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.

## 2 comments:

So in two dimensions, two lines intersect in a point:

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 ?

It's really three half-planes in the plane whose intersection forms a triangle. I think there is something useful in that fact, that could somehow be extended to the negative-dimensional case.

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.

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