I am a University of Michigan grad with a BA in English Literature. I am currently attending GVSU to complete a Mathematics minor. In the fall I will begin Grand Valley’s Graduate Teaching Certification Program (the GTC).

The Shape of Social Criticism:

Swift and Abbott’s Use of Geometric Imagery to Dissect Their Worlds

by

Jessica Knoll & Dawn Orbeck

Math 341

Winter Term 1999

Most of us would not commonly consider the subjects of literature and geometry to be close relatives of one another. Whereas literature is considered to exist within the general realm of "the humanities," geometry is considered a pragmatic and logical discipline that would fall into the scientific realm of mathematics. However, when we sharpen our focus, when we truly entertain the question: how are geometry and literature related, we find that they coincide with one another in many of the arenas of our everyday life.

On a superficial level we can easily observe that a narrative piece of literature might have mathematical content. Take, for example, "The Lottery," a short story by Shirley Jackson. This tale revolves around a small village that holds an annual stoning of one member of the village. The ritual's roots are based on superstitious beliefs, and the selection of the particular victim is based upon a complex lottery that operates with several rounds of elimination. This story can easily be enjoyed for its literary components, as a cultural study, or can be broken down on a mathematical level. We can use the facts given in the story to determine what the probability is for a particular member of the community to be selected as the victim. We can determine whether or not the lottery is fair; if everybody has an equal chance of being selected, or whether its result is biased. In any case, there is no doubt that Jackson’s tale demonstrates an intersection of literature and mathematics. This would also be true of any story that incorporates geometric concepts and principles. Among the most famous are Alice’s Adventures in Wonderland by Lewis Carroll, Flatland by Edwin Abbott Abbott, Sphereland by Dionys Burger, Planiverse by A. K. Dewdney, and Gulliver’s Travels by Jonathan Swift.

Conversely, we might ask ourselves if we observe any literary elements inherent in mathematics? The ability to write well, literary knowledge, and an appealing composition style are all, undeniably, necessary components of communicating mathematical ideas, concepts, and proofs both effectively and in a manner that maintains the interest of the intended audience. This is yet another clear indication that the two subjects are not independent of one another.

We also observe relationships between geometry and literature on more abstract levels. Just as letters and words are the building blocks of any literary narrative, points, lines and geometric shapes are the building blocks of any visual narrative. We can easily think of numerous examples that demonstrate visual narratives in the world of art. Take, specifically, the art of quilting. This is perhaps one of the most patent areas where basic geometric shapes are used to create images and scenes that, in turn, work together to reflect a story, theme or idea.

Artist Michael Cummings recently exhibited a collection of his "Narrative Quilts" at the Bates College of Arts in Lewiston, Maine. He uses shapes, textures, patterns, new fabrics, and some with personal or historical significance to applique his "vivacious, provocative and sensitive narratives" upon a brilliantly colored background (Glass, 2).

Historically, a quilt’s inherent ability to communicate a narrative or information has even been exploited for covert purposes. During the Civil War era, a common way for runaway slaves to navigate the Underground Railroad was by means of quilted maps. Sympathizers and abolition activists skillfully disguised maps into the designs of everyday quilts. These quilts were then conspicuously hung in windows and doorways. Only those who knew what to search for were able to divine the messages right before their eyes.

The mathematical language of geometry is also essential in the analysis of poetry and literature, especially when considering the subject of literary structure. Peter Schwenger, a literary critic, highlights the geometric nature of the structure and devices of poetry. He explores the thematic implications of an unnamed and undated painting by the poet William Blake. This painting is commonly referred to as "Elisha in the Chamber on the Wall," or "The Inspiration of the Poet." What is striking about the painting is that although its central focal point is a small scene, (depicting a figure in a tiny room and a man sitting at a table), the real object of the drawing is the creation of ambiguous space. The tiny central scene is near the vanishing point of the piece, where several walls pull the eye toward the tiny scene. The dominant subject of the painting are these walls that seem to "frame" the scene, as if it exists within several surreal boxes. What further complicates this drawing is its shading. The shading of the walls surrounding the scene follow no natural law, making the source of light and true nature of this "box" dubious.

Schwenger cites several critics who throughout time have interpreted this painting as a metaphor for the poetic experience. "The painting makes visible the relationship between ordinary experience and that of poetic inspiration; the two are closely related and in communion with each other, indeed one is in a sense inside the other, but they are also separated by the profound shift of gears necessary to move between them" (100).

The discussion of this painting and Blake’s use of geometry to depict the poetic experience on canvas is then used as a springboard into a larger analysis of the poetry of Blake and Coleridge, relating the "framing" techniques that poets use in actual poems. The painting by Blake is a visual metaphor for the many geometric experiences of poetry and the geometric properties that operate on several levels. Schwenger asserts that there is an inherent square-like nature of Blake’s poetry. Blake structurally creates poetic squares framed inside squares, inscribed within squares. Similarly, Coleridge’s poems utilize circular patterns. As we swirl and cycle inward to the heart of a Coleridge poem, circles appear within circles.

Take, for example, Coleridge’s elusive and mysterious poem "Kubla Khan," (also probably his most famous). The entire atmosphere is framed in enigma, beginning with the title. What then proceeds is a series of figurative circles; each serves to distort the last. Each frame brings the reader a step further removed from reality and transmutes the original meaning and purpose of the text.

The body of "Kubla Khan" mimics this circular pattern of regression that is set up by the poem’s title. Coleridge even goes so far as to embed fragments of his earlier poems into this one. A fragment is embedded within a fragment as Coleridge frames ten lines from his poem "The Picture; or, The Lover’s Resolution" within the brief anecdote of the conversation with Porlock in the first section of "Kubla Khan." Through this technique, Coleridge is able to express the loss of a vision as a simile (a simile being a framing technique in and of itself). The images and words themselves further reinforce the rippling and concentric circling that Coleridge is engaging in on both a structural and stylistic level:

with the exception of some eight or ten scattered lines and images, all the rest had passed away like the images on the surface of a stream into which a stone has been cast, but alas! without the after restoration of the latter:

Is broken—all that phantom world so fair

Vanishes, and a thousand circlets spread,

And each mis-shape[s] the other. Stay awhile,

Poor youth! who scarcely dar’st lift up thine eyes—

The stream will soon renew its smoothness, soon

The visions will return! And lo, he stays,

And soon the fragments dim of lovely forms

Come trembling back, unite, and now once more

The pool becomes a mirror

(Coleridge, excerpt from "Kubla Khan").

Through examples such as Coleridge, we readily observe that geometry exists within literature in terms of the structure of poetry as it appears on a page, as poems are thematically constructed, and in the absolute necessity for geometric language as a descriptor of literary structure and technique.

On a level that is subtler yet, authors may exploit geometric shapes, language, and geometric properties to create moods and even to formulate arguments on a symbolic level. Semantically, an author can use angular and acute descriptors to create a mood of anxiety or, conversely, curved and circular implications to dispense an atmosphere of calm. Stylistically, a writer might choose to write in abrupt and curt segments of sentences, or instead, use a fluid and Joycian stream of language that seems without limit or constraint.

In his 1884 novel Flatland, Edwin Abbott Abbott takes this idea of using geometric properties as an extended conceit (a complex and extensive metaphor that surpasses the normal conventions for a metaphor) to an unprecedented level. Swift, as well, masterfully uses geometry as a symbolic tool for commenting on his society.

Another writer who relies upon mathematics as a model for literature is Lewis Carroll. Most of us are familiar with his famous children’s’ stories Alice’s Adventures in Wonderland (1865), Through the Looking Glass (1871), and What Alice Found There (1871). However, few of us realize how heavily the pages of these stories are saturated with mathematical content.

Carroll lived in the restrictive society of Victorian England for the entire span of his life (1832-1898). Although he is best remembered as an author, he excelled in Mathematics. He was a lecturer and tutor in Mathematics (his primary area of study) at Christ Church in Oxford where he was eventually ordained, although he never proceeded to priests’ orders.

Children’s literature of the Victorian period was riddled with "elements of nonsense and fantasy," and Carroll’s work was by no means an exception (Green, 2). It is very probable that this intense need and love for the fantastical is a reaction to the very strict and formal society of that time. "The Alice books sold steadily through (his) lifetime," and they were popular "for their lack of moralising," (in an extremely moralistic society), "together with their intelligent tone and whimsical quality" (Green, 2). Both of the Alice books have a "commonsensical protagonist, a child insistent on the rightness of the values of middle class society and of elementary education that she takes with her into landscapes which warp, overturn and subvert ordinary perception" (Green, 2).

Buchanan, in his book Poetry and Mathematics, points out that Alice’s Adventures in Wonderland is rife with allusions to mathematical ideas and logic. "The Caterpillar in Chapter V...is a typical teacher of mathematics, and Alice is a typical pupil, in this case doubly surprised and bewildered because she is at the same time the figure to be understood" (Buchanan, 48).

Furthermore, Buchanan asserts that Alice herself is a geometrical figure. "A figure in geometry is the sort of thing that retains an identical character throughout a series of possible transformations…Alice is obviously a geometrical figure on several occasions in Wonderland...This is a highly generalized form of geometry. It is often called positional or projective geometry…interested in the transformations themselves: how a figure retains certain properties such a proportionality of lines and angles throughout given transformations" (Buchanan, 48-53).

Although Carroll clearly exploits the geometric principles of proportion in his work, the author that is probably most associated with projective geometry in literature is Jonathan Swift.

Jonathan Swift, author of Gulliver’s Travels is perhaps the author most identified with mastering the use of sarcasm in literature. Yet, he too uses geometry as a vehicle for his sharply pointed observations.

Swift was not Irish, but seemed to be by some strange influence. "He is Irish by his power of satire and by that reserve that has made the love-interest in his social gift and by his concentrated animosity; he is Irish by his ability for enlisting the populace and of carrying on a propaganda as one might carry on a battle-- by horse, foot and artillery," (Colum viii). Swift did not like Ireland as a residence, but in Ireland he was looked on as a native, while in England he was a stranger. He hated the fact that he was born an Irishman. But he would not accept that the English were in anyway superior to the people they dominated, "the Irish peasantry." He wrote, "the poor cottagers," had, "much better natural taste for good sense, humour, and raillery than ever I observed amongst the like sort in England," (Colum viii). The cause for Irish ruin to him was clearly objective: "Their trade was deliberately crushed purely for the benefit of the English in England, all valuable preferences were bestowed upon men born in England as a matter of course, and finally, that in consequence of this the upper classes, deprived of all other openings, were forced to rack-rent their tenants to such a degree that not one farmer in the Kingdom out of a hundred could afford shoes or stockings to his children, or to eat flesh or to drink anything better than sour milk or water twice in a year," (Colum ix).

Swift was the "first writer in English who was able to mould public opinion,--who was able to do in a pamphlet what the greatest orators were able to do in a speech-- state a public policy with authority, with invincible logic and in a way that would have popular

appeal," (Colum xii). Swift’s most famous work, Gulliver’s Travels, is known as "a satiric masterpiece; it satirizes man’s abuse of human reason as reflected in his political, social and academic institutions; at best, man is foolish; at worst, he is nothing more than an ape; ...it is a bitter denunciation of mankind; ...[a reflection] on man’s corruption of his highest attribute, reason," (Landry). Swift claimed to "hate and detest that animal called man," but to "heartily love John, Peter, Thomas, and so forth," (Landry). Jonathan aimed his witty, imaginative, and often bitter satire at such subjects as politics, literature, and human society.

The authors first adventure was to a country called Lilliput. It is a country where everything is an inch to our foot. The average man stands six inches; the trees, animals, etc. are all of proportional size.

During his stay there, Gulliver slowly gained the trust of the Lilliputians and the Emperor. They feed him as best they could, due to his size. The Emperor took the liberty of appointing learned men to teach the "man-mountain" their language. Tailors were assigned to make Gulliver clothes after the fashion of the country. For the safety of the Lilliputians, he was searched for any weapons that might be conceived dangerous. Gulliver eventually earned his liberty to roam about the country, but under certain circumstances of course.

Here Swift is showing how the little Lilliputians are gentle enough to feed, clothe, and teach Gulliver. Yet for some reason, they don’t completely trust him. It doesn’t seem fitting that such small gentle looking creatures would be cable of such things as imprisoning Gulliver.

There was a time when the Emperor decided to entertain Gulliver with several of the country's shows. There was one especially that caught his eye, it was that of the rope-dancers. This feat was "performed upon a slender white thread, extended about two foot, and twelve inches from the ground," (Swift 29). This "is only practiced by those persons who are candidates for great employments, and high favour , at court," (Swift 29). They are trained for this from youth, and are not always of noble birth, or liberal education. When a great office is vacant, for whatever reason, "five or six of the candidates petition the Emperor to entertain his Majesty and the court with a dance on the rope, and whoever jumps the highest without falling, succeeds in office," (Swift 29). Sometimes the people already in office are called upon to exhibit their skills, and to show that they have not lost their talent. It is mentioned that there are sometimes fatal accidents that do occur. Gulliver himself saw two or three candidates break a limb.

There was another diversion, that was only shown before the Emperor and Empress, and first minister, upon particular occasions. "The Emperor lays on a table three fine silken threads of six inches long. One is blue, the other red, and the third green," (Swift 30). The threads are to be prizes for the persons whom the Emperor deems worthy. The ceremony takes place in his Majesty’s great chambers of state, where the candidates undergo a trial of dexterity very different form the rope-dancing. "The Emperor holds a stick in his hands, both ends parallel to the horizon, while the candidates advancing one by one, sometimes leap over the stick, sometimes creep under it backwards and forwards several times, according as the stick is advanced or depressed," (Swift 30). Whoever performs the best and longest in the leaping and creeping is rewarded "with the blue-coloured silk; the red is given to the next, and the green to the third, which they all wear girt twice round about the middle," (Swift 30-31).

In Gulliver’s Travels, the climbing, creeping, and rope-dancing are ways of winning court-favors. In England blue, red, and green, ribbons represent the great orders of the Garter, the Bath, and the Thistle. These are religious representations. The way that Swift put the Lilliputians to win honors in court is ridiculous. People are willing to hurt themselves to deem themselves worthy. In real life a lot of the time people are that way, and they don’t see the bending-over-backwards that they do to be silly in any way. Other people might see their actions differently, and would never do such a ridiculous thing.

There had been conflict in Lilliput, and Gulliver was called upon to help. It seems that there had been two struggling parties in the empire, "under the names of Tramecksan and Slamecksan, from the high and low heels on their shoes, by which they distinguish themselves," (Swift 41). "The animosities between these two parties run so high, that they will neither eat nor drink, not talk with each other," (Swift 42). The Tramecksan, or High-heels, exceed the Slamecksan, or Low-heels in number. The Lilliputians, being of Low-heel, exceed the High-heels in power. The other great empire Blefuscu (High-heel) are threatening Lilliput with an invasion.

The two mighty powers have been engaged in an "obstinate war for six and thirty moons past," (Swift 42). It began such that the primitive way of breaking eggs before they ate them was upon the larger end. The Majesty’s grandfather, while he was a boy, opened an egg according to the ancient practice, and happened to cut one of his fingers. Whereupon the Emperor, his father, "published an edict, commanding all his subjects, upon great penalties, to break the smaller end of their eggs," (Swift 43). The people so highly resented the law that there had been six rebellions, raised on the account. These commotions were said to have been instigated by the monarchs of Blefuscu. When the people were exiled for their beliefs, they sought refuge to that empire. "It is computed, that eleven thousand persons have, at several times, suffered death, rather than submit to break their eggs at the smaller end," (Swift 43).

Many books had been published about the controversy, but the books of the Big-Endians had been long forbidden. During the course of all of the troubles, the emperors of Blefuscu frequently accused the Lilliputians of "making a schism in religion, by offending against a fundamental doctrine of [their] great prophet Lustrog, in the fifty-fourth chapter of the Blundercral (which is their Alcoran)," (Swift 43). It is thought to be a "great strain upon the text: for the words are these; That all true believers break their eggs at the convenient end;" (Swift 43), which is the convenient end is left to each man.

The Big-Endians and the Little-Endians disputed in Lilliput as the Catholics and the Protestants disputed in England in Gulliver’s time. The Alcoran, is a sacred scripture of Islam. Muhammad, who told of his revelations, applied the name. Secretaries would write the revelations down, and his followers memorized them. The collection as it is now was compiled by his followers a few years after his death in 632. The Alcoran is the earliest known work in Arabic prose; it is divided into 114 suras (chapters) of various lengths and contains the Islamic religious, social, civil, commercial, military, and legal codes. The chief doctrines laid down in the Alcoran are that only one God and one true religion exist. The Alcoran is above criticism and it is a work not to be proved but itself the standard of merit.

Gulliver’s second adventure took him to a land known as Brobdingnag, were everything is a foot to our inch. The average man there stood a good seventy-two feet high, and everything else in the country was proportional. Gulliver found himself running to hide in some corn so that the giants wouldn’t find him.

Gulliver tried to advance himself through the corn, but it was so tightly intertwined at parts that it was impossible to pass through. Gulliver was forced to lay down between to ridges, and wish that there was where he would end his days. He thought back to his time spent in Lilliput, whose inhabitants looked upon him as though he was the greatest prodigy they had ever seen in the world. Then he thought about how inconsiderable he was in this nation, as compared to a Lilliputian in ours. He conceived that was the least of his misfortunes, for "as human creatures are observed to be more savage and cruel in proportion to their bulk, what could I expect but to be a morsel in the mouth of the first among these enormous barbarians that should happen to seize me,"

(Swift 86). "Undoubtedly philosophers are in the right when they tell us , that nothing is great or little otherwise than by comparison," (Swift 86).

Scared as he was, Gulliver found himself to be in the path of one of the reapers. With the next step he could be squashed by a mammoth foot, or cut in two with the reaping-hook. So he let out a scream as loud as fear could make him. The creature stopped, and looked round about under him for some time, when at last he found Gulliver. The giant took his time in deciding how to pick up Gulliver, for the fear of getting bitten or scratched by this small animal. Finally he was taken up between the forefinger and thumb, and was brought in about three yards from the eyes so that the giant could get a better look at him.

Gulliver did not put up a fight for fear of being tossed upon the ground, as one would do to "any little hateful animal which we have a mind to destroy," (Swift 88). The giant seemed pleased with Gulliver’s voice and gestures, and began to look at him as a curiosity. He then put Gulliver in the lappet of his coat, and immediately ran along to his master. Who happened to be a farmer, and the same person he had first seen in the field.

Having been taken to the farmer, Gulliver was curiously examined. The farmer decided to keep him as sort of a pet. He was fed, and got to know the family. The daughter ended up taking care of him mostly. She made him clothes, and taught him the language. Gulliver called her his Glumdalclitch, or little nurse. Word spread fast that his master had found a strange animal in the field, exactly shaped like a human only smaller. It seemed to imitate all of the humanistic actions, "speak in a little language of its own, had already learned several words of theirs, went erect upon two legs, was tame and gentle, would come when it was called, do whatever it was bid, had the finest limbs in the world, and a complexion fairer than a nobleman’s daughter of three years old," (Swift 96-97).

The great brute size of the giants led Gulliver to believe that they would treat him as a brute. Just as the Lilliputians thought Gulliver would treat them. He soon comes to find that they do not, in any way, intend to harm him.

Another farmer that lived near by came to visit on purpose to enquire into the truth of the stories. Gulliver was immediately produced, and placed upon a table, where he did everything that was commanded. The neighbor suggested that Gulliver should be brought about as a sight. The next market day, Gulliver was brought about, with his little nurse close at hand. The reception that the farmer got from showing Gulliver was so great that he continued to show him at different towns frequently. Until one day a Gentleman Usher came from court, commanding the farmer to carry Gulliver immediately to the Queen and her ladies. Whereupon she was so delighted that she proposed to buy Gulliver if the farmer was able to sell him. Taking the Queen up on her request, Gulliver was sold. Now having some knowledge of the language of the people, Gulliver requested that his little nurse might be able to stay and take care of him. With the farmer’s consent, the little girl was now a part of the court.

Gulliver was a part of the court now too. It was made so that he had every comfort that they could provide for him. A small box was made for him to reside in. It was "a wooden chamber of sixteen foot square, and twelve high, with sash-windows, a door, and two closets, like a London bed-chamber," (Swift 110). The ceiling was made to be lifted up and down by two hinges. The chamber was also quilted on all sides so as to resist jolts that might come to him when he went in a coach. The Queen ordered the thinnest silks that could be gotten to make him clothes. Although they were a little thicker than an English blanket, he became accustomed to them.

Something that he did not become accustomed to was the Queen’s dwarf, "who being of the lowest stature that was ever in that country... became insolent at seeing a creature so much beneath him, that he would always affect to a swagger and look big as he passed by me in the Queen’s antechamber... he seldom failed of a smart word or two upon my littleness; against which I could only revenge myself by calling him brother, challenging him to wrestle, and such repartees as are usual in the mouths of court pages," (Swift 111-112). The dwarf played many tricks on Gulliver; dropping him in a bowl of cream, and wedging his legs into bone with the marrow knocked out. It seems as thought the dwarf could not deal with the new reality that he was not the smallest in the kingdom anymore. He was soon sent away, after many whippings for the harm he had done to Gulliver.

The comparison between Gulliver and the Queen’s dwarf is amusing. The dwarf does not know what to do now that he isn’t the littlest. The tides are turned when the littlest of the big people turns out to be the cruelest.

Many other hardships were encountered by Gulliver in Brobdingnag. "I should have lived happy enough in that country, if my littleness had not exposed me to several ridiculous and troublesome accidents," (Swift 120). One day while he and his little nurse ventured into the garden, the dwarf followed behind. Gulliver noticed some dwarf apple trees nearby, and couldn’t resist making the comparison to the dwarf himself. Upon which the dwarf took it upon himself to shake one of the trees as Gulliver walked under it. Causing dozens of apples to come tumbling down about him, one of them hitting him on the back and knocking him down flat on his face. Another day while Gulliver was left by himself in the garden, a sudden violent shower of hail fell upon him. It immediately struck him to the ground, and gave him many cruel bangs all over his body, as if he had been pelted with tennis-balls. Yet another time when he was left to himself, a small white spaniel belonging to one of the chief gardeners snatched Gulliver up in its mouth. The dog had been so well taught that he was carried between his teeth without the least hurt. The poor gardener was in a terrible fright as his dog brought back his prize. Many other misfortunate accidents happened, but to tell them all would be of great length here.

Gulliver is like an ant in Brobdingnag. He is constantly being overlooked. There are a great many dangers for him to inquire do to his size. He always finds help from the great friendly people of the country.

The King of Brobdingnag was a prince of excellent understanding. He many times requested that Gulliver be brought to him in his little box so that they may converse. One day Gulliver took the liberty to tell his Majesty "that the contempt he discovered toward Europe, and the rest of the world, did not seem answerable to those excellent qualities of the mind he was master of," (Swift 131). Gulliver then told his Majesty that tallest of men in his country were usually least provided with such thought. Also, "among other animals, bees and ants had the reputation of more industry, art and sagacity, than many of the larger kinds," (Swift 131). As inconsiderable as Gulliver took himself to be, he hoped that he might live to do his Majesty some signal service. The King heard him with attention, and began to conceive a much better opinion of him than he had ever before. In taking into thought everything that Gulliver had told him, the King spoke these words to him; "My little friend Grildrig, you have made a most admirable panegyric upon our country; you have clearly proved that ignorance, idleness, and vice, are the proper ingredients for qualifying a legislator: that laws are best explained, interpreted, and applied by those whose interest and abilities lie in perverting, confounding, and eluding them... It doth not appear from all you have said, how any one virtue is required towards the procurement of any one station among you; much less that men are ennobled on account of their virtue, that priests are advanced for their piety or learning, soldiers for their conduct of valour, judges for their integrity, senators for the love of their country, or counselors for their wisdom," (Swift 132-133). The King then concluded from what he had gathered that the "bulk of your natives to be the most pernicious race of little odious vermin that nature ever suffered to crawl upon the surface of the earth," (Swift 133). After all that was said, Gulliver requested his freedom. to which he was not granted.

For this next adventure, Gulliver went to a very strange land, an island that floats. He stayed there for a while only to leave soon after his arrival. He ended up on an island named, Glubbdubdrib. As nearly as he could interpret the word, it signifies the Island of Sorcerers or Magicians. The name fits the island on the account that, "the Governor and his family are served and attended by domestics of a kind somewhat unusual. By his skill in necromancy, he hath a power of calling whom he pleaseth from the dead, and commanding their service for twenty-four hours, but no longer; nor can he call the same persons up again in less than three months, except upon very extraordinary occasions," (Swift 201-202).

Now that you have an idea of how strange this place was, I’ll explain the bit of geometry that I found to stick out. The people of the land have what they call, "Struldbrugs," or "Immortals." Very rarely, a child will happen to be born in a family with a red circular spot in the forehead, directly over the left eyebrow, which was an infallible mark that it should never die. The spot, as described, "is about the compass of a silver three pence, but in the course of time grew larger, and changed its colour; for at twelve years old it became green, so continued till five, and twenty, than turned to a deep blue: at five and forty it grew coal black, and as large as an English shilling, but never admitted any further alteration," (Swift 207). By the time the strudlbrugs reached thirty years old, they grew melancholy and dejected, increasing in both till they came to fourscore. When they come to fourscore (80 years old) they grow dead to all natural affection. Finally realizing that they will never die, and they are considered dead in law.

The example of the strudlbrug is typical of Swift. The spot on the forehead singles the person out from society, it changes as the person changes. As the person grows and gets older, so too does the spot. The spot is smaller and bright colored in youth, gradually changing and growing bigger and uglier with age. It signifies how we change with age. In youth we are fresh and impressionable. The older we get, the more set in our ways we are and the more stubborn we are. Only to come to a miserable end, with little memory, and ready to die. The strudlbrugs never get to die. Their memories decline like the rest of ours do, and so do their bodies. To top it off, they are treated like they don’t exist once they reach the age of eighty years old. Much like we put a lot of our elders in "homes" once they are a menace for us.

This last adventure in Gulliver’s Travels was to a land of the Houyhnhnms. It is essentially a land of horses. The horses in this land are not the domestic animals that we have made them today. They play the role of the rational animal, while a human like creature, called a Yahoo, is the irrational animal or beast. Gulliver stayed there for about three years. In his stay there, he learned the language of the Houyhnhnms, and had many conversations with his master there.

Upon one conversation, Gulliver’s master asked him to explain more about this thing called war. The Houyhnhnms were peaceful creatures, so they did not understand the concept of war. Once Gulliver was done explaining what war was, his master thought he was lying to him. He thought that due to our beliefs war is justified, but that the shame is greater than the danger and that nature had left us utterly incapable of doing much mischief. "For your mouths lying flat with our faces, you can hardly bite each other to any purpose, unless by consent. Then as to the claws upon your feet before and behind, they are so short and tender, that one of our Yahoos would drive a dozen of yours before him," (Swift 259).

The example that swift is making here is that the shape of an animal does not matter as to it rationality. The way in which an animal puts it mind to use is what matters. The horses in that country looked just as those in our country, but the Yahoos looked somewhat different from human beings of today. That is in part due to the evolution of the human in that country not evolving as far as we have today. With the horses running the country, nothing "bad" was going on. They settled their differences upon reason and thought, not with war and blood. So maybe we as humans need to take a step back and look at what we really are doing with our minds.

Travels of Lemuel Gulliver into remote Nations of the World (1726) or more commonly known as Gulliver’s Travels , Swift’s masterpiece, is commonly considered a children’s story but was originally intended as a satire on humankind. Swift’s satire was originally intended as an allegorical and acidic attack on the vanity and hypocrisy of contemporary courts, statesmen, and political parties. The writing of this book was said to have taken more than six years, in which he incorporated his ripest reflections on human society. This is a savagely bitter work, mocking all humankind. "Nonetheless, it is so imaginatively, wittily, and simply written that it became and has remained a favorite children’s book.

Gulliver’s Travels is all an illustration of one idea firmly held: Swift looks from above down upon men; he looks from below up at them, and then he looks at them from behind. There are four voyages in which particular institutions and policies, special crimes and vices are exposed to our mockery and our disgust. These exposures are only side-attacks. The satire is in the world which Swift creates over and against our own; a world in which there is no place for our affection. By creating such a world he has made a great general satire.

He creates this world most triumphantly when he alters the measurement of man; he creates it most horribly when he gives beasts the reason of men and gives men the habits of beasts. He creates it again when he makes human reason occupy itself solely with frivolous things. The world in which there is no room for our affection creates a satire upon our world, the atmosphere of which is human affection. We can feel nothing for beings so small "that they would be infants to our infants, and so large that they would be higher than our houses; so pedantic that they would weary our reason and so foul that we should shrink away from them," (Colum xiv). If the Lilliputians measure up to humanity inch for foot and if the Brobdingnagians tower above them foot for inch, everything in their respective countries are exactly to scale.

A set of imaginary voyages modeled after the real accounts of voyages written by English seamen. "It has been read by children as a wonder-tale and by statesmen in exile as a piece of secret history," (Colum xix). The voyages have been made over, as often before, into a story for children. "For them Gulliver will bestride a country, and will walk through streets taking care that his clothes do not brush down the houses; he will be put into a giant’s pocket, and will be carried away, house and all in an eagle’s beak; he will come to the flying island, and afterwards will be spoken to by the wise horses," (Colum xix).

This story book is a "fairy tale inverted." In this fairy tale the little beings have beauty and graciousness, the giants are dull-witted, and the beasts are helpful, and humanity is shown as triumphant. Yet, "the little beings are hurtful, the giants have more insight than men, the beasts rule, and humanity is shown, not as triumphant, but as degraded and enslaved" (Colum xvii).

Children’s literature, in general, has many authors that use scale and size to say what it is they want to get across. Roald Dahl, a British writer of novels, short stories, and film scripts, but best known for his children’s books is such an example. He was born in Wales, and educated at Repton, a boarding school for boys. "His harsh treatment while a student there led him later to write stories about cruelty and revenge," (Microsoft). Dahl did not attend a university, rather he worked for Shell Oil Company in 1933. Later he enlisted in the Royal Air Force (RAF) at the start of World War II, and served as a fighter pilot and as an air attaché in Washington D.C.

During his years in the RAF, he wrote such things as Gremlins, and some adult fiction books. Dahl was the author of 19 children’s books, the best know of which were James and the Giant Peach, and Charlie and the Chocolate Factory. His most recent works included Fantastic Mr. Fox and The BFG.

The BFG is a fantastic story of a little girl being snatched up out of her bed by a big friendly giant, and their adventures. This is a story for children of all ages to enjoy. Sophie, the little girl, was fortunate that she had this nice jumbly giant snatch her up, and not one of the other giants. All of the other giants eat "human beans!"

There are few good examples of geometry used in this book. For instance, the big friendly giant (BFG), is merely half the size of the other giants. He is much much kinder and gentler than the other giants. (This is the opposite concept that was held in Gulliver’s Travels.) The BFG is never going off to eat "human beans" at night like the rest of them. The other giants are fifty-some feet in height. They are disgusting with only a loincloth covering their burnt brown skin. Their bellies were large, they had long arms and big feet, their eyes are tiny black holes, the noses were small and flat, but their mouths were spread almost ear to ear.

The descriptions of the people eating giants is fitting. They need the long legs and big feet for running, the long arms to snatch people up quickly, and the huge mouths to eat so much. Size description sets the mood for the gruesome giants.

One thing that troubles me about this book is that there are no female giants. Giants just seem to appear out of nowhere in thin air. They have no mother, much less a need for one. Is this lack of femininity due to the colossal size of the giants? The size of the giant does not make up for such a loss in my mind. I have yet to figure out why this is so.

A trait that stands out about the BFG is that he has enormous ears. With these ears he can hear anything. He goes about during the day catching dreams in dream country. From the aid of his ears he can hear them whizzing by. Once he has caught a dream he can tell what the dream is about by listening to it very carefully.

The BFG’s ears lend him help in his job of dream catching. Just like height helps basketball players jump for a rebound, and long legs help runners run faster.

Size and scale are great tools for writers to use in children’s books. It can help to clarify an idea or emphasize something that would otherwise be overlooked. Comparing things is the only way to tell if something is big or small. Imaginations run wild, especially in children. Giving them something to start with, and then letting them find a way to put it to use stimulates their creativity.

In his 1884 novel Flatland, Edwin Abbott Abbott takes the idea of using geometric properties as an extended conceit to an unprecedented level. He constructs for his reader an entire two-dimensional world that symbolically represents his own Victorian world.

Abbott, "a classicist, theologian, and Shakespeare scholar whose hobby was the study of higher mathematics" (Banchoff, jacket), lived from 1838 to 1926 (roughly the same time as Lewis Carroll). He worked as the headmaster of the City of London Schools, and in addition to his work as an author, he was an outspoken intellectual radical for his times. "Abbott was a social reformer who criticized a great many aspects of the limitations of Victorian society. He was a firm believer in equality of educational opportunity, across social classes and particularly for women" (Banchoff, xvii). He strove to understand the connections between what might appear to be radically different subjects. Abbott promoted higher education for women and access to an education for all people. He disapproved of the stronghold that the aristocracy maintained upon the arts, refinery and wealth. He even suggested that "the techniques of literary criticism developed by classical scholars should be used with the same vigor to examine the Scriptures" (Banchoff, xxi). His interest in mathematics and its relationship to society and literature, resulted in Abbott’s classic 1884 novel of a two-dimensional world in which the narrator, A. Square, comes into contact with beings from different dimensions, Flatland. In this novel, Abbott uses the restricted two-dimensional world as an allegorical representation of his restricted Victorian world. His novel challenges readers to explore complex mathematical concepts, exercise their ability to interpret his literary allusions, and to think critically about his social world, and in turn, our own.

Mathematically, Abbott demands a high level of abstract conceptualization from his readers. He reminds us that we normally do not put much thought into our three-dimensional world. We take its properties, freedoms and restrictions for granted. In Flatland, the reader is forced to concentrate on the properties that make up three-dimensional life, as well as, imagine what living beings would experience living in two-dimensions, (and even one dimension or none).

For the greatest portion of the novel, the reader is invited to envision life in two-dimensions. Appropriately, this land is named Flatland. However, we also visit Pointland (a world of no dimension), Lineland (a world in one dimension), Spaceland (our familiar three-dimensional world), and Thoughtland (the land(s) of four or more dimensions). In each case, the reader will note the elaborate and complex nature of relating a world of different dimension to a being that lives within a different dimension. We see that just as our narrator from Flatland struggles and fails to enlighten a being from Lineland, so a being from Spaceland struggles with the narrator. Abbott’s discussion of Thoughtland is actually self-reflexive in nature because as A. Square struggles to articulate thoughts and concepts relating to four or more dimensions, we see that Abbott is struggling to articulate these thoughts to his audience in an accessible and lucid manner. He brings his readers through several patterns, explanations and analogies, yet still is conspicuously at a loss for the right words. He describes the elusive nature of glimpses into extra dimensions through the narrator’s own gradual loss of three-dimensional visualization.

Flatland is Abbot’s exposition of the quarrels with the society and time in which he lived. His narrative follows several strands of criticism in making an argument against the restrictive nature of Victorian England. Abbott is able with incredible insight and precision, to step back and away from the society in which he lives and critique it as if with the benefit of removal in time and space. He sees the greater repercussions over time of the existing inequities of the Victorian era and tends to think in terms of prescriptive long-term solutions.

Flatland is divided into two sections. In the first section, A. Square, describes Flatland and its inhabitants. In the second section, A. Square describes his experiences and thoughts regarding worlds of different dimensions.

The purpose of the first section of Abbott’s novel is twofold. Abbott’s description of life in two dimensions is entertaining and ingenious. It coaxes his audience into pondering the restrictions imposed by the number of dimensions in which a being lives. However, Abbott’s description of Flatland and its inhabitants is not as innocuous as it might first appear. Every facet of Flatland that Abbott explores, parallels a facet of Victorian England that he wishes to condemn. He parodies his society through overstatement, sarcasm, and exaggeration, translating the absurdities and restrictions of his world into the more absurd and restricted world of Flatland.

The second section of Flatland is more mathematically and conceptually challenging for the reader than the first. We are forced to jump from one level to another in time and space. Allegorically and symbolically, this section has its purpose too. It explores the value and difficulty of independent thought and challenging a society’s status quo. As A. Square learns about other dimensions and modes of existence, his own life takes on greater meaning and significance. However, as he begins to publicly espouse his ideas and knowledge as a result of his enlightenment, he is ridiculed and eventually considered a criminal.

In the first section of Flatland, Abbott gradually eases his audience into a two-dimensional world through the narrator’s detailed and thorough descriptions of his world. A. Square begins by introducing himself and his land:

Based upon knowing the restrictions that living in two-dimensions would impose on a three-dimensional being, the narrator describes every facet of his world that would be foreign to us. He explains the different shapes and corresponding different classes of people. He tells us how homes are constructed, what the weather is like. He describes social interactions and how people are able to recognize each other given the fact that in a two-dimensional world, every figure will appear as a straight line. We find out what becomes of irregular shapes in Flatland. We learn of the strict and often outright ridiculous social codes that govern the land. The narrator also gives us some pertinent historical information about his land, especially pertaining to place of color in his world.

Abbott condemns the rigid and hierarchical social system that he lives in through his explanation of the social structure of Flatland. In Flatland, a person’s social class is determined by how many sides they possess. (The greater the number of sides one possesses, the greater is one’s social standing).

Isosceles triangles are the lowest class of men. The more acute their angle, the less intelligent and socially valuable they are. The members of the military and law enforcement are comprised of this class of men. They are considered, because of their low intelligence, to be a threat socially, but mostly to one another. Because of their sharp angles it is easy for them to impale and murder one another. They are visually the most threatening-looking and, therefore, it is the stereotype among the upper classes that they are indeed more "inhuman" and dangerous. The argument is made that their lives are also more "dispensable."

Abbott’s use of cold and harsh language, coupled with the overstated sentiments of the narrator, serve as flags to his reader that his personal beliefs do not parallel those expressed by the Flatlanders. He is, conversely, pointing out the ignorance of those in his own aristocratic world that stereotype lower class people as simple-minded and less worthwhile human beings. Abbott further reinforces his point by making a distinction between how the upper class and the lower class recognize one another in Flatland. While the upper class uses sight to recognize and identify one another, the lower class uses feeling. Metaphorically, Abbott is pointing out that the aristocracy uses superficial and surface criteria to judge other people, while the working class defines people by their essence. They are able to feel and live more completely, maintaining a fuller existence and understanding of each other.

Equilateral Triangles are typically tradesmen, and considered to be middle class. "Professional Men and Gentlemen are Squares...and Five-Sided Figures or Pentagons" (Abbott, 8). The Nobility, "of whom there are several degrees, beginning at Six-Sided Figures, or Hexagons, and from thence rising in the number of their sides till they receive the honorable title of Polygonal, or many-sided" (Abbott, 9).

Abbott, through his description of the middle-class, is not nearly as acidic and harsh. His descriptions are less sarcastic and condemning than those used in the description of the working class. However, he does point out the manic ambition of the middle class Flatlanders, who are obsessed with superficially promoting themselves to a higher class; in some cases, to such an extent that they kill themselves trying. A. Square describes "Circular Neo-Therapeutic Gymnasiums," to which many Flatlanders send their newborn children. In such an institution, the delicate sides of an infant can be broken and reset in the hopes of adding extra sides to the child and, therefore, adding a social degrees of status (Abbott, 45). Despite the fact that "scarcely one out of ten survives…many a promising child is sacrificed…so strong is the parental ambition among those Polygons" (Abbott, 45).

This dangerous fight against nature and these desperate measures taken to live up to a superficial social ideal are Abbott’s means of pointing out the ludicrous and inherently damaging nature of a society defined and driven by social status. His own hierarchical society was just such a place and Abbott warns his reader that we kill our very human core by refusing to be the individuals that we were naturally born to be.

Finally, the highest socially ranking individuals are Circles. "When the number of the sides becomes so numerous, and the sides themselves so small, that the figure cannot be distinguished from a circle, he is included in the Circular or Priestly order; and this is the highest class of all" (Abbott, 9).

All women in Flatland are straight-line segments (not shapes at all). They possess the lowest social standing, command the least respect, and have little, if any, power or authority in Flatland. They are thought to be merely emotive creatures, without the necessary cognitive and logical skills to grasp any abstract ideas or engage in academic pursuits. "About three hundred years ago, it was decreed...that, since women are deficient in Reason but abundant in Emotion, they ought no longer be treated as rational, nor receive any mental education. The consequence was that...they sensibly declined during each generation in intellectual power" (Abbott, 49).

A. Square’s description of Flatland females (straight-line segments, with a sharper point at one end than the other, and possessing an eye at one end), is suspiciously similar to how one might describe a sewing needle. This is no coincidence. Abbott intentionally described Flatland females in this manner because the needle is a powerful symbol of Victorian femininity. It forcefully invokes a notion of constrained potential.

Abbott, an advocate for higher education for the women of his times, obviously saw women as potentially beneficial contributors in the realm of academia and his society at large. Unfortunately, due to the social prescriptions and restrictions, women were not given an opportunity to develop in this realm.

Wolfert, of the University of Toronto, writes that needlework was the only "finer art" in which women could participate (and it was exclusively women who practiced this art). A woman’s skill in needlework was extremely important. It "represented gentility and a feminine capacity to beautify surroundings" (Wolfert). On a more sinister note, it also "represented the confinement of women to the domestic sphere" (Wolfert). She also states that women were severely judged based upon their level of skill in embroidery. She notes that it was such an important measuring tool of a woman’s social worth that there are recorded instances in which engagements were called off by disillusioned suitors that found out that their prospective brides were lacking in their needlework skills.

This restriction of women to one sphere of artistic expression, and judging them so heavily by this one permitted, (or more accurately required), area of production, reduced them to one-dimensional creatures (in terms of the constructs of Victorian society). Abbott visually manifests this position by portraying women as line segments.

A. Square also devotes a great deal of attention to the inherent danger of women because of their shapes. Women are able to impale and kill other figures with the greatest ease because of their needle-like form. Metaphorically, Abbott reminds his reader what has been demonstrated across time and throughout the history of peoples: The more steps that a society takes to undervalue, underestimate and restrict a segment of its population, the more dangerous this group becomes to the society. It becomes inherently more dangerous precisely because it does possess powers and abilities greater than the society gives it credit. When this group decides to rise up and use those powers and abilities, the assumption and foundation upon which the society has built itself, will crumble from below. Abbott implies that by reducing women to one-dimensional creatures, the society turns them into dangerous weapons against itself.

This point is further reinforced by Abbott’s conspicuous recycling of misogynistic Renaissance tropes and his conquering of them in "Part Two."

Our narrator points out that another inherent danger of women’s lack of multiple sides is the possibility of "hidden irregularities." If a woman in Flatland possesses any type of geometrical irregularity, it cannot be observed due to her lack of shape. However, this irregularity could be passed on to a son in whom the irregularity would become manifest. For this reason, Flatlanders trace back the lineage of women across several generations to make sure that a woman is safely regular before a marriage takes place. This danger is cited as yet another reason that women are inferior creatures to be feared.

This discussion echoes an entire genre of literature obsessed with the imperceptible sexual status of women. The Elizabethan tragedies are riddled with such concerns. In fact, in many plays, it is the dubious sexual status of a female that sets off the chain of tragedies. During this historical period, women were considered little more than pieces of property whose commercial values were directly tied to their sexual purity or lack thereof. Due to this absolute need to keep women "pure," or at least appearing so, men at this time became obsessed with attempting to gage what is unable to be physically observed: a woman’s chastity. This, and the consequent frustration and impossibility of this task, led to hyper-critical resentment and frustration with the female sex as a whole and this sentiment of female deception and duplicity is readily observed throughout Renaissance literature.

Abbott calls upon these notions through specific literary allusions to Shakespeare, in particular, Hamlet and the suicide of Ophelia. However, it becomes clear that Abbott does not condone these misconceptions of women in his second section. When A. Square visits a higher dimension (symbolically a world of more enlightened thought), he learns from his guide that in this higher dimension, the stereotypical characteristics that have historically brought women under fire, are considered higher levels of cognitive operation. Through this example, Abbott rejects the historical notion that women, as more emotional and less rational creatures than men, are socially less valuable. Instead, the point is that women are more readily able to transcend the logical and pragmatic world of limited thought and experience cognitively at a more complex level rather than a less complex one.

In Flatland, irregular figures are considered subhuman. Many are without human rights and are used as dispensable subjects for scientific studies or for any other purpose that can be dreamed up:

Abbott lived in a world that was particularly cruel to individuals with mental and physical disabilities. Individuals were placed into "lunatic asylums" at alarming numbers. Moreover, the asylums became a zoos of sorts for the aristocracy. They would plan weekend visits to tour the asylums and observe with disgust and delight the dysfunctions of those incarcerated.

Through Abbott’s Swiftian exaggeration and overstatement regarding the status of Flatland irregulars, his distaste with the cruel treatment of disabled individuals in his society becomes clear.

Section Two of Flatland begins as A. Square describes a dream that he has experienced in which he visited Lineland, a one-dimensional world. In his dream he struggles to explain two-dimensional life to the inhabitants of Lineland who experience everything on a straight, flat line. He is frustrated with his inability to make the Lineland creatures understand. He feels that this dream was a premonition for the visitor that he receives in the second section. Throughout this section, A. Square narrates his experience of coming into contact with a Sphere. With the Sphere, A. Square travels into three-dimensional space, ponders four or more dimensions and even briefly visits Pointland, a world without any dimension whatsoever.

Allegorically and symbolically, this section explores the value and difficulty of independent thought and challenging a society’s status quo. As A. Square learns about other dimensions and modes of existence, his own life takes on greater meaning and significance. However, as he begins to publicly espouse his ideas and knowledge as a result of his enlightenment, he is ridiculed and eventually considered a criminal. Because the other members of his land are constrained in their shallow thoughts and unable to see beyond the horizon of their short existence, they are unequipped to grasp A. Square’s meaning. The consequence of his beliefs is that he is perceived to be mad and dangerous and he is condemned to spend the rest of his existence in prison. Through the fate of A. Square, Abbott expresses his own fear and frustration. As a member of the society who "thinks outside of the box," Abbott realizes that most of his contemporaries are unable to view the world with the same perception that he does. The lenses that he wears see beyond the constraints of the present and see through the superficial edifice of the Victorian world. Yet he must remain alone, a prisoner of his own insight and unable to shift the attitudes of his contemporary peers. "Yet I exist in the hope that these memoirs…may find their way to the minds of humanity in Some Dimension, and may stir up a race of rebels who shall refuse to be confined to limited Dimensionality. That is the hope of my brighter moments" (Abbott, 102-3). Through these closing words of A. Square’s, Abbott cries out for social action and societal reform.

In conclusion, we have only touched the tip of the iceberg concerning the relationships between Geometry in Literature. Indeed, our everyday language is a testament in itself to the great debt that literature and language have to geometry in helping articulate and express everyday concepts and ideas. When we make a conscious effort to make ourselves aware of this presence, we can observe it every day.

Abbott, Edwin Abbott, Flatland. Princeton, New Jersey: Princeton University

 Banchoff, Thomas "Flatland: A New Introduction" (an introduction to Flatland).

Bohning, Gerry and Williams, Rebecca, "Quilts and Tangrams: Linking Literature

Buchanan, Scott, Poetry and Mathematics. Chicago: University of Chicago

Colum, Padriac, "Introduction To ‘Gulliver’s Travels" (an introduction to Gulliver’s Travels. New York: The Macmillian Company, 1917.

Dahl, Roald, The BFG. Middlesex, England: Puffin Books, 1987.

Glass, Marc, "Cummings to Discuss ‘Narrative Quilts’ at Bates Museum of Art."

Green, Ruth, "Lewis Carroll: a Celebration." BBC Online Author Factfile,

Jackson, Shirley, "The Lottery."

Landry, Peter, "Biographies, Jonathan Swift."

Microsoft Encarta 98 Encyclopedia, "Jonathan Swift," and "Roald Dahl.",

Poincare, Henri, Science and Hypothesis. Dover Publications, Inc., 1952. 

Schwenger, Peter, "Blakes’s Boxes, Coleridge’s Circles, and the Frame of

Swift, Jonathan, Gulliver’s Travels. New York: The Macmillan Company, 1917.

Wolfert, K., "Needlework and the Construction of Femininity." University of

"Jonathan Swift, 18th Century English Satirist"

One useful way for students to explore the relationships between Geometry and Literature is through trying their own hand at interpreting geometrical literary metaphors.

Using the lands of different dimension described by Abbott in Flatland, divide students into five groups so that each group corresponds to one land: Pointland, Lineland, Flatland, Spaceland and Thoughtland, respectively. Providing each group with Abbott’s metaphorical description of the land, ask the students to decide in a group what the characteristics and properties of their land are. In particular they should pay attention to what beings in their world "see" and how they could explain their world to a being from a different dimensional world. Students should be provided with props that can aid them in exploring, characterizing, conceptualizing and explaining their world.

After the groups have had some time for investigation, ask each group to share their findings with the class.

A wall display that can help in creating thought can contain several questions about off the wall things. Some of the questions can have various answers, and some of the questions can have definite answers that are a little harder to get to. For instance, our wall display for Geometry in Literature asks four fairly easy questions. Some have definite answers, and some don’t. Here they are: