This was one crazy, opium fuelled, brilliant book about geometry and different dimensions and I am going to explain it the best way I can but Edwin A Abbott does it so much better.
Here is a story of Square who is a square and lives in a two dimensional world of geometrical figures. The first part of the book talks about the social breakdown of the Flatland and it is a thinly disguised satire on the Victorian society. People are divided into classes according to their geometry and the worst off are women who are not even figures; they are just straight lines. They have few rights and no one actually takes their intellect seriously. On the other hand they are dangerous because being straight lines they can easily pierce any figure. A woman from behind looks just like a dot, you might miss her until it’s to late and she has stabbed you. Different parts of Flatland developed different strategies for dealing with the danger, from not allowing women to leave their houses, to forcing them to constantly wiggle their bums, so they are visible from far. They should also sound a ‘peace-cry’ when out and about, in case anyone missed the wiggling bum. Seriously children, don’t do drugs. It makes you write things like that.
The second part of the book gets more interesting as it delves deeper into the concept of dimensions. As I said, our hero lives in a two-dimension reality. Try to imagine such a world. You probably see it as a piece of paper with various figures drawn on it. Of course, that’s how a creature from 3D world would see it. You’re looking at it from above, i.e from the third dimension. If a 2D world was your entire reality you would only be able to see lines and dots. Your eyes would be on the same level as the figures and you would see everything in one dimension and infer the second dimension because you can move in it and you have learnt it through experience.
The same way we can’t actually see the third dimension but we can tell it’s there. We know we can move in three dimensions and we know about perspective, light, shadow, etc. It is easier for us to understand a two-dimension reality than it is to imagine a four-dimension one. We can see it perfectly when our Square visits a one dimensional land and he laughs at it and tries to explain to the King that there is more to life than just looking at a dot in front of you. There is another dimension where there are not only dots but lines as well. The King of course laughs him off. Yet, when Square is confronted by Sphere who tells him about the third dimension and shows him ‘tricks’ that the third dimension allows him to do, Square is just as incredulous.
Even though the mathematics tells him there must be another dimension (and another, and another), he can’t quite believe it until Sphere shows him a little bit of a 3D world. Then he is a convert, and he quickly assumes there must be more dimensions. Fourth and fifth and ad infinitum. I think while reading this I got as close as I would ever get to understanding and imagining a 4D world. If in a 3D world we can see the insides of everything of a 2D world, then I suppose a in 4D world we would be able to actually SEE all three dimensions, all the insides of everything. My brains hurts. Am I making any sense? I thought I could see it but now it’s been a week after I finished reading the book and had those vivid dreams about the fourth dimension. The vision pales. I still believe in it but I can no longer grasp it. Just like the poor Square, back in his 2D-Land, thrown in prison for preaching revolution, still believes in the third dimension, but can no longer conjure the image of a Sphere in his head. Sometimes he feels he can almost see it again for half a second, and then it’s gone.