# Introduction

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.

# Interactive Section

Equation: ,

 Setup: Play:

## Explanation

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 co-ordinates 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.

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 non-convergent regions show more structure. However the WebGL version appears to deal better with repeated roots.

## Images

 An image from the fractal generated by z3−2.51133z2−1. This is just on the well-behaved side of the transition from non-convergence to convergence. This unusual colouring in the centran region is caused by the small number of iterations (100) being unable to resolve the detail. Another image close to the location of the previous one, this time though the number if iterations is 10,000 (and I've zoomed in a bit and changed the colour scheme). A lot more of the detail is filled in but there are still unresolved areas (at the centres of the spirals). You can make out the boundary of the Julia set like region, the transition from the non-convergent region to convergent seems to be very abrupt but it is difficult to see this with a finite number of iterations. You can use the animation controls above to narrow down on this.

## Other pages

(c) John Whitehouse 2010 - 2021