"[The universe] is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures..." — Galileo Galilei
By Joshua Bell. Last updated July 29, 2007.
"How does gravity curve space? Well, imagine that the world is two dimensional, like a rubber sheet..."
Sure, you've heard that before. But have you really thought about life on that two dimensional sheet, or wondered where that analogy came from? When I was 8 years old an older cousin of mine introduced me to Flatland and Sphereland, and I was hooked. Of course, at the time, I missed the satire and was merely enthralled by the animate polygons, but it was a start.
While there are many books on the mathematics of higher dimensions, many of which use lower-dimensional analogies to explain various points, several works - both classical and modern - stand out as taking place in a fictional two-dimensional universe.
This first foray into a two-dimensional world doubles as both a lesson on the concept of higher dimensions and as a social satire on Victorian England. The narrator is A. Square, a four-sided polygon in a world of two-dimensional creatures (and thinking). A rigid social structure is in place, from women (mere lines) through the lower classes (isosceles triangles), regular polygons (such as our narrator) to the many-sided or nearly circular priest caste. All is orderly, rigid and unchanging - until a higher dimensional caller comes to visit.
In some ways an homage to the previous work, but set in a distinct - and more realistic - two-dimensional world known as Astria. The triangular inhabitants walk about on legs across the surface of their circular planet. The story introduces the world and some of the challenges facing two-dimensional creatures. Along with some of Hinton's other writings, this appears to portray the concept of higher dimensions in a spiritual or metaphysical light, and the story reads as a bizarre form of morality play.
A sequel to Flatland, although written nearly a century later. This follows the adventures of A. Hexagon (grandson of A. Square) who lives in a world that has changed both socially and scientifically. Explorers proved that the world was indeed round, and the theory of higher dimensions is well known. But Flatland is still unready to cope with concepts such as the curvature or expansion of space.
Unlike earlier works, which use the technique of simplifying the world to teach an abstract concept (e.g. visualizing higher dimensions, the curvature of space), this work attempts to portray a plausible two-dimensional world and is best thought of as a thought-experiment towards that end. Physics, chemistry, biology, computation and culture are explored in detail. Whereas in Flatland the inhabitants are polygonal shapes sliding about on an idealized tabletop world, in The Planiverse the denizens walk on the one-dimensional surface of a circular planet (much like Hinton's Astrians), must climb over one another to pass, and build complex machines of springs and hinges. The story lasts just long enough to introduce the reader to the world and take them on a grand tour of a possible two-dimensional civilization.
Another sequel to Flatland (but not to Sphereland). Victoria Line, the great-great-granddaughter of A. Square, discovers her ancestor's suppressed writings. In her time, higher dimensions are well accepted in theory but not as a physical reality. With her guide she goes on to explore such notions as fractional dimensions, hidden spatial dimensions, hyperbolic geometry, quantum weirdness, space-time, singularities and time-travel, and perhaps the ultimate question - what is a geometry?
What if the story of Flatland took place with a four-dimensional being visiting our real three dimensional world, at the turn of the millennium (Y2K) during the dot-com craze? And what if there was something sinister going on in hyperspace? That's the story of Spaceland, and unfortunately that's pretty much all of the story. Just as in Flatland, the character visits lower dimensions to learn about hyperspace by analogy, and stumbles over issues such as orientation and perspective. The author's attempt to spice things up with supposedly three dimensional characters falls flat. (Sorry!) There is nothing new in this book; even the cameo at the end by a strange visitor from another two-dimensional universe seems out of place and unrewarding.
Wikipedia lists two short stories which I have not yet read:
I haven't read this one, but Ben Sandler wrote in with a short review:
It's sort of halfway between a textbook and a popular-level book, and it teaches all about different sorts of curved space, typically with examples in which Flatlanders explore a toroidal universe, a Klein bottle universe, and so on. (The book also deals unusual three-dimensional spaces. I found the "two-sided Moebius band" to be particularly interesting. The description involved ants like in Escher's famous Moebius band lithograph, but there were red ants on one side and black ants on the other...) In any event, although the latter chapters get harder, it's definitely worth reading as far as one can get.
Poor Ian Stewart - once he caught the Flatland bug while writing Flatterland, he couldn't stop himself. In The Annotated Flatland, Stewart dissects the people, history, meaning, and mathematics behind the original. A biography of Abbott starts things off, and practically every sentence in the original work spawns an annotation exploring some aspect of the environment which brought Flatland to be.
Two-dimensional worlds seem to call out for animated treatments, but surprisingly there aren't many and they seem to be multiplying! Here are the ones I know about:
Apparently it featured the voice talents of Dudley Moore. Disambiguation between this and the next adaptation c/o http://www.flatlandthefilm.com/faqs.html.
I only know about this one from a reference in Wikipedia and an entry in IMDB.
Based on Jeffrey Week's book (see above). Check out the web site: http://www.geom.uiuc.edu/video/sos/
A 15 minute short film - sadly, not available for public distribution, although the script is available at the home page: http://www.math.purdue.edu/~meiken/flatland/flatland.html
Not yet completed (as of March, 2007) but the trailer looks promising. Visit the web site at http://www.flatlandthemovie.com to view the trailer.
R. Perrin Ehlinger, voice artist for B-Square in the film, contacted me to let me know about this one. From the site (http://www.flatlandthefilm.com) it looks great! I have the DVD on order... more news when it arrives.
Since someone asked, here are my thoughts, circa mid 2001:
My view of Sphereland is inevitably biased since I read both Flatland and Sphereland together at a young age. It always seemed the "canonical sequel" to me. Flatterland is also a sequel to Flatland both in terms of plot and intent, and definitely a product of the early 21st Century.
Flatland uses a (contemporary) Victorian setting to introduce the hot topic of the day - the notion of abstract higher and lower dimensions. The inhabitants of the 2D world and their culture are depicted through a social satire on the world of that day. It's a very concise book in that it doesn't actually try to teach you much, but - very much in the writing style of the day - merely slips a few new ideas in while you're enthralled by the tale.
Sphereland takes a "gentle" sequel approach; the story updates the setting with social reforms, bringing a (mostly) 20th Century equality, knowledge-base and technology to the culture, but plays the lessons it intends to teach about higher-spatial geometry against the strict environment of scientific dogma. Whereas Flatland, boiled down to the core, teaches the single lesson that other dimensions are conceivable as mathematical constructs, Sphereland introduces the notions of curved and expanding space and how they apply to the real world. Directly after reading Flatland, Sphereland seems the "obvious" sequel; same sort of tone, same sort of pacing, same use of analogy to introduce concepts.
Flatterland takes a "radical" sequel approach; it's a "hip and with it" 21st Century sequel; the protagonist isn't a lone scribe but a teenage girl who emails friends on the InterLine and gripes about her parents. The story moves quickly and is a wide-ranging overview of different geometries and even questions like "what the heck is a geometry anyway?", as well as diving into notions like space-time, quantum mechanics, etc. Rather than a gentle analogy or investigation by the characters, a helpful guide gives lessons about each new facet of mathematics.
Sphereland feels more like a timeless sequel to Flatland, but having had time to ponder, I think Flatterland is the true contemporary follow-on; it addresses pertinent questions now and using concepts familiar to the current audience.
Put another way: Would someone write Flatland the same way now? No - the protagonist fighting a repressive and close-minded system was a valuable tool at the time, but it's not the culture of the "educated elite" today and would get in the way of the lesson. So Flatland-the-book is an artifact of the 19th Century; Flatterland is a artifact of the 21st and does it rather well.
On the down side, it's so hip and with it that it won't age as gracefully. Already some of the cute bits are strained - the Space Girls, for example. In another year or so no-one will even get the joke. (That's a problem affecting much of our cultural output at the moment, however - c.f. the signaling protein named sonic hedgehog.) Also, it attempts to do too much and in not enough detail. I'd love to have spent more time visiting Platterland, visiting the Moobius Cow or pondering the plight of the two-and-a-half-gon and less time talking with the quantum cat or bartering with the Hawk King for a wormhole. But that's the 21st Century for you - the solution is left as a problem for the reader.
At one point I'd idly started toying with writing a sequel to Sphereland, called Fractalland. I'd barely gotten started and some notes made up about the rest of the book when I stumbled across Stewart's Flatterland, which was pretty much the book I was going to write. (Or, to be realistic, to start but never finish writing.) I was a tiny bit disappointed to see that someone had beaten me to it, but thrilled that another story had been added to the genre.
The story was to take the form of an epic quest to another planet, in the grand old tradition of science fiction. Voyage of the Space Beagle by A. E. Van Vogt would be an example of the micro-genre. Along the way the intrepid explorers would learn about new types of regular shapes, fractional dimensions, hidden dimensions and time dilation. Here's what I'd gotten jotted down before I hung up my keyboard: