The signed-dimension D of a set should be the length exponent at which that set can be measured.
This is different from the standard notion of dimension, which is the length exponent that part of the set can be measured.
The difference is significant. Normally the dimension of a line is said to be 1, but infinitely long lines can't be measured, only counted. The real-dimension is 0, and the same for infinite planes and volumes. It is only finite extent geometry that has the dimension one usually assigns to it. For example:
- Line segments have D=1 so their lengths can be measured
- Disks have D=2 so their areas can be measured
- Points D=0 so they can be counted.
By this reasoning the set of integer vectors in 2D space can be measured in points per square length unit, D=-2 and similarly for higher order spaces. Notice that these densities can't have dimension less than the negative of the topological dimension, which mirrors the case for positive dimension shapes. Notice also the consistency that in positive dimensions, if you measure assuming D is higher you get 0 and assuming D is lower you get infinity. For negative dimensions the same is happening but the other way around. For instance the number of integer vectors in 2D space per length unit is infinity.
So negative dimensions refer to densities as positive ones refer to quantities. It is somewhat analogous to contravariant (e.g. vectors) and covariant quantities (covectors or one-forms) in physics. But negative dimensions don't have to just be point-like. For instance a 2D grid of lines spaced a metre apart can be measured as two metres of line per square metre, which is 2m^-1. Nor do they have to take on integer dimensions, for example a set of points equidistant along the Koch curve can be measured as the number of points per length^1.26. Another way to think about negative dimensions is that they represent extrapolations of a replacement method, and positive dimensions represent interpolations:
As you can see from the last two, if the replacement scheme acts on lines then the negative dimension version reflects from the 1D position. The replacement scheme can also act on fractal geometry:
Here the negative dimension version reflects from the 1.26D position of the Koch snowflake. In essence the negative dimension geometry is measuring the density of Koch curves within the infinite extent version of the Koch fractal on the right side.
For replacement schemes, let's call the initial shape in the iterations the tile. Then finite geometry uses the usual Hausdorff dimension:
This shows that geometry can be split into a positive dimension (quantity) part and a negative dimension (density) part, and so it is still possible for shapes of infinite extent to have positive dimension, for example an infinite row of disks measures the disk area per metre, so has dimension 1.
In the real world the examples are more stochastic but the definition applies equally. For instance, if you were to measure a country's coastguard capability, you might try to measure the number of coastguards per kilometer of coastline, but this will change depending on what length ruler you use to measure each kilometer. Instead the coastguards can be measured in number per km^1.3 (i.e D=-1.3) where 1.3 is the fractal dimension of a representative piece of the coastline. Just as real coastline is not fractal to infinitely small scales, the coastguards do not cover infinitely large extents. Both are just approximations in the real world.
Aside from the Hausdorff dimension, there are several other dimension measures (box-counting/Minkowski dimension, information dimension, correlation dimension), there are all generalised in the so called q-dimension formula, normally written:
We can modify this to give a generalised version of the signed-dimension:
q=0 is the box counting/Minkowski dimension, q=1 is the information dimension and q=2 is the correlation dimension.
In all cases we can either write the dimension of a set directly, e.g. -0.26D, or leave it in its component form, e.g. 1-1.26D for clarity.