Click on the image to see its parameters.

Tutorial: Breaking Newton (Real) 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. In this formula, the equations use real numbers. That makes it easier to keep them from having any solutions. For example, the equation

x2 = 1

has two real solutions, x = -1 and x = 1. The regular Newton's Method algorithm has no trouble finding them. However, the equation

x2 = -1

has no real solutions, and the Newton's Method algorithm will wander around forever searching for them. For our purposes, this is good. The fractal formula takes two such equations, say,

x2 = -1 and
y4 = -2

and runs them both through one iteration of the algorithm. Then, it will combine both x and y into a complex variable z, manipulate z in some fashion, and then break it up into its real (x) and imaginary (y) parts, to iterate again.

More precisely, the x and y equations are of the form:

f(x) (or f(y)) = c,

where f(x) (or f(y)) is a function specifically chosen because its equation may not have a solution, like x2 or y4. The value c is a real parameter, like -1 in the above examples. For more interest, c can be a constant or be allowed to vary each iteration. Varying is accomplished by adding a value, multiplying by a value, or raising c to a value. Once x and y are iterated, they are combined into the complex value z (which is used with the coloring formula). Like c, z can be manipulated each iteration, in the same basic ways.

Newton's Method works by determining an amount by which to change the variable each iteration. How much the variable is changed can be altered through the "Newton factor," which is normally 1. This formula allows you to set your own value, as well as change the value each iteration, like you can with c and z.


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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