Circle Inversion Fractals

Inversion Limit Sets

Although we did not emphasize this point when describing the random IFS algorithm, the fractal generated is not the orbit of a point, but rather the limit set of the orbit.

Suppose X is a set of points in the plane.

The limit set L(X) of X is the set of all points that can be approximated arbitrarily closely by points of X.

That is, a point q belongs to L(X) if for every distance d > 0, there is a point of X closer to q than d.

More precisely, q is a limit point of X, if for every d > 0 there is a point w in X with 0 < dist(w, q) < d.

Note that 0 < d(w, q) implies that w is not equal to q.

Here dist(w, q) denotes the Euclidean distance between w and q.

The set of all limit points of X is the limit set L(X).

Here are three examples of limit sets.

Example 1 If X is a finite set of points, then L(X) is empty.
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.
Example 3 If X is the Cantor set, then L(X) = X.

Return to inversion limit sets.