Life In N Dimensions

This piece is part of an occasional series: Mathematics: Where The Wild Things Are, about the weirdness of mathematics, and how we can use visualization and our imaginations to help us to *see*. If I get it together, I want to do a podcast on visualization, imagination, and mathematics.

A point is dimensionless but can be used to denote an exact location. This gets confusing the real world, since you usually make a point with a pencil, and it appears to have length and width, however small. But in the mathematical sense, points are dimensionless.

Being a point would be pretty boring: nothing to see, nothing to do. How can you see anything, if there are no dimensions with which to see? You would just kind of hang out there, in space somewhere. The Architect could move you around the plane if you are in one. But there would still be nothing to see, nothing to do, nothing to attach to. Perhaps you could be placed in an infinite collection of points, and then you would at least know your point-friends are near.

A line has only one dimension, its length. Life as a point on a line would mean you can look forward (seeing infinitely forward, unless you are on a line segment, in which case you would not see past the point on the end. Your peripheral vision would not exist, you could not look left or right. There is no left and right, on a line. Remember, you only…