Today is the birthday of Edwin Abbott Abbott, who was born in 1838 in London. The first thing you’ll notice about him, is that his middle name and surname are the same. This is because his mother and father were both called Abbott, they were first cousins. This isn’t the only odd thing about him. Abbott was a schoolmaster who did extremely well at Cambridge in theology and mathematics. This could have been quite dull, but he did something really interesting with it. In 1884, he published a novella called ‘Flatland’.
The first half of Flatland describes a two-dimensional world and the second explores the nature of dimensions. It is often categorised as a science fiction novel, but whoever wrote his wikipedia entry describes it as ‘mathematical fiction’. The narrator of the story is a square, who is sensibly called A. Square. He tells us about his world, where all the men are polygons with various numbers of sides while all the women are just lines. The more sides the men have, the more important they are. The lowest classes and soldiers are all isosceles triangles, middle class men are all equilateral triangles, while the professional, gentleman classes are squares and pentagons. Hexagons are the lowest rank of nobility. The more sides a man has, the closer he is to approaching the perfect shape, a circle. All perfect circles belong to the highest, priestly class.
In a two dimensional world, everyone (apart from the women) has length and width but no height and cannot be viewed from above. This means that the only way to perceive what shape a person had, was to either feel the angles of their corners, or to observe them through a sort of fog, that was quite common in Flatland and see the way their edges faded from view. The women (and Abbott quite rightly received some stick for this) presented a different problem. Being a line, they were quite easily observable from the side, but end on, they were practically invisible. This made them very dangerous, as they were liable to stab people either accidentally or on purpose. For this reason, every Flatlanders house had a separate door for women, just in case. Additionally, all women had to constantly sing a ‘Peace-Cry’ as they moved around so that all the men knew when they were coming.
Flatland society was very rigid and only equal sided polygons were tolerated. A. Square explains that it would be very difficult in Flatland to discern what sort of a person you were dealing with if, what you thought was a perfectly ordinary triangle coming towards you: “drags behind his regular and respectable vertex, a parallelogram of twelve or thirteen inches in diagonal”, if you invited him into your home, he might get stuck in the door. On the whole, they equate irregular polygons with criminality and immorality, which probably says as much about the Victorian world Abbott inhabited as it does about Flatland.
In the second half of the story, Square dreams that he visits a land that has only one dimension, where everyone is a line, but they only observe one another as points as they are unable to see their own length. Then, in the year 1999, on the eve of their millennium, square is visited by a sphere from a world where there are three dimensions, Spaceland. He cannot comprehend what he is looking at because what he observes, as the sphere passes through the plane of his existence is a perfect circle that changes in size. If you’re having trouble grasping this, there’s a great little video narrated by Carl Sagan here. Square is taken by sphere on a journey to a dimension which is unknown to him called ‘up’. From there he can see inside all the houses and inside all the people, he calls this ‘omnividence’. Square tells Sphere that being able to see everything makes him feel like a god. Here is Sphere’s reply:
“… if a pick-pocket or a cut-throat of our country can see everything that is in your country, surely that is no reason why the pick-pocket or cut-throat should be accepted by you as a God. This omnividence, as you call it—it is not a common word in Spaceland—does it make you more just, more merciful, less selfish, more loving? Not in the least. Then how does it make you more divine?”
Sphere takes square to visit Spaceland and, once there, he begins to wonder if it is possible that there is another world above that one, where there is yet another, fourth, dimension and everyone there can look down on Spaceland and see everything. And why not a fifth or a sixth or even more? Sphere dismisses this as a nonsense that is only talked about by madmen and Square is returned to Flatland. There, he dreams that Sphere takes him to a world with no dimensions at all. It is occupied by a single point. Sphere observes: “That Point is a Being like ourselves, but confined to the non-dimensional Gulf. He is himself his own World, his own Universe; of any other than himself he can form no conception…” Square tries to communicate with the being but, because it has no concept of anything other than itself, it assumes his words are it’s own thoughts. It is perfectly happy because it is omniscient and omnipresent in its own non-dimensional world, but it has no aspirations.
Back in Flatland Square tries to explain the third dimension to its inhabitants. Meanwhile, it turns out that the priestly class have known about the third dimension all along, but they keeping it to themselves. Square is imprisoned as a madman and it is from prison that he is writing this book.
It was a pretty wild story in the 1880s and was not well received, reviewers just didn’t know what to make of it. The Athenaeum said: “That whimsical book Flatland by a Square (Seeley & Co.), seems to have a purpose, but what that may be it is hard to discover.” While the New York Times said: “It’s a very puzzling book and a very distressing one, and to be enjoyed by about six, or at the outside seven, persons in the whole of the United States and Canada.” Abbott’s strange book was largely forgotten until Einstein published his general theory of relativity. After that, Abbott was hailed as some sort of prophet with his concept of a fourth dimension. His ideas have been much revisited by science fiction writers. It you wanted to read the original for yourself, you can do so here. What I really like about it are his ideas about omnipotent beings who either see everything but just don’t care, or who are so infinitely small that they fail to see anything other than themselves.