The image below shows a generalised Newton Raphson fractal constructed from the polynomial z3 − 2.55z2 − 1, Beneath it are a set of controls that allow you to modify the equation and the drawing parameters and to set up animated sequences. See Explanation for more information about the fractal or Instructions for details of how the controls operate.
If you only want to solve polynomials, rather than draw pictures, visit the polynomials page.
Equation: ,
Setup: 
Play: 
This page allows you to explore the Generalised Newton Fractal for polynomials with real coefficients up to order 9. Patterns are generated by picking a starting point and iterating until it converges on a root or we reach the number of iterations. The potential starting points are combinations of a seed (Sx,Sy) and a constant I'm calling alpha (Ax, Ay), so for any polynomial we could construct a 4−dimensional pattern based on the 4 coordinates Sx x Sy x Ax x Ay. As we are limited to a 2−Dimensional screen we construct patterns planar slices by keeping two of these parameters constant and varying the other two.
Currently we have two modes, keeping the seed constant and varying alpha or keeping alpha constant and varying the seed. There are four other potential slices that could be obtained by varying one 'S' and one 'A' which I may add at a later date. I have implemented these options in the similar 4−D Mandelbrot explorer page.
For more information about the Newton Raphson fractal or to see some images generated with my earlier Python generator please visit My Original Newton Raphson page. The Python version was able to detect cycles (something I didn't include in the WebGL version) so the nonconvergent regions show more structure. However the WebGL version appears to deal better with repeated roots.
These select between the five configuration panels, only one of which can be visible at a time. Tick the boxes to expand the section(s) you are interested in:
The area allows you to choose where in a particular polynomial you would lile to explore. The first row contains two images:
The second row allows you to vary the seed and α values. One of these will be "inplane" and act like a pan, the other will be an "outofplane" change that will affect the shape of the fractal. The arrow move the attribute by the amount in the edit box in the direction indicated. The circle in the middle will restore the value to the start value (see animation). The third row allows you to configure how zoomed in the image it, or the number of iterations that are used to construct it. The default value of 100 iterations is adequate for most well behaved equations but there are regions close to some of the nonconvergent locations that require more effort. Some of the "examples" show regions like this. 

∣≪  Return to the start 
≪  Fast rewind 
<  Step back one frame 
▶  Resume playing from the current position 
∥  Pause 
>  Forward one frame 
≫  Fast Forward 
≫∣  Skip to the end 
(c) John Whitehouse 2010  2021