What's the Point? The Layers of Flatland

Two dimensions--more to it than you'd think.


Flatland cover by Wikimedia CommonsSo. Flatland. Flatland is an imaginary world, imagined by the great imaginer Edwin Abbott Abbott (that's kind of an unimaginative name, though), who wrote a book called "Flatland" in order to explain the mathematical concepts of dimensions. It's a pretty good book. Check it out. Also, it's a romance. Oh yeah, and also, he wrote it in 1884.

Math is Hard

Let's break it down. Flatland is a two-dimensional universe in which women are line segments, soldiers are triangles, gentlemen are squares, and nobility are regular polygons. The ever-rare circles are part of the priesthood. Every thousand years, a sphere descends upon Flatland from the third dimension in order to educate an apostle about the reality of extra dimensions. Every freaks out, and the priests conduct purges and suppress the information. Cool story.

That's pretty much all you need to know for now, and I'm not going to ruin all the fun stuff for you. Really, you should just read it yourself. It's a pretty short book. We've got a copy here at the library that's only 124 pages long, and that's WITH illustrations, notes, and an introduction.

But Reading is Hard, Too!

Okay, okay, fine, I get it. You don't want to read the book. I'll make it easier for you. How about a comic strip? This excellent webcomic has a joke about Flatland that you might enjoy. Although, really, you'd probably enjoy it even more after you've actually read the book.

I Must Have Moving Pictures!

STILL too many words for you? Well, you're in luck! Because the Peninsula Library System is here for you! We have exactly what you need: a movie version of Flatland, called Flatland: The Movie!  You are so very, very welcome. References to Flatland are more prevalent than you may have previously known. For instance, in Pulp Fiction when Uma Thurman and John Travolta are sitting in the car outside of a diner, Uma Thurman makes an explicit reference to Flatland.

The interesting thing is that when Abbott originally wrote Flatland, he used it to satirize the Victorian social mores that permeated the very world in which he lived, so when Uma Thurman calls John Travolta a "square" in this scene, she is actually, not-so-secretly, casting judgment upon his class and upbringing. How rude, Ms. Thurman!

This Moving Picture is a Real Horrorshow

I can only do so much more for you at this point. How about I point you in the direction of children's television programming. The PBS show "Arthur" has an episode called "Falafelosophy" in which a character writes her own story. Unfortunately for her, that story is suspiciously similar to Flatland, populated as it is by angry triangles and beleaguered circles.

But the episode is quickly redeemed by the fact that it guest-stars Neil Gaiman! Author of excellent books like Stardust, Coraline, American Gods, and the Sandman comic books!

So What Was the Point?

"The Square then has a dream in which the Sphere visits him again, this time to introduce him to Pointland. The point (sole inhabitant, monarch, and universe in one) perceives any attempt at communicating with him as simply being a thought originating in his own mind."


Author Bio:

A regular Josh Pearce cannot be constructed using compass and straightedge.


Flatland is total fiction, even conceptually

I have read any number of discourses on Flatland in various attempts to introduce the concept(s) of dimensions in excess of the 4 normally sensed by humans.

Well, I object to all of these thought experiments on the ground that they all, and I mean ALL, are dishonest.

They all "pretend" that 2-dimensional entities can exist (can't, there's nothing there), and sense each other (quite impossible - no 3rd dimension into which sense organs project), and communicate (impossible, again, since there is nothing there).

In Flatland, even entities one atom thick cannot exist. The depth of Flatland is infinitely small as in non-existent.

But repeatedly, without criticism, all Flatland entities are discussed exactly as if they have a very, very, very tiny 3rd dimension (thus 3-dimensional). There is simply no such thing as a 2-dimensional physical entity. Just as a point has zero substance (how can it?) and is an imaginary concept, a 2-dimensional "world" cannot exist because it has zero depth: substance requires volume. Area times zero depth is zero. Thus, one pretends lines and circles and shapes can exist, but how? There is nothing there for them to be expressed in; they have no depth and thus no substance and thus don't exist.

All the physics popularizers that re-tell the Flatland analogies start with this false pretense that such "worlds" exist but they cannot, as I say, even conceptually. So what does that say about the various extrapolations and analogizing to multiple dimensions?

It bothers me.

I know, I know. It's just a good thought experiment that seems to help us get a grip on the the various effects of adding dimensions to a picture. It offers some fascinating metaphors and imagery.

But the whole edifice is founded on a perfect falsehood that any sort of Flatland can exist at all. Flatlanders don't exist, even conceptually. The book "Flatland" is based on a profound falsehood and is suspect from page one.

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