The Annotated Flatland

Knowing that Abbott was a believer who sought ways to describe theological truths in fresh ways made me eager to read his book. His biting commentary on how women were treated in his patriarchal culture (not to mention many others in the lower tiers) feels validating to me, and an important reminder that truth is supposed to set us -all- free, not bind some while elevating and entitling others. Abbott was a brilliant student, winning the highest honors available in the classics, mathematics, and theology. He became a fellow of his college by 23 and a year later took orders as an Anglican priest. After only a couple of years as a master teacher, he rose to headmaster of a prestigious public school in London at the incredibly young age of 26. Abbott was a prolific writer of serious books on theology, but his best-known work is this novella called Flatland, published when he was 46, in 1884. A fascinating mathematical construct, Flatland features beings living in a two-dimensional world who encounter a three-dimensional being. Abbott follows their philosophical and mathematical process in trying to make sense of what otherwise would be considered an impossibility in their two-dimensional world. A Satire One level of Flatland is a commentary on the hierarchy of Victorian culture, such as the illustration below of two separate doors, one for women and one for men. Women are mere line-segments, whereas men of even the lowest caste are a full shape—a triangle. The “author” of Flatland (the pseudonymous “A Square”) is a gentlemanly square, having received a small inheritance which added to his triangular stature, making him the statelier square. Climbing the ladder of social influence and recognition, of wealth and power, has strict rules, and not everyone will make it through. Some people—those with irregular sides to begin with—will never be able to climb, nor will their children or children’s children. Social activism is considered criminally dangerous, and those who show potential beyond their class are either quietly promoted or just as quietly done away with (euthanized, a form of social and family planning that was being popularized in some circles in the late nineteenth century). Roles within society are rigidly maintained, particularly the dichotomy between men and women, with women always and permanently remaining of an entirely other and second class. Women by their very shape as mere line-segments can never attain anything more than their shape will allow. A Story of Dimensions All of Flatland is ruled by circles, a priestly cast of shapes formed by ancient lineages. Their inheritances down. coming through countless generations, have added so many lines to their forms they have become indistinguishable from the perfection of a circle. On New Year’s Eve, A Square has a gripping dream in which he encounters beings from a one-dimensional world where the men are lines and the women are mere points of light. These beings are unable to experience the fullness of A Square because he is two-dimensional, and they can only “see” one of his sides at a time. A Square tries mightily to describe the two-dimensional world he comes from to the king of this one-dimensional land (and therefore the wisest and most intelligent of all the line beings), but the Line King simply does not have the capacity to understand what A Square is trying to convey. But then, A Square has been visited by an equally-to-him incomprehensible being—a sphere. The Sphere reveals itself to A Square as a circle, then as a circle that becomes a point, growing sizes of circles, diminishing sizes of circles, then back to a pinpoint as the Sphere passes through A Square’s the two-dimensional realm. Talk about a mind-bender! A Square does not have the capacity to perceive all three dimensions of the Sphere, but what he does see is amazing to him. And then the Sphere brings A Square to Spaceland, a three-dimensional world, that so expands A Squares vision and understanding, he will never be the same again. Censure A critique ensues on the ramifications • of intellectual and scientific leaps forward in understanding. • on philosophical questions of what creates and maintains society. • on what we might today call group-think. • of what those with power are willing to do to retain that power. • of what happens to innovators and visionaries. Of particular interest is the censure on "revelations from another world." Though the top leadership of Flatland secretly know other dimensions exist, they suppress any discussion of it, and stentorian laws are in place to summarily silence any who speak of such things. What this means for A Square becomes painfully poignant as he wrestles with the powerful new concepts he has embraced and the stiff laws and culture of his world. Spiritual Plane A bifurcation has happened in Flatland that has split certain elements of society and humanity with something of the same results that perhaps splitting an atom might have. Women and men, for example, are kept carefully apart except under certain highly regulated circumstances. Women are deemed unnaturally dangerous because of the shape of their bodies and must cry out any time they enter a room, or walk about in public lest they be mistaken for a point, or a man is accidentally sliced clean through by one of their lines. Intellect is split from emotion, so that reason becomes the man’s sole domain, and feeling the woman’s sole domain. Woman are considered, in fact, incapable of reason though they receive educations, and men are considered specially skilled in avoiding emotion and in conversing rationally. As A Square receives revelation after revelation, he at first thinks he is gaining in reason and becoming even more devoid of emotion, but the Sphere shows A Square how important feeling is, how compassion, for instance, is a God-like quality. The book itself shows how a society’s dogma, based on what is perceived to be truth, can become oppressive. Those in power use dogma to suppress new discoveries which undermine their authority and continued claim to power. Knowledge, then, revelation, becomes an enemy to dogma when it questions the “truths” up-on which dogma has been founded. This is edgy stuff! Those who became believers and spread the Gospel as described in the Greek scriptures met this kind of opposition head on. They were visionaries, religious revolutionaries, worshiping Christ Who is fully God and fully human, a concept no one had ever heard of before, and proclaiming literal resurrection, a feat never before achieved. Their testimony disrupted the dogma of every religion in the known world and suggested a spiritual-physical revelation that is flatly impossible in the world as we know it. Societal structures were and still are called into question. Those in power then, and today as well, seek to suppress aspects of the Gospel, such as freedom and full equality for all people, men and women of all shapes and sizes. Some seek to suppress the Gospel entirely. Others forbid the interpretation of the scriptures in any other than the currently established way. Even translations of the scriptures come under rigorous demand to conform to certain concepts such as complementarity for some translations, traditional use of male-oriented words over and above gender-neutral language for others, the translation of numbering, of certain adjectives describing categories of people or actions. The list is long. Annotation and Commentary Ian Stewart enriches the reading of Flatland greatly with his own explanation of Abbott’s references to various Victorian references, and explication of Abbott’s mathematical allusions. Abbot’s novella is a wonderful read even without these annotations, but with them the reader can enjoy the richness Abbott intended. As a point of reference, I had all three of our daughters read this book when they were in their variously high school or early college years. They all not only thoroughly understood the story and the meaning, they had enough math knowledge to smile at Abbott’s witticisms. And that is my personal recommendation as well. This is a great book for youths to really examine their own faith, and to prepare them for how the world really works.

With the Alice in Wonderland books, the late 1800s seem to have been the time for really creative mathemetical writing. Although not as frequently read, Flatland, the Edwin Abbott Abbott story of a little square coming to understanding that higher dimensions do indeed exist outside his world is a delightful read. For those seeking to understand what life is like in other dimensions, Flatland is very comprehensible with clear writing and simple, easy to understand illustrations that help drive home Abbott's points. Originally written with many sly references to the then existing state of British culture, Abbott's invitation to try and understand higher dimensions was also an invitation to society of his time to try to re think its views on a myriad of issues...including its openness to women in education. In this way, Abbott converted viewing higher dimensions into both a mathematical and social challenge...points Stewart was sensitive to in his annotations and his own homage, Flatterland. Although other editions of this work exist, the annotated Flatland is the one to buy both because of its faithful reproduction of the original and its thought provoking and helpful footnotes that give the work broader meaning.

'Flatland' is amazing. Dimensions are the point of the tale. And the line, the square and the solid cube. (Sorry about being so oblique, but I often angle for a laugh at the beginning of a review, no matter how circuitous.) The author Edwin Abbott Abbott with a wink and a smile introduces us to the science of geometry in the Victorian Age in this (a)cute story about A. Square. To understand the concepts that these surprisingly charming fantasy characters who live in a two-dimensional world illustrate, I think you need to be either good in math or be a high school graduate. Nevertheless, if you have some mathematics education, this is a fun read which becomes something bigger on the inside. But I was very dubious, initially and I was afraid of a something coincident with the most boring math class I'd ever taken and too much eccentricity. I circled my living room going around and around (at least 360 degrees, I think). Do I want to read 'Flatland', even for a monthly book club selection? A Victorian Romance, equal to no less than 0 in my estimation, about math? But, a distant chord projected tangentially into my attempts to square the circle and I realized I had transcended my doubts. So, I reached a crossroads of sorts, an intersection in my thoughts. I devised a postulate for myself: I'm at a point between two directions - do I make a line towards the library, or use up valuable space on my Kindle downloading this book? Plus, I'm geometrically opposed to wasting my time. A pain, as if a ray was bisecting the plane of my forehead, warned me I was overthinking this. After all, an endpoint to reading is expanding my universe into different dimensions of ideas. I'd function on a higher plane, I mused. So, after I folded myself into the curvature of my couch, I grabbed my Kindle and made the calculus of how I would squeeze in time to read this dense concept fantasy. Gauss what? I like it! The topologies of 'Flatland' (Pointland, Lineland, Flatland and Spaceland) were surprisingly easy to understand. I believe the biographers of Abbott who described the author as an outstanding teacher and writer are correct! He also was surprisingly liberal for a preacher, standing up for women's rights and the extension of upper-class social freedoms to everyone. Additionally, for many Victorians including Abbott, the new scientific discoveries about physics, especially about the possibility of a fourth (and more) dimension, revived a lot of theories about whether discovery of this dimension was the solution to where heaven and/or spirits might live. A Victorian pop culture grew up around the new kinds of proto philosophical- and mathematical- based social sciences. Abbott manages in about 100 pages to tie in all of these social questions while his character A. Square tells us about his life and friends in a two-dimensional city. When he is visited by a sphere, a physical impossibility, it turns Square's mind hyperbolic. However, once the sphere shows him the wonders of his three-dimensional world, two of which being that as a Spaceland citizen, Sphere can see Square's intestines and take money out of safes in Flatland as easy as pi. It seems three-dimensional creatures see straight into two-dimensions because there is no 'roof' to a two-dimensional world, only length and width. It would follow fourth- or other multi-dimensional beings could see into our three-dimensional universe, doesn't it? Boggles the brain proportionally to the black hole of ignorance one possesses before reading this.... The wonders of Flatland go on and on infinitely. For example, Square's wife is a line, as are all women, and the King is a circle, as are all priests. Lower forms (workmen, soldiers, middle-class) are triangles and the noble classes above squares are pentagons and hexagons. Squares are professional men and gentlemen. Intermarriages can be arranged, and babies are born naturally upgraded into a higher rank. The more regular and sided an individual, the better the individual's class, so Equilateral triangles are of a higher class than Isosceles triangles. Women are the lowest class (which was a satirical device - the author actually believed in women's rights). Execution for crimes and rebellions, as well as knowing too much, are common (politics in Flatland sounds very harsh to me, but then I can be quite obtuse). Frankly, I think there must be a transference of information between universes as anyone can tell you they have discovered me humming a Peace-cry often while cleaning my house. I must admit I am in awe of the unusual symmetries between the deadly needling talents of two and three-dimensional women. I wonder at the timeless qualities of real and irrational human nature. My mind is a torus of spinning speculation, even turning inside out. This novel has no parallel in modern times, but it was a bit lacking in depth. P.s. The annotations were fantastic and the Kindle handles all of the endnotes with no problems.

I love this addition. I agree with the reviewer that it is best to read just the text completely during the first read,to capture the story as it was first intended. Then go back and soak up all the great mathematical ,historical and philosophical insights the annotations provide. It will expand your imagination,no matter where land on the educational spectrum! Still can't believe I had not heard of it before. Jon Haidt mentioned it as one of his favorites in thebook "The Happiness Hypothesis".

A very witty book that has a touch of dark humour and all a fair amount of geometry/maths involved. I first read this book in secondary school, so it's very accessible for a variety of readers. This is a special hardback edition version where the novel section is in the middle, and you have annotations/notes on the sides. Definitely for the fans of the original!

Les annotations donnent à cette oeuvre un intérêt accru.

One might think that all the comprehensive annotations are superfluous or far off-topic. Well, in a way they are ;), but it's actually the very stuff you'd be googling sooner or later yourself - if you're the reflective/contemplative type, that is. Which you probably are, if you already intend to buy this book.

My son-in-law has been wanting to read this for years. Now he can.

A very nice edition and a quality one and a very interesting book especially for mathematicians!And the commentary very helpfull.