Flatland

There is a fun old book that was written in 1874, by Edwin A. Abbott, called Flatland: A Romance of Many Dimensions. In this book Edwin Abbott pretends that he is a two dimensional square who lives in a two dimensional world called Flatland. (See Flatland cartoon.) As a square in Flatland, Edwin is writing this book from Flatland prison. To explain why he is in prison, a little background is in order. Flatland is a class-conscious society made up of lines, triangles, squares, pentagons, hexagons, and other figures. They all move around on a two dimensional plain like a huge piece of paper, living out their lives. Women are the lowest class and are lines. The next class is extremely acute triangles and as the base of the triangle lengthens the class status rises. Then there are squares, which Edwin is, and they are the professional class. As the number of angles increase, the class goes up until it forms a circle. The circle is the highest class, a priest.

Everyone in Flatland looks like a line to everyone else. The main way that they tell one another apart is by voice or touch. In order to understand why everyone looks like a line; imagine a quarter lying on a flat table. If one looks down from up above (three dimensions) the quarter would look like a Flatlander priest or circle. Then if one were to move their eye down so as to view only across the surface of the table, all that one could see of the quarter would be a line. Keep in mind that Flatlanders cannot rise up or go below the flat two-dimensional surface they live on.

The story begins one night when Edwin is helping his Grandson, who is a very smart pentagon, do his math lessons. As they work on the math Edwin shows his Grandson that by squaring a number one can represent the two dimensional reality, they live in, with math. This is because any single number, like 11 inches, represents a straight line that is one-dimensional. Then the square of any number, like 11 inches times 11 inches, represents two lines and so it is two-dimensional. As they worked Edwin then showed his Grandson that it was possible to cube a number, i.e. 11 inches time 11 inches time 11 inches. The Grandson was fascinated with this and said, " if a single number represents one dimension, and a squared number represents our two dimensions, then can a cubed number mean that there is a third dimension?" At this Edwin scolded his Grandson for talking nonsense, and told him to go to bed and get such silly ideas out of his head.

This may sound strange to us because we live in a three dimensional world; however, if someone were to say to us that raising a number to the forth power (11 inches, times 11 inches, times 11 inches, times 11 inches) means there is a forth dimension, we would think it is silly to assume that meant that there is a forth dimension. The same is true for two-dimensional Flatlanders, for they cannot imagine three dimensions.

Then that night Edwin had a dream in which he came across a strange land that was only one-dimensional, called Lineland. He then tried unsuccessfully to convince the people of Lineland as to what two dimensions looked like. After all, if your world has only one dimension there is no way you can imagine two dimensions. Just as we in a three-dimensional land cannot conceptualize four dimensions.

The next night he suddenly saw a very short line appear in his house and then grow longer in length. The stranger then spoke to him and told him that he was from Spaceland, which was a three-dimensional world, and that he was a sphere. Edwin could not conceptualize such a thing as three dimensions or a sphere, so the sphere spent most of the night trying to convince Edwin that there was a place called Spaceland that existed in three dimensions. Finally, in the midst of a fight, the sphere lifted Edwin up out of his flat world into three dimensions. At first Edwin could not believe what he saw but then he understood and could conceptualize three dimensions. Once back on Flatland he tried to convince his countrymen of the existence of Spaceland but eventually ended up in prison for teaching lies.

The point of the book was to make us aware that we may be doing the same thing as the Flatlanders. Because we cannot see more than three dimensions does not mean that more dimensions do not exist. In fact, there is little doubt today that there are more than three dimensions.
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