Critical Values   2 comments

Critical values are used as initial values for generating fractals that are expected to diverge or converge, values other than a critical value produce “perturbed” fractals. The most famous fractal is of course the Mandelbrot set and its critical value is zero, using other values produce perturbed Mandelbrots.

Note: critical values do not apply to Julia fractals, i.e. none of the fractals parameters is set to the location in the complex plane, the location in complex plane is used as the initial value.

Many fractal generating programs do not make it easy to set the initial value for a fractal, Saturn allows the initial value to be set to any fixed value, the location in the complex plane or the location in the transformed complex plane.

There is a reason the critical value for the Mandelbrot set is zero, there is a method to determine the critical value for a fractal formula and using that method produces zero for the Mandelbrot.

In general a fractal formula is as follows:

zn+1 = f(zn)

The critical value is determined by solving:

f′(z) = 0

What this means is that you have to differentiate f(z), for the Mandelbrot set,

f(z) = z2 + c
f′(z) = 2z

So solving 2z = 0, z must equal zero.

I do not intend to explain the rules for differentiation there are plenty of websites that will do a better job than me.

The higher power Mandelbrots have the general form:

f(z) = zn + c


f(z) = zn + c
f′(z) = nzn-1

The solution f′(z) = 0 for these higher power Mandelbrots is always the single value zero. These fractal are generated using Saturn’s Zcpac fractal type, “cp” for complex power and “ac” for add constant.

There are other higher Mandelbrot power fractal types provided by Saturn: Quadratic, Cubic, Quartic and Quintic these have in turn one solution, one or two solutions, upto three solutions and upto four solutions. It is possible that the parameters used result in a single solution which will result in the same fractal as is produced using Zcpac only the size and location will be different. It is easiest to chose parameters such that two solutions can be found, such critical values will give a wide range of different fractals, it gets much harder to find solutions when the number of solutions is greater than two.

Not all fractal types produce f′(z) = 0 formulae that can be solved and the differentiation of f(z) gets more difficult the more complicated the fractal formula.

If a critical value can not be determined you can use whatever fixed initial value you like or you can use the location on the transformed or untransformed complex plane.

Posted 24 November 2012 by element90

2 responses to “Critical Values

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  1. What I really like is that I have not the slightest grasp of the maths (or math) and not the barest understanding of the images, but ever since learning botany (the endless unfolding fronds etc) plus seeing that documentary on ‘how long is a piece of string’, this whole world has fascinated me. Feel free to explain more …

    • Thank you for comment. Most of the mathematics I’ve seen in several books on fractals is incomprehisible to me. The subject of critical values may have been covered in those books but not in such that it is obvious. I can’t remember where I first encountered the method I use for critical values, it was probably somewhere on fractal forums. From time to time I’ll post articles on particalur fractal types with examples using various values for their parameters.

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