Click on the image to see its parameters.

Tutorial: Breaking Newton (Complex) Formula

Text and Images © 2018 Kerry Mitchell


If you've played with Newton's Method fractals before, you're probably used to seeing this type of image:

This formula doesn't do that.

Newton's Method is used to find solutions to an equation. The above image has four colors because the underlying equation has four solutions, and each pixel represents a complex number that winds up at one of those four solutions. With this formula, the underlying equations generally don't have any solutions and the resulting images come from the routine's wandering around, trying to find one.


The standard Newton's Method fractal uses an equation with several complex solutions. This formula uses an equation with complex solutions, too, but interferes with the process so that the solution is never found. For example, the equation

z4 = c,

where c is a constant, has four complex solutions, the four complex fourth roots of c. The regular Newton's Method algorithm has no trouble finding them, as shown in the image above. It works by taking an initial guess at a solution (initial z), determining an adjustment to it, and combining the adjustment with the old z to make a new z. The three images below show three ways in which this process is sabotaged such that a solution is never found. In the first, the value of c is changed each iteration. In the second, the new value of z is tweaked each time. And in the third, the weighting of the adjustment (via a factor I call the Newton factor) is varied each time. This is what this formula does.

The equation that is to be "solved" is of this form:

lefthand side = righthand side

Each side can be: c, a power of z, an exponential or logarithmic function, or a trigonometric function of z (circular or hyperbolic).


The parameters are in three groups: general, c, z, and Newton factor.

General Parameters.

C Parameters Z Parameters Newton Factor Parameters


Sample Image

Here’s how to make the image at the top of this page.

  1. Start a new fractal, with these characteristics:
  2. On the Formula tab, set:
Your image should look like the example. If not, go snag the parameters from the examples page.

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