Negative Dimensionality

So these are kind of what two positive dimensions look like. We generally continue to the left, and down, to reach negative infinities, but in a sense this is only because where we place the origin is arbitrary; regardless, the negative infinities, reached by going down or to the left, are still part of the positive-dimensional space.

If we added a third dimension here, it’d be orthogonal to these two; generally, we’d have it, from our perspective here, passing through the origin away from or towards the viewer; the choice there doesn’t matter for our purposes.

Okay, so, what does a negative dimension look like?

Well, the positive dimensions all go outward from the origin. Fractal theory tells us some properties of the negative dimensions - if you cut a negative linear geometry in half its measure doubles, cut the sides of a negative square geometry in half and its measure quadruples.

Compare to positive geometries; double the length of a line, and its measure straightforwardly doubles. Double the sides of a square, and the area quadruples.

But this doesn’t tell us much about how to think about them.

Well, a dimension is in some respect a line from an origin to infinity; a positive dimension connects the origin to an infinity whose distance - whose measure - from the origin is positive. A negative dimension is a negative distance away from the origin. It’s not just negative, mind, but let’s hold off on that for now.

Okay, so how do we visualize negative distance?

Well, if we visualize the positive dimensions as extending away from the origin, we don’t have much room “outside the origin” to think about the negative numbers. Also, the infinities the positive dimensions reach are each a positive infinite distance away from each other. Perhaps we could think of the infinities the negative dimensions reaching getting a negative infinite distance away from each other.

What does our considered-as-negative distance entail?

A distance is the measure you would have to traverse to get from one point to another. For a point A and B, as the distance increases, you have to traverse more “measure”, whatever exactly that is, to get from A to B. Until you get to 0, when they overlap.

Does this relationship break when distances get into the negatives?

Let’s ponder the mathematics, here, paying particular attention to what happens to distances as you approach the origin.

For the points [0,1] and [1,0], the distance between the two points is sqrt(1^2+1^2). Well, we get two values out of this; the square root of 2, which can be positive or negative. Ah, so merely thinking in terms of positive and negative isn’t sufficient; you still have to traverse this space, regardless of its sign.

Well, what if we half the sides? So [0,.5] and [.5,0]. Well, now we get sqrt(.5^2+.5^2), or sqrrt(.5), which ends up being, unsurprisingly, one fourth of the square root of 2.

So this equation isn’t going to work for negative dimensions. [0,1n] and [1n,0] have a distance of … what? It’s a quarter the “distance” as [0,.5n] and [.5n,0].

Well, we end up with an equation more like sqrt(1/x^2 + 1/y^2).

Which ends up producing a somewhat odd topology; there are, effectively, two overlapping coordinate systems, which I’m going to call the unit coordinate system, and the measure coordinate system.

The unit coordinate system goes from 0 to infinity_n; the measure coordinate system, on the other hand, goes from infinity to 0.

The measure coordinate system is measuring the distance between two points of equal unit on different axes - the measure - the equivalent to positive-dimensional distance - in negative dimensional space between [0,0n] and [0n,0] is infinite. So whereas in positive dimensional spaces, we may say that all possible dimensions cross the origin at the same point, in negative dimensional space, the situation is more complicated; in one sense, they all meet at the origin, but in another, they’re infinitely separated at the origin.

On the other hand, the measure between all the negative dimensional infinities is 0; in one sense, all negative dimensions meet at the same infinity. In another, well, they don’t.

Which is correct? Both. And neither.

In the sense that the origin is a single point, however, it’s notable that all negative dimensions meet at the origin. And in the sense of the measure, it’s notable that all negative dimensions meet at infinity.

Take all the infinite possible negative dimensions, form them into a bundle of fibers, and use them to connect the origin to infinity; you can organize them more or less arbitrarily, but of particular interest is that you can shape an infinite number of dimensions into a two-dimensional surface: A sphere. An infinite number of negative dimensions is not analogous to an infinite number of positive dimensions; they behave fundamentally differently.

We have a name for this strange topology, in point of fact, and the construction of it is very similar to what we have described. It’s the Riemann Sphere.

I suspect, but am not going to attempt to prove, that a given negative dimension, as represented in this spherical geometry provided some kind of consistent relationship of measure to the other negative dimensions, inscribes a particular helical spiral around this sphere; this is because .5n is twice as “far” from the origin as 1n, which is twice as far from the origin as 2n. I’m reasonably certain we can turn this into a more sensible spherical geometry by wrapping each negative dimension in a particular “distance-preserving” helix around the sphere.

Given all this came from my musings on crackpot physics, it should be unsurprising for me to note that I strongly suspect the curvature of this helix to be in some sense significant from a physics-oriented perspective, and in particular, I suspect that general relativity may fall out of all of this, if one orients the geometries involved such that infinity_n is the future, and the origin is the past, where each particle is an “instantiation” of this particular geometry; converting between the coordinate systems involved looks like a nightmare of tensors, to me, so, if somebody wants to pursue that insane path, good luck.

Probably somebody more rigorously minded could either prove all this out properly, or, even more intriguingly, show whatever differences exist between the topology implied by the negative dimensions and a Riemann Sphere proper. If there aren’t meaningful differences, and if I’m correct that general relativity and spacetime curvature falls out of coordinate transformations between particles, then there’s a correspondence theory whose acronym fails me for the moment that probably just kind of unifies physics, maybe.