-1 dimensions?

[Mr Skeptic]

Is it possible for something to have a negative number of dimensions? Or is that just meaningless? How about imaginary dimensions?

 

The reason I ask is that there seems to be something missing in the table of simplex elements to make a complete Pascal's triangle.

[Shadow]

Isn't this the same as asking how a line with negative/imaginary length looks...?

Edited December 16, 2008 by Shadow

[ajb]

The Lebesgue covering dimension of the empty set is -1.

 

I have also come across people talking about negative dimensions in other contexts, but I have never looked into it. But for sure such an idea exists.

[Mr Skeptic]

I found another one:

http://www.newworldencyclopedia.org/entry/Dimension#Negative_dimension

The negative (fractal) dimension is introduced by Benoit Mandelbrot, in which, when it is positive gives the known definition, and when it is negative measures the degree of "emptiness" of empty sets.

 

Hmm, degree of emptiness. Does that mean that some empty sets are more empty than others?

[ajb]

All I can suggest is read the literature.

 

You should also be aware of supermanifolds where the odd dimensions behave like "negative dimensions" in some formulae. In essence, this is due to the extra minus signs associated with Grassmann odd objects.

[MolecularEnergy]

Well yes it is. In certain row matrix representations, some express the time dimension as -1.

[ajb]

MolecularEnergy (who is banned by the looks of it) is referring to the signifier of the (pseudo)Riemannian metric used in relativity. This is not the "dimension".