Circle Inversion Fractals

Inversion Limit Sets

Example 2 If X is the set of all points (x, y) in the plane, with both x and y rational numbers, then L(X) is the entire plane.

We must show that for any (a, b) in the plane, and for any d > 0, there is a point (x, y) in X with dist((a, b), (x, y)) < d.

In a square with one corner at (a, b) and diagonal of length d, all points lie within a distance d of (a, b).

By the density of the rationals in the reals, there is a point on a horizontal side of the box and having x-coordinate w, a rational number.

Similarly, there is a point on a vertical side of the box having y-coordinate v, a rational number.

Then the point (w, v) lies in the box, so is within a distance d of (a, b), and belongs to X.