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Math Trek

A Stranger from Spaceland

4:53pm, June 3, 2002
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"It was the last day of the 1999th year of our era. The pattering of the rain

had long ago announced nightfall; and I was sitting in the company of my

wife, musing on the events of the past and the prospects of the coming

year, the coming century, the coming Millennium."

Those words appear near the beginning of the 15th chapter of a remarkable

book titled Flatland: A Romance of Many Dimensions.

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 books 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 below it. All Flatlands

inhabitants–straight lines, triangles, squares, pentagons, and other

figures–are trapped in their planar geometry.

On the surface, Abbotts narrative appears to be simply an entertaining tale

and a clever mathematics lesson. From Flatlands 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 womens

rights, and he couldnt resist taking a satirical swipe at his class-conscious

societys 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


When he wrote Flatland, Abbott was headmaster at the City of London

School, an institution that prepared middle-class boys for places at

universities 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 Bards plays.

At first glance, Flatland appears out of place within this collection, but a

closer look shows that it combines elements of Abbotts broad range of

interests, from the reform of mathematics education to the nature of


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.

Abbotts group considered such exercises a waste of time and felt that they

narrowed the study of geometry unnecessarily.

Abbotts 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 Flatlands

university "Wentbridge"–a sly dig at Cambridge.

Flatland also represented one of Abbotts attempts to reconcile scientific

and religious ideas and to illuminate the relationship between material proof

and religious faith.

In the New Years 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 (as a line) at any

given moment. Rising out of Flatland, the sphere looks like a line that gets

shorter and shorter until it finally dwindles to a point, then vanishes


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


After a harrowing but eye-opening adventure in Spaceland, A Square

awakes on New Years 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. 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!

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