Edwin Abbot: The Genius of Flatland

Edwin A. Abbot was born April 16, 1838, in Somerset, England. He was educated at the City of London School, and quickly distinguished himself as a teacher, and later headmaster of his alma mater. One of the foremost Shakespearean scholars of his time, Abbot also produced nearly 200 books dealing with the Transcendental. A fifty-year writing career spanned every topic from miracles to mundane hints for tutors on how to teach recalcitrant and slow children.

The foremost theme of Abbot’s work is simple: understanding the mystery of Divinity is not possible with the finite human mind. On those rare occasions when we do perceive something of transcendental nature, our language is often imperfect and cannot communicate our visions precisely. Yet we must continue to try, risking being misunderstood, persecuted and rejected. “Indeed,” wrote Abbot, “this is the central message of the Gospels.”[1]

Contrary to modern opinion, Abbot was not a mathematician. He was, instead, the epitome of the Victorian schoolmaster. Beyond his ground-breaking book on multidimensionality, Flatland, he addressed little else of note to the study of mathematics. However, within the earlier work of such shining stars as Kant, Tyndall and the like (whose works Abbot apparently read), Abbot was struck with the concept of higher dimensions and their relationship to our dimensions. He theorized that what we perceived as miraculous, events which were not subject to the laws of nature, may actually be the interaction of a heretofore unperceived higher dimension. Drawing upon this analogy, Abbot encased a withering satire upon his time in the midst of one hundred pages of mathematics.

Written from the point of A. Square, a prosperous businessman in the second dimension, Flatland reduces the complexities of Victorian society to a series of graduated geometric shapes. Such difficulties as the Color Revolution (reminiscent of the Puritan Revolt which placed Cromwell as Lord Protector), methods of determining regularity of shape, and recognition of individuals are dealt with in a serious (to the square) treatise. However, the square’s entire worldview changes upon receiving a visit from a member of the third dimension, a sphere.

Within Flatland, all shapes, regardless of their nature, perceive each other strictly as lines. Abbot explains this phenomenon thusly:

“Take for example an equilateral Triangle – who represents with us a Tradesman of the respectable class. Fig. 1 represents the Tradesman as you would see him while you were bending over him from above; figs. 2 and 3 represent the Tradesman, as you would see him if your eye were close to the level, or all but on the level of the table; and if your eye were quite on the level of the table (and that is how we see him in Flatland) you would see nothing but a straight line.”[2]

To further complicate the matter, all women are Straight Lines, and thus, seen end-on, are mere Points. The damage which can result from hurried contact with females then is undoubtedly considerable. Among other things, then, females are required to sway side to side and to emit a continuous “peace-cry” to warn unsuspecting males of their presence.

However, upon being taken in vision to Lineland, the square perceives a new conception: one in which all inhabitants can view each other only as points. The King of this land, in discourse with the square, states that the square’s voice originally came to him from somewhere near his stomach[3]. The nature of this discourse does nothing to convince the Lineland Monarch of the nature of a Second Dimension, nor in truth, does the emergence of a Sphere into Flatland and the subsequent discourse alter the Square’s belief that there is something beyond the plane in which he lies.

This is in very truth the center of dimensional math. Each dimensional figure can perceive in one less dimension than it inhabits. The Sphere, for instance, will perceive a Cube as a square, while the square perceives another square as a line. By this reasoning, Abbot assumes humankind to exist fourth-dimensionally, since our perception is truly in the third dimension.

Furthermore, as the Sphere explains to the Square, each indicative shape for each dimension is related in arithmetic and geometric progression. (Dimension ,Representative , Shape , Number of sides , Number of Corners , Perception)

The number of sides are related in arithmetic progression (s= 2d, where s is the number of sides and d is the level of dimension). The number of corners is related in a geometric progression (c=2d, where c is the number of corners and d is the level of dimension).

It is this conceptualization of the extrusion of higher dimensional objects into lower dimensions that led to the formulation of quantum string theory and to the introductory work on quantum mechanics. If the analogy holds true for fifth dimensional objects as for fourth, there may be one day an equation which allows for access to the fifth dimension and thus travel at supralight speeds.

Flatland is recognized as one of the greatest works of fiction of the Victorian era. Since its original publication, in both is uncorrected and unannotated form (October 1884) and its later annotated publication (December 1884), it has enjoyed 80 editions in English and has been translated into over 30 languages, including German, French, Hebrew and Japanese. Forewords for the books have been written by such stars in the scientific cosmos as Isaac Asimov (1983), Ray Bradbury (1982) and Banesh Hoffman (1952).

Abbot spent a great deal of time in contemplation not only of the Divine, but also of his society, in which he perceived a fantastic degree in injustice and inequity. Hence, the satirical nature of Flatland may actually outweigh even its mathematical prowess. Social observers and writers of the time, including the renowned Leigh Hunt, saw dimly what Abbot wrote so clearly: that Victorian society was designed by the aristocracy to perpetuate its morals, and paid absolutely no attention to the needs or desires of the underclass. The outrage with which the book was received indicates that more than one home truth is concealed beneath the veneer of the incredible text on multidimensional concepts.

The world in which Abbot found himself was one of temporal and spiritual confusion. England had just recently recovered from the depredations of the Napoleonic wars, and the spiritual world was still reeling from a movement which culminated in the expected return of Christ in 1844[4]. Thus, the spiritual and moral handicaps of the age were no less than the intellectual limitations which the Victorians imposed upon themselves. Shortly after writing Flatland, Abbot commented, “It is very true that the Gospels are neither studied nor accepted among the scholars, for they cannot understand the miraculous and cannot separate the message from the miracle.”[5] It was the intent of this philosopher, scholar and theologian to create, in a work of fiction, a means of separating the message of the Gospels from the miraculous nature of the setting in which the message occurred.

He succeeded beyond all of his wildest dreams, and won for himself a perpetual place about the greatest thinkers of his era.

Edwin Abbot died on January 17, 1926, at his home in London, England.

Works Consulted

Abbot, Edwin A. On Miracles. Chicago: Anon Press, 1982. Original Publication 1882
Abbot, Edwin A. Flatland. New York: Penguin Press, 1988. Original Publication 1884
Abbot, Edwin A. The Kernel and the Husk. London: Chatwicke & Sons, 1887.
Banchoff Thomas F. 1989. From Flatland to Hypergraphics: Interacting with Higher Dimensions [online]. Cited 17 March, 2003. Available on the World Wide Web: http://www.geom.uiuc.edu/~banchoff/Flatland/
, Robert. The Nineteenth Century, A History. Norwood, PA: Norwood Editions, 1989.

[1] On Miracles, p. 17

[2] Flatland, part 1.1 “On the Nature of Flatland”

[3] Flatland, part II.13 “How I Had A Vision Of Lineland”

[4]The 19th Century, A History, p. 135

[5] The Kernel and the Husk, p. 39

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