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.

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

Instructions

The "Setup" buttons:

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:

  • Clicking the image on the left allows you to view the fractal created by choosing a specific value of alpha and varying the seed. Sx is horizontal, Sy is vertical.
  • Clicking the image on the right allows you to view the image created by choosing a specific value of the seed and varying alpha. Αx is horizontal, Αy is vertical. The fractal only converges for ∣α−1∣ <1.

The second row allows you to vary the seed and α values. One of these will be "in-plane" and act like a pan, the other will be an "out-of-plane" 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 non-convergent locations that require more effort. Some of the "examples" show regions like this.

The "Play" Buttons

∣≪ 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

Images

an image 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.
an image 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