Cubic Observations

Fractal Art is where art and mathematics meet, in this post the mathematics has intruded, so apologies to those whose only interest in fractals is artistic.

Although this is a post about cubic Mandelbrots, I’ll start with a note regarding the standard Mandelbrot. The formula for the Mandelbrot has only one term involving z and that is z squared, what happens when an extra term of z is added?

So, using the formula:

z_n+1 = αz_n² + βz_n + c

where α and β are constant real, imaginary or complex numbers,

what do the pictures look like? When the critical value is determined and used for the initial value of z the resulting picture is always a Mandelbrot, only its position in the complex plane and its size change. When a value other than the critical value is used the result is simply a perturbed Mandelbrot.

Now what of Cubic Mandelbrots? With the extra terms added for z squared and z and constants multiplied to each term, is the result using the critical value as the initial value of z always the distinct Cubic Mandelbrot?

No, not a bit of it!

The example pictures that follow will use formulae with both z squared and z present, z only, along of course, with the z cubed term. The first formula is:

z_n+1 = z_n³ + z_n² + z_n + c

So, what is the critical value?

f(z) = z³ + z² + z + c
f′(z) = 3z² + 2z + 1

the critical value is the solution of the following:

3z² + 2z + 1 = 0

there are two values of z when used in the quadratic formula will result in zero, the values are:

z = -0.333333333333 + 0.471404520792i
z = -0.333333333333 – 0.471404520792i

Both these are critical values and can be used as the initial value of z. The values are irrational i.e. the digits after the decimal point will go on for ever, the values shown are good enough to illustrate the generated image.

Example 1

The above picture is for the first critical value, the second critical value is the same but the other way up.

Example 2

Example 2 above zooms in on the branching structure at the top of Example 1, it show a number of well formed Mandelbrots of the sort you get with quadratic equations this confirms that the critical value used is correct.

Next equation:

z_n+1 = (1/3)*z_n³ – z_n + c

So, what is the critical value?

f(z) = (1/3)*z³ – z + c
f′(z) = z² – 1

So the critical values are 1 and -1.

Example 3

Example 3 uses the critical value -1, the critical value 1 produces a mirror image.

Example 4

Example 4 zooms in of the “rocky” end of Example 3 and shows a distorted but still well formed Mandelbrot surrounded by “rocks”.

Third equation:

z_n+1 = (1/3)*z_n³ – 2z_n + c

So, what is the critical value?

f(z) = (1/3)*z³ – 2z + c
f′(z) = z² – 2

So the critical values are root 2 and minus root 2. An approximate value of root 2 is 1.41421356237.

Example 5

Again the images produced by the two critical values are mirror images of each other.

Finally a formula with all the terms present:

f(z) = (1/3)*z³ + z² + z + c
f′(z) = z² + 2z + 1

which factorises nicely to give

f′(z) = (z + 1)(z + 1)

and hence just a single critical value of -1.

Example 6

Example 6 is the Cubic Mandelbrot. This leads me believe that if there is only one critical value the result is always the Mandelbrot of the relevant power which will explain why quadratic equations always generate a Mandelbrot when the critical value is used since quadratic equations can only have one critical value.

I haven’t studied the mathematics of fractals and what I’ve seen in various publications is mostly incomprehensible to me, so my conclusion is probably already known complete with a mathematical proof.

When I first prepared this post I had intend to include formulae without a z term, I forgot, however as I’ve only scratched the surface of this family of fractals there is scope for an other post…

Advertisements