Negative Dimensionality

[Nick]

How is it possible for their to be negative dimensions? I always thought of the number of dimensions as the number of axis's perpendicular to each other.

[wendy]

The lowest dimension i have heard of is -2, but i never been there to see what it means. Ok.

[moonlord]

What does it represent, though? I've heard of -1D as a empty set, but I don't see why...

[houserichichi]

Negative dimension of what? The "axis" argument doesn't allow negative numbers so wherever you heard about them doesn't refer to the dimension definition you're used to. There are many other "types" of dimensions out there since no single one can describe every mathematical object. Fractals have fractional dimensions but that doesn't mean they have a partial axis...it's a totally different definition altogether.

[bo198214]

If we have a look at the inductive dimension (that already can be defined on any topological space), we see that it is simply useful to assume the dimension of an empty set to be -1.
Roughly because the empty set is the boundary (boundary can be defined in any topological space) of every point, and every point should have dimension 0. And the dimension of the boundary (of subsets) should be (at least) one less than the dimension of the regarded set.

[thigle]

The Adventures of Discriminant Boy In The Negative Dimension, ( from a footnote in Kent Plamer's 'Negative Dimensionality and the General Schemas Theory' from holonomic.net )