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:

Parallel to Sx x Sy.

Parallel to Ax x Ay.

Seed (Zx,Zy)

Δ
Current:

Alpha (Ax,Ay)

Δ
Current:

Zoom:
Current:

Iterations:
Current:

'α' is the constant in the damped iteration z → z − α × f(z) / f'(z). Setting this to "1" will reproduce the standard equation.

The seed is the starting point of the iteration. In normal mode changing the seed is the same as moving the centre of the image, in alpha mode it alters the shape of the image.

"Current" shows the parameters used to generate the image above. "Start" is the set of parameters used to initialise an animation. "End" is the target point of the animation, where it will stop. The controls in the "Action" column allow you to manipulate these.

Parameter Current Start End Action

Z9

Frames:

Frame 0

Z8

Z7

Z6

Z5

Z4

Z3

Z2

Z1

Z0

Alpha

Seed

Zoom

Steps

Colour

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:

Explore

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.

Colouring

The colours assigned to the roots are derived from their location on the complex plane. These controls work like those in the "explore" section and allow you to change this mapping. There is no clever explanation, you'll just have to use trial and error.
"Contours" and "Tightness" map how the number of iterations is mapped to a contour band.

Settings

This section is mainly used for two reasons:

Defining the polynomial

Setting up the end-points of an animation

but it can also be used as an alternative to the "explore" section to locate the fractal you would like to draw.
The first column identifies the parameter being set, the next three labelled "Current", "Start" and "End" identify the parameter values being used, and the final column: "Action" has a number of buttons for manipulation of these values. The three main columns are:

Current This column contains a number of edit controls where you can type in new values. It is also used to display the values currently on display, so if you change the displayed pattern using any of the other controls your typed value will be overwritten.
Start This column shows the values used for the start of an animation. These values are also used when you select one of the "reset" options in the "explore" or "colour" sections.
End This column shows the values used for the end of an animation. When an animation is running it interpolates all the values from the start to the end. Positions and polynomial coefficients are interpolated linearly, zoom and iterations are geometric. Mixing pan and zoom in the same animation can have unexpected results as if you are zooming in the pan will start off slowly and then speed up towards the end as the range of the image decreases.

The action buttons are:

Update Current Updates the values used to construct the image from the edit boxes in the current column. This will cause the image to be redrawn.
Current to Start This copies the values currently being displayed from "current" to "start". Note: If you have edited any values you should select "Update Current" to apply them first.
Current to End This copies the values currently being displayed from "current" to "end". Note: If you have edited any values you should select "Update Current" to apply them first.
End to Current This copies the values from the "start" position back to the current. This is the same as clicking the "to start" button in the animation controls. The image will be redrawn.
Start to Current This copies the values from the "end" position back to the current. This is the same as clicking the "to end" button in the animation controls. The image will be redrawn.
Swap start & End This swaps the "start" and "end" values, useful if you want to reverse your animation.
Frames Not actually a button. This controls the number of frames in an animation, and when the animation is running the number labelled "Frames" shows you where in the animation you are.

Load/Save

This area allows you to save interesting patterns so that you can recover them at a later date. The saving and reloading is done using JSON syntax. There are two buttons and a drop-down list:

"Display": This will render the settings used to construct the current image as a JSON string in the editing area below the buttons. To save this you can copy it to the clipboard and paste it into a text editor of your choice.

"Apply": This will take the text in the window below the controls, which is exected to be in JSON format, and use it to update the current pattern. You can use this to reload a pattern you saved earlier, or if you are confident you can edit the text already in the window. The parameters all have the same or similar names to the corresponding values in the settings.

"Examples": This contains some example locations that you can view by selecting them from the list.

Resolution

This section allows you to set the size of the image as it appears on the screen and the resolution of the underlying image. The higher resolution image is intended to provide a higher quality image when printing. You can view this higher resolution view by using the browser's "open image in another tab" functionality or by saving the image to a file.
This functionality is in two parts, there is a list of screen sizes you can choose from and then a multiplier that will expand the underlying image, so if you choose "640x480" and "5" you will have an underlying image with resolution "3200x2400".
Starting an animation will set the multiplier back to 1.
The WebGL precision value sets the precision mode in the WebGL code that constructs these image. I've never seen it make any difference.

Hide

Just hides all the controls.

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 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 central 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.

Image Size:		This is the size in pixels of the image shown on the screen.
Canvas Size Multiplier:		This allows you to render the image to a higher resolution to that shown on the screen. Only use when you intend to save the image, possibly for printing. Very large images might be too large for the canvas's buffer, in which case your image might be truncated. Animation will set this back to 1.
WebGL Precision:		This allows you to change the WebGL floating point precision. I think most graphics cards don't implement 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

Newton Fractal