Thursday, September 27, 2007

Flatland- the annotated edition: experience hyperspace first-hand

Flatland: A Romance of Many Dimensions, is a pretty famous book, at least in certain circles. It is a book about space, in two and three dimensions, and was published in 1884 by Edwin A. Abbot. This is a great book to read to kids of nearly any age (depending on the details, maybe 4th graders, maybe 7th graders, maybe revisiting the book with 9th graders, or 14 year olds, who are studying geometry).

Flatland describes the world of A. Square and how his two-dimensional world is shaken up by a strange visitor who takes A. to the world of three dimensions, and beyond! The story touches on four dimensions, but also on one- and zero-dimensional matters. It is at once a fairy tale and a math lesson, with geometry being the focus, yet it is written in an accessible and amusing style (many parents will be able to follow it, at least most of the way- your kids might help!). You do have to be willing to handle the hyper-Victorian manner of speech adopted by the characters to enjoy the book.

A. Square's familiar two-dimensional world is similar to the world we see every day on flat sheets of paper, but its society is extremely rigid: class distinctions amongst men are made by the number of vertices their shapes posses: soldiers are triangles (with a vertex count of three) and gentlemen are squares (having a vertex count of four). The more vertices one has, the higher the social class. At one extreme are priests, who are circles: imagine a polygon increasing from square to pentagon to hexagon to ... an infinite number of vertices, and then you have a circle! Women, on the other hand, are all line segments in A. Square's world, though they are of different lengths. Before you get angry, please note that the author was an early proponent of educating women equally to men, so Flatland has to be read as the satire on Victorian society that is was, while simultaneously being a fairy tale and geometry lesson, and even science fiction. In fact, the extreme rigidity of rules for social behavior is pretty funny, but it can make the reader very glad not to live in such a world (I certainly hope you don't!).

It is a shocking but ultimately very pleasing experience for A. Square to learn about "spaceland", where three dimensions exist. We have the fun of accompanying him on the journey. One of the remarkable things about going from a two dimensional world to a three dimensional world is that many of the same techniques are used if one wishes to go from three to four dimensions, or from four to five to n dimensions. This progression is described up to four dimensions and a bit more, at least partly, in Flatland, and a number of modern "sequels" take us well beyond the fourth dimension. One such sequel is Flatterland by Ian Stewart. I had some trouble understanding Flatterland about a third of the way in, which made it difficult to read to the kids- I need to work at it some more.

The amazing thing about mathematics is the ability to treat multi-dimensional shapes with the same ease as their two- and three-dimensional counterparts. So, while it may be difficult to "see" what a 6-dimensional hypercube looks like, especially since we are mostly limited to two- and three-dimensional representations, it turns out to be pretty straightforward to calculate the surface area and volume of that 6-D hypercube (using the equations for surface area and volume of a 3-D cube as a starting point). If you can calculate all of the properties of a hyper-dimensional object, you do understand it quite well, even if this is hard to believe at first. We tend to have a sometimes crippling dependence on what we can see and hold, but math can free us to "see" far more.

Before you click off to another page, just consider that you already know a 4-D world in great detail: the 3-D world with time added as a fourth dimension. In fact, huge numbers of dimensions may be present in problems or hobbies you currently explore, but it may take a few moments with A. Square to realize it.

As I have recently come to learn, Edwin Abbott traveled in exciting intellectual circles, being friends with H. G. Wells, for example. This connection, and others to Victorian life and events, to a certain George Boole (of Boolean Algebra fame), the novel Frankenstein, and other cultural icons, are explained to us in the The Annotated Flatland: A Romance of Many Dimensions, with extensive notes and pictures provided by Ian Stewart. Mr. Stewart, who wrote the previously mentioned Flatterland, is a celebrated mathematics professor and author, with titles including Does God Play Dice? and Nature's Numbers to his credit. The additions by Stewart include marginal notes, a preface, an introduction, a chapter on the math of four dimensions, and an extensive bibliography, all of which help us to understand better both Abbott's primary tale and his subtext that satirized Victorian society.

The Annotated Flatland is highly recommended!

Note: E. A. Abbot passed away in 1926, in Hampstead (find notes and maps for Hampstead Heath by clicking on the link).

© James K. Bashkin, 2007

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