Negative Dimensional
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:
2+2=3+1.
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:
2+1=3+0.
The point is zero dimensional.
What do two lines intersect in?
1+1=3+x.
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:
1+-1=3+y.
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.
Comments
1+1-2=0
and a point and a line intersect in a -1 dimensional object:
0+1-2=-1
But in 2 dimensions, three arbitrary lines typically form the boundary of a triangle. Perhaps
1+1+1-2=1 ?
But I don't think your suggested form of the equation can be consistent. The intersection of three lines in the plane doesn't typically form a line, which is what that equation says.
Mandelbrot's approach uses a rule involving codimensions, which might be a better approach for some people.