Julia and Mandelbrot Sets

Alternate Parameter Planes

This is part of the Introduction to Julia and Mandelbrot Sets. For more information on complex numbers, see Dave's Short Course on Complex Numbers.

The µ-plane

The collection of quadratic polynomials can be parameterized in different ways which lead to different shapes for the Mandelbrot sets.

The typical parameterization is in terms of a complex parameter µ, and the function being iterated is f(z) = z2 - µ. If the set

0, f(0), f(f(0)), ...

is bounded, then µ lies in the Mandelbrot set. With this parameterization, the most notable feature of the set is a cardioid studded with circles.

The 1/µ-plane

This isn't so much a different parameterization, but a different view of the usual parameterization. If alpha = 1/µ, then the function being iterated is f(z) = z2 - 1/alpha. With respect to this parameterization, a point belongs to the Mandelbrot set in the 1/µ-plane if its inverse belongs to the Mandelbrot set in the µ-plane.

The inverse of the cardioid is the exterior of a teardrop shape. The circles on the outside of the cardioid are inverted to circles on the inside of the teardrop. The cusp of the cardioid becomes the cusp of the teardrop.

The 1/(µ+.25)-plane

A similar modification of the µ-plane can be found by inverting the µ-plane after translating it right 0.25 units. The cusp of the cardioid is then moved to the origin, and that inverts to infinity.

With this transformation, the cardioid is sent to the outside of a parabola. The circles on the outside of the cardioid are inverted to circles on the inside of the parabola.

All three of these planes, the µ-plane, the 1/µ-plane, and the 1/(µ+.25)-plane are planar representations of the same complex sphere. These transformations preserve the features of inversive geometry. In particular, circles remain circles under inversive transformations.

The lambda-plane

Another common parameterization of the quadratic functions is by a complex parameter lambda. The functions being iterated are f(z) = lambda z(1-z). If the set

1/2, f(1/2), f(f(1/2)), ...

is bounded, then lambda lies in the Mandelbrot set.

The correspondence between µ and lambda is µ = lambda2/4 - lambda/2. Note that two values of lambda correspond to one value of µ, except that only one value of lambda corresponds to the cusp of the cardioid in the µ-plane. In effect, the cardioid is doubled into a pair of circles in the lambda-plane. The precise relation between lambda and µ is that lambda is 1 plus or minus the square root of (1+4µ), and µ is 1/4 of lambda squared minus 2 lambda.

Each circle has radius 1. The left circle is centered at lambda = 0 while the right circle is centered at lambda=2. They meet at lambda = 1, which corresponds to the cusp µ = -0.25 in the µ-plane.

One particularly nice aspect of the lambda-plane is that the circles are attached to the unit circle (the one centered at lambda=0) at the roots of unity. There is a big circle attached at 1, a smaller circle at -1, two smaller circles at the two primitive third roots of 1, two smaller circles at the two primitive fourth roots of 1, that is at i and -i, and so on.

The 1/lambda-plane

If the lambda-plane is inverted, everything outside the unit circle is moved inside the unit circle and vice versa. The left circle in the lambda-plane becomes the outside of the unit circle while the right circle becomes a circle inside the unit circle attached at the point 1. This image also shows circles attached at the roots of unity.

The 1/(lambda-1)-plane

Perhaps the most interesting presentation based on lambda is found by translating the lambda plane left one unit then inverting. The point where the two circles meet is moved to zero, then inverted off to infinity. The interiors of the two circles are mapped to the outside of an infinite vertical strip. The strip is one unit wide centered at 0. Small circles are tangent to the sides of the strip and inside the strip. Note that the circles are not of uniform size and they aren't uniformly placed.

The 1/(µ-1.40115)-plane

The point 1.40115 is called the "Myreberg point" of the Mandelbrot set. The sequence of circles attached to the right of the main cardioid get smaller and smaller and approach this point. That point is not the end of the Mandelbrot set since there's a path leading off to the right.

Inverting on that point makes all these circles larger and larger instead of smaller and smaller. Exploring this inverted plane can be quite interesting. The original cardioid is turned around and distorted a bit. It appears near the center of this image. The big circle to its left is the inversion of the small circle to the right of the original cardioid. The little bit of a line moving off to the right of the image is the end of the path in the µ-plane ending at z = 2. Here's a big view of this plane, where x and y each vary from -400 to +400.

The 1/(µ-2.0)-plane

The end of the Mandelbrot set occurs at the point 2.0. We'll invert the µ-plane there for this last version. The original cardioid looks smaller, and the first circle larger. The finite line studded with cardioids that used to end at 2.0 now goes off to minus infinity.

© 1994.
David E. Joyce

Department of Mathematics and Computer Science
Clark University
Worcester, MA 01610

These pages are located at http://aleph0.clarku.edu/~djoyce/julia/