Non-Linear Dimensions

It is said that for dimensions larger than the first four( length, width depth, time) are in fact “curled up” on the order of the Planck Scale.

I had always believed a dimension to be a linear measurement: does the mathematics exist for angular dimensions?

If so, could something like spin be described as an angular dimension?

For two traditional dimensions and one curled up dimension, the coordinates should look like (a, b, θ), where a and b are any real numbers and θ is in [-π, π]. In fact, any point on a circle can be specified with one angular dimension. Remember that the simplest notion of the dimensionality of an object is just the number of coordinates needed to specify any one point in that object.

Do any large dimensional Grand Unification Theories incorporate angular dimensions in their mathematical architecture?

You can also use linear measurements; they’re just restricted to the range from 0 to the circumference (or -1/2 the circumference to 1/2, or the like).

Most of the theories with more than the standard 3+1 dimensions (which includes more than the GUTs) have the extra ones curled up small like this. It’s the standard explanation for why we don’t perceive them directly.

Suppose we wish to quantize General Relativity…


…to fully describe the hypothetical graviton, believed to posess Spin 2.

Would we need to add a fifth angular dimension to the four linear ones already in GR?

You don’t have to have integer numbers of dimensions, either, at least not at really small scales, according to a recent article in Scientific American. The gist of this article was that computer models of atoms of space (that is, fundamental building blocks of space, not matter) could use some simple expected behavior rules and wind up predicting the number of dimensions we see in the universe, and they predict 3 plus or minus something like a percent; they also predict that the number of dimensions changes gradually on smaller and smaller scales. At least, this is how I recall it. This past summer, I think.

Maybe. Depends on whom you ask. We don’t have a working theory of quantum gravity yet, so we don’t know what it would need. The string model (or some variation thereof) is usually considered to be the front-runner, and it requires a bunch of extra dimensions, but there also exist attempts at quantum gravity which don’t. None of them really works, though.