Could we have negative dimensions?

[MathematicalPhysicist]

So I had a discussion with my brother about this (it was sort of a joke, but I am a mathematician and every joke I turn into a theorem and vice versa... :-)), it was kinda of short.

But if we already have dimensions which aren't whole integer numbers (he didn't know that, and didn't seem interested about it) couldn't we define negative dimensions?

I sort of trying to visualise such a thing, not sure what to define here. obviously this notion should generalize Hausdorff dimension definition.

 

[chiro]

Hey MathematicalPhysicist.

One thing I think you should realize is that anything negative is just something that is relative to something else.

A negative number is relative to an origin just like an angle (or its cosine) is relative in terms of orientation.e

We also have integrals and other measures that are relative and for this we call these "signed" measures which are apparent in areas, volumes, determinants, and other measures.

So in the same spirit, you could define a negative dimension is something that has some form of relativity to something else in that it has less dimension, information, or something else that maintains its meaning but capture the spirit of this relative reference.

All negative things must have this form of relativity and I don't think that its not possible to allow dimension to extend to this in the way that we do to determinants, areas, volumes, and numbers.

 

[hilbert2]

So I had a discussion with my brother about this (it was sort of a joke, but I am a mathematician and every joke I turn into a theorem and vice versa... :-)), it was kinda of short.

But if we already have dimensions which aren't whole integer numbers (he didn't know that, and didn't seem interested about it) couldn't we define negative dimensions?

I sort of trying to visualise such a thing, not sure what to define here. obviously this notion should generalize Hausdorff dimension definition.

You should probably read this article: http://users.math.yale.edu/mandelbrot/web_pdfs/123negativeFractalDimensions.pdf .

If a single point has dimension zero, then the only way for a set of points to have negative dimension is that the set has to be empty. Apparently sets can be empty to different degrees.