Iterating Inversion
This is a step by step introduction to the fractals obtained by inverting a point to multiple circles.
Inversion to 4 tangent, congruent circles.
Inversion Fractals
The above fractal picture was created by placing 5 circles in the plane and seeding the plane with a point. The point is then inverted to each circle; the resulting points are inverted; and so on, rapidly creating a very large number of points in a dramatic pattern.
If, as in this circumstance, points end up bounding around following a definite pattern, we say the points are attracted to this pattern. The region of initial points that create this pattern is called the basin of attraction. For these patterns, any starting point in the plane will produce the same pattern..
Invert a point to a circle, you get a point.
Invert a point to two circles, you get points distributed on the line connecting the centers.
Invert a point to three circles, you get points distributed on the orthogonal circle.
Invert a point to four or more circles, you get the fractal patters show in these pages.
Invert to shiny spheres, you get analogous fractal patterns.
inversion defined
Inversion defined
The inverse of a point to a circle is to the radius as the radius is to the original point; i.e., if P is the twice the radius from the center, then the inversion is half the radius from the center. The coordinate formulas are shown in the figure.
The center of the circle inverts to infinity.
A point in the interior of the circle inverts to the exterior; and vice-versa.
A point on the circle is its own inversion.
inversion of a straight line into a circle
Inversion of lines and circles
The inversion of a line is a circle through the center of the circle and vice-versa. In the picture the black circle is the one we are inverting to. Note that all three graphs intersect at the same points. This makes the line the radical axis of the two circles. The inverted circle goes through the center of the circle of inversion, which under inversion becomes the infinite point on the line.
The inversion of a curve is done point by point.
Here is the inversion of a line that does not intersect the circle.
These figures were done 20 years ago on an original mac using Microsoft Basic.
A straight line inverts to a circle and the inversion to three lines is the orthogonal circle.
Inversion to 3 circles
A point has an inversion to each of three circles, which new points invert to the same circles, which invert to the same circles, and so on, defining the iterative process. The result, which is independent of the starting point, is a set of points distributed on the orthogonal circle of the three circles.
a particularly beautiful inversion fractal for 4 circles
4 circles
The points are attracted to the orthogonal circle, if it exits, otherwise a fractal is created. A fractal is always created for 5 circles, which is bounded by the mutual orthogonal circles.
invert line to circle
same
hyperbola to circle
to 3 circles
