everyday experience or intrusions from higher dimensions, he argued.
Flatland raises the fundamental question of how to deal with something transcendental, especially when recognizing that you would never be able to grasp its full nature and meaning. Mathematicians face such a challenge when they venture into higher dimensions (see . How do they see multidimensional objects? How do they organize their observations and concepts? How do they communicate their insights?
Flatland serves as a provocative and informative guidebook for pondering those questions.
Happy New Year!
Copyright 1999 by Ivars Peterson
Abbott, E.A. 1991., Flatland: A Romance of Many Dimensions. Princeton, N.J.: Princeton University Press. (This edition has an introduction written by Tom Banchoff.)
Banchoff, T.F. 1996. Beyond the Third Dimension: Geometry, Computer Graphics, and Higher Dimensions. New York: W.H. Freeman.
Burger, D. 1965. Sphereland: A Fantasy about Curved Spaces and an Expanding Universe, trans. by C.J. Rheinboldt. New York: Barnes & Noble Books.
Peterson, I. 1998. The Mathematical Tourist: New and Updated Snapshots of Modern Mathematics. New York: W.H. Freeman.
The full text of Flatland is available at http://www.geom.umn.edu/~banchoff/Flatland/.
Biographical information about Edwin A. Abbott, prepared by Thomas F. Banchoff of Brown University, can be found at http://www.geom.umn.edu/~banchoff/ISR/ISR.html. Banchoff and Laura Dorfman are working on a multimedia biography of Abbott.
Comments are welcome. Please send messages to Ivars Peterson at firstname.lastname@example.org.