Reflections: Sphere-Land

Sphere-land is the sequel to Abbot's classic book Flatland. I'd been looking forward to read this book and it didn't disappoint me. The two books explore the concept of geometric progression and our conception of the number of possible dimensions in space. In Flatland, Abbot conceptualizes how life would be in two-dimensions, and how completely appalled the two-dimensional creatures would be, if they were told there was a universe bigger and more complex than they knew existed. Abbot hilariously brings out the utter confusion and disbelief of the 2D people in trying to visualize and come to terms with a new dimension, a new direction and completely different species and structures. It's an ultimate satire on how cloistered we 3D folks really are, in assuming that there can't be anymore dimensions, or more complexly developed species other than us. In Sphereland, Burger continues the discussion by adding more layers of geometric complexities that the 2D people toil to solve. He extends Abbot's narration by showing the scientific, societal and cultural evolutions in the land of Flatland. Scientific revolutions were at their hilt, through space and world expeditions to learn more about their world, as well as test their hypotheses on the 3D land. Two scientists who were ahead of their "flatanders" started observing intriguing things that completely altered their basic scientific knowledge - people who traveled straight along the east, somehow started meeting their friends who traveled along the west, there were mind-bogglingly startling triangles, whose angular sum exceed 180 degrees, and it seemed like the distance between their world and other worlds kept increasing for some reason! An expanding spherical universe?!! What does it mean?! With a benevolent 3D sphere paying visits to the Flatlanders every New Years' Eve to offer hints on their perplexing geometric problems, the prime scientists grapple with constantly having to revise their notions of scientific plausibility that go against some of their fundamental concepts!

Yet again, the book reinstates that our beliefs and scientific knowledge are very limited and are subject to constant revision. Just as the Flatlanders brandish the two scientists as being insane trouble-makers who indulge in black-magic with supernatural creatures from the other world, so is our perception on many baffling and inexplicable phenomena in our world. Burger also brings out the sad and frustrating truth about how even the intellectual scientific community ostracizes the two intelligent scientists and refuses to try and comprehend their research and solutions...it goes to show that it takes a lot to be open-minded to consider hypotheses that stretch the limits of one's current imaginations. To a 2D creature, it is unfathomable to visualize or conceive of a third dimension and the meaning of "perpendicular".

The book is well-written and while it may not be as revolutionary in inspiring to think about multiple dimensions, the book yet again hammers in the basic humility that we human beings are probably not the only sophisticated and complexly developed creatures, our current knowledge of science ultimately reduces to a set of plausible hypotheses, and there are definite limits to our mental and cognitive faculties. Just because we cannot understand a phenomenon because of this basic limitation, it doesn't rule out the truth about the phenomenon.

Reflections: Flatland

A classic that needs no introduction. I have seen this book on our bookshelf for as long as I can remember and yet I never ventured into even turning a page, for I had seen this book in the hands of intellectuals, of mathematics geniuses and heard them talk about a new abstract dimension in mathematics. Me and mathematics, and that too abstract, never go together in a positive sentence. And naturally I shied away from the book as long as I could. But I'm glad that I finally "faced my fears" by not only reading this book but also marveling at how simple it now seems to try and conceive of another geometric dimension and perhaps many more.

Imagine a land of two dimensions, a world existing inside a geometry book, where triangles, lines, squares and other polygons have a life, a personality, intricate social structures, legislatures and laws...all within the constraints of their two-dimensional flatland. Ha! how bizarre does it seem to us, elevated people of the third dimension. How silly that there could be a degradable world in mere 2 dimensions and how amusing to read about utterly ignorant and pompous flat shapes who assume that their world is the only one to exist and that space beyond 2 dimensions is pure gibberish! The smirk on the faces of 3 dimensional-creatures soon gets wiped away with a mere question - what if you were ignorant of a fourth dimension? How silly would us 3 Dimensional know-it-alls appear to a more intricate creature looking down upon us from a greater dimension and witness all our hullabaloo over a small speck of space?

This reminded me of Plato's allegory of the Cavemen, in his Republic. Plato claimed that what we humans perceive are mere shadows on the walls of the deep caves inside which we reside. Seldom do one of us climb out of the cave to see the sun and the beautiful land beyond the caves, whose mere shadows are the ones we perceive as being real. Yet a person who comes back from such a land of "true shapes" gets ridiculed at, due to pitiable ignorance and tight walls of brick built around one's sense of reason, power of thoughts and imaginations.

The author with his tickling style of satire casually brings out the plausible notion of further dimensions with his charming tale of a frustrated Square trying to grapple with the truth he witnessed in another dimension and faltering at conveying what he witnessed, to his closed minded fellow men. The cruel fate of innovators and thinkers, starting from Copernicus. Since the book is more than a 100 years old, the author's depiction of the social system in the land of 2 dimensions reflects the common norms of society and culture that existed in most parts of the world then (on reflection we can still see remnants of those social structures even in today's contemporary societies).

The descriptions and narrations of the inhabitants of the land of 2D really appealed to the child in me. If I had read this years before, I would have easily gotten distracted, fantasizing about the squares and triangles every time I opened my geometry textbook. I'm probably the only loony who does this, but when I was little I associated each alphabet to a personality of their own, living in their land of English. Somehow the shapes of the letters (and probably the phonetics of some of the words which begin with them) paint a general impression of their personality. The letter "A" for example is an arrogant intellectual, B is the henpecked husband of C, who is an incorrigible spendthrift...and A-H forms one family of the upper class, with I-S forming the middle class etc. I used to have a similar imagination with vegetables and fruits, but that could be the influence of cartoons! But to read about personalities and characteristics of Isosceles, Circles and their little houses, in a "grown-up" book dealing with math-fiction, was just so refreshing to the imagination! And to see it all being strung into a significantly intense concept is like watching Mickey Mouse explain the theory of relativity and me understanding it! How much more creative can an abstract book get? :)