Negative-numeric and imaginary dimensions

I doubt there's defined anything along those lines. Rotated 3D space is still 3D space, not 3iD or -3D. The amount of dimensions of space don't change in rotation or reflection. Perhaps you would be intrested in more insightful description of dimension?

The dimension of a vectorial space is equal to the size of the biggest free family of vectors in that vectorial space (or, if there is no maximal size, it is defined as infinity).

Because of that definition, only integers (and infinity) are valid dimension values. Using negative integers or complex numbers as dimension values would first require to propose an alternative definition of dimensions, because there is no such thing as a family of -1 vectors or a family of i vectors.

I agree w/ the previous postings but will express it a little bit different:

Negative numbers as well as imaginary numbers are valid values for _co-ordinates_ of dimensions (w.r.t. a suitable space, of course) but not for the _count_ of dimensions. Counting starts at zero and continues up to infinity.

A complex number, to show the difference, consists of a real part and an imaginary part. It can be understood as a vector, where the value of the real part denotes the co-ordinate on the real axis, and the value of the imaginary part denotes that of the imaginary axis. So a complex number could be interpreted as a vector in a 2-dimensional space. It is often illustrated so, w/ the real axis horizontal and the imaginary axis vertical.

I don't know about negative- and complex-valued dimensionality (though I doubt it exists), but the notion of dimensionality can be extended to positive real numbers.

Depends on what space you are referring to. If it is real coordinate space then only non-negative integer dimension is viable (not including infinite). If you are talking about inner product space or metric space then you can infinite dimensions, and if you are referring to fractal dimension then any positive real number is acceptable.

Mind you, real coordinate space in itself is still an abstract notion of the space we perceive, so abstract away!

Thanks for replies.
I have a clearer understanding of dimension now...