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Newton's Method and Halley's Method Fractals

These fractals are derived from methods for finding numerical solutions to equations. The Convergent Fractals page in the Technical section has information on the mathematical details. The fractals show how quickly each initial estimate converges to a solution.

These fractals are very nice, but the underlying equations are not complex enough to give a good variety of structures at higher magnifications. That means the fractals are truly self-similar, and look exactly identical at all levels of magnification, so there is nothing more to see by zooming in.

Choice of Function

By choosing different functions to solve, we can create different fractal images. One of the earliest fractals made many years ago was based on finding values of z that have z3-1=0, which are the cube roots of 1. In the complex number plane, there are three values that solve this equation, and depending on which initial starting point is chosen, Newton's method will converge to one of these three roots.

I have made a few images applying Newton's method to the sine function, and also to finding the 10th roots of 1.

Halley's Method

Halley's method is a more complicated and more rapidly converging technique that is similar to Newton's method. It works better than Newton's method for its intended purpose, which is to find solutions to equations, but its more rapid convergence means the fractals it generates are actually less interesting.

The Secant Method makes more interesting fractals, but is not as useful in numerical analysis.


Newton's method is the method I chose to use in my high-precision division function. This is a standard approach, as it converges more rapidly than the simpler approaches and doesn't require much more calculation time.

Newton's Method For Complex Roots of 1
Newton Fractal

Newton's method applied to z3-1=0. This is a classic image, one of the first fractals ever made back in the early days of fractal exploration.

Newton Fractal

Zooming in to the center, we see the theme that is repeated infinitely throughout this fractal.

Newton Fractal

Zooming in again. Can you tell how deep? This is all you will ever see in this thing, no matter where you look or how much you magnify.

Newton Fractal

Each point in the plane converges to one of the three solutions to z3-1=0. Here I have made the classic red-green-blue rendering showing which root each point goes to. Getting my program to do this was a hassle, but now I have the ability to create very intricate colorings with combinations of different palettes.

Blue represents starting values that converge to z=(1,0). Red and green represent starting values that converge to the complex roots of 1, which are
-1 + i 31/2 and -1 - i 31/2

What's remarkable about this is how complicated the boundary between the different domains is, and how you can start close to the domain of one root but end up converging to a different one.

Newwton's Method For Zeros of the Sine Function
Newton Fractal

Newtons' method applied to sin(z)=0 in the complex number plane. This view extends horizontally a little more than -5pi/2 to +5pi/2.

Newton Fractal

The sine function again. No matter where you look this is all you will ever see, possibly with some asymmetry between the sizes of the loops.

Tenth Roots of Unity
Newton Fractal
Newton Fractal

Newton's method applied to x10=0, zoomed in a little. These were rendered with a very nice smoothing formula by David Makin. The top one uses his formula unaltered; the bottom one includes an exponential smoothing step.

Halley's Method
Halley Fractal

Halley's method for x3-1=0. Note that the more efficient root-finding method gives a less interesting fractal.

Halley Fractal As with the Newton fractal, this is a Julia set type of rendering and will look exactly the same no matter where or how much you zoom in.