Ever since I learned about fractional dimensions, I wondered what negative dimensions would be. There is an answer, and a paper by Mandelbrot is how I learned about it. The basic idea is simple:
In 3-dimensional space, the intersection between two planes is a line. You can calculate this as follows:
The planes on the left of the equation are each two-dimensional. The space is three-dimensional, leaving one dimension left over for the intersection: a line.
A plane and a line intersect in a point:
The point is zero dimensional.
What do two lines intersect in?
Solving for x, we find that two lines intersect in a negative-one-dimensional space.
You can find the intersection of this space with yet another line:
In this case, y must be -3. So the intersection of three arbitrary lines in 3-dimensional space is -3-dimensional.
What I'm trying to figure out is what corresponds to polygons and polyhedra in negative-dimensional spaces.