Kaluza-Klein theory adds one dimension to GR to make gravity PLUS electromagnetism
Add two more dimensions to describe nuclear forces
And four more to describe matter.
A periodic table of forces.

What if we want to describe a periodic table of dimensions?
Over in this thread

I explain why I believe time to be the first dimension.

Take one uni-directional dimension of time,
over three bi-directional dimensions of space,
over seven microscopically curled-up dimensions of the quantum world.

Displayed something like this:
Physically or mathematically, am I in error?

It’s not needed, since all of the spatial dimensions are interchangeable. There’s the time dimension, and then there’s some number of spatial dimensions, and that’s about it.

You could further split it up into the three obvious spatial dimensions and the n non-obvious ones, but given that we don’t know the value of n (it may well be zero), or, if n is non-zero, why they’re non-obvious, or, if the non-obviousness is due to compactification, what the scale(s) or topology of the compactification is, there’s not really anything more to say.

Ah, but I was arranging the dimensions according to the non-interchangable qualities.

Level one - Time. Unique in its quality of only moving forward at a steady pace.
Level two - length, width, and depth. essentially identical to each other.
Level three - curled up dimensions the size of the planck length. Are they all the same size? Uni- or Bi-directional? perhaps that is determined by what side of the “sphere pyramid” they’re on?

But your level one and level two are certainly interchangeable. Your “width” dimension might very well be part of someone else’s “time” dimension depending on your relative orientation and velocity (and whether there are big gravitational sources in the area).

There’s no need, all these 'dimensions’are described in the context of the mathematical theory of manifolds (usually with some other additional struure imposed as well, usually differentiable structure and a metric).

As their are infinitely many manifolds, differing from each other in very many ways (and sometimes differing only by ‘infinitessimal amounts’) they don’t really lend themselves to kind of hierachy imposed by a perodic table-type schema.

And I keep reading that the laws of physics work equally well with one spatial dimension and three time dimensions.

There is only one thing, and that is space-time. We can represent it mathematically in a variety of ways, and we’re unsure of the number of dimensions needed to express all its properties, but I don’t think that trying to separate dimensions out gains you anything in understanding.

You can’t even say time is unidirectional. We don’t know enough about time to know its properties. It’s an actively violent argument.

It’s works perfectly well if you use 4 null ‘dimensions’ (read ‘coordinates’). The one thign to say abou sapcetime is that if you insist on an orthogonal basis you will end up with 3 spacelike basis vector fields and one timelike basis vector field.

I also forgot to add on my original post that the “unidirectionailty” of time is usually an emergant property in such theories and doesn’t eneter in to the equation when considering the “dimensions” of spacetime.

That’s mostly just a manifestation of Murray Gel-Mann liking nice tidy diagrams. You could organize those so they didn’t particularly resemble a hex lattice.

That’s really just a re-labeling of the same dimensions that we have. Basically, you can say that there are three dimensions with positive signature and one with negative signature, or you can say there are three with negative signature and one with positive. Either way, the important thing is that there’s one with opposite signature from all the others.