What is a Dimension?

July 23, 2023

There's a legend about Rene Descartes that describes him sitting in his bed watching a fly on the ceiling and wondering what the patterns of its movements were. He imagined if he divided the ceiling into a grid of rows and columns, he could keep track of the fly and look for patterns in its movements analytically. And this is how Cartesian Coordinates were invented; the x/y grid axes we use in mathematics were the first dimensions.

While this story is most likely apocryphal, Descartes did conceive of the idea of a "dimension" in the way we use it today. A dimension is a mathematical construct we use to describe the space around us. We can look around us and see that space (and the objects within it) can be boiled down to three independent properties: length, width, and height. If we want to do math with these objects, we can imagine them on a grid like Descartes' fly but with one more axis for depth. Then each object can have a size and position described with just numbers.

This hopefully makes sense so far, but things get weird after the first 3 dimensions.

Einstein contributed a 4th dimension in General Relativity. The Kaluza-Klein theory introduced a 5th dimension. Then its successor string theories added more, all the way up to 10 spatial dimensions and 1 time dimension. Like the first 3 dimensions, the newer dimensions are mathematical constructs: we take all of our formulas about fields and particles that we did in 3 dimensions, update them with however many dimensions we want, crunch the math, and see if it checks out. We don't necessarily know how to interpret the math into a physical description, but some interesting properties of the dimensions emerge when we try:

• General Relativity treats time as a dimension. The time dimension does get some special treatment in the math, but otherwise it's like the spatial dimensions in that objects should be able to move in either direction. Why do we only see things move in one direction through time then?

• The 7 extra dimensions in string theory are just like the spatial dimensions, but with the awkward property that they're completely undetectable to us. Oskar Klein suggested these extra dimensions might be circular and extremely small, with a radius near the Planck length where we can't see them. This isn't a requirement for string theory though, there could be other reasons we don't see them. Maybe we live in some 3D subspace within the larger 10D space, like a living cartoon on a 2D sheet of paper in a 3D world.

Technically a dimension can be anything: it's a parameter in math equations that describe space. There could be a binary dimension where all points of space are either "red" or "blue." There could be two time dimensions with infinite directions time can flow on the plane it forms. Or maybe there's a consciousness dimension with lower and higher realms of experiencing reality. The question with speculative dimensions remains the same: if they exist, then why don't we see them?