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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…

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At first glance the Mandelbrot appears to be a strange knobbly shape. It is much more interesting than that, it is incredibly complex and intricate with miniature “copies” of itself appearing again and again in its spirals, fronds and dendrites. This post is the start of a sequence of images from the Mandelbrot set each succeeding image being a magnification of the preceding image. I first produced a set of increasingly deep zooms into the Mandelbrot starting with a relatively deep zoom with the centre of the image adjusted slightly as the depth increased.

The easiest way to show the location of that first image was to zoom out decreasing the magnification by a factor of 25 (dimensions by a factor of 5) until the location in the Mandelbrot set was clear.

Here is a picture of the Mandelbrot set:

The Mandelbrot Set

The first picture in the sequence zooms in on the filaments protruding from the topmost bud of the main body of the Mandelbrot.

Into the Depths No. 1

The width of the first image was 4, the width of this image is 0.5 giving a magnification of 64 times.

Into the Depths No. 2

Into the Depths No. 3

Into the Depths No. 4

Into the Depths No. 5

Into the Depths No. 6

Into the Depths No. 7

Into the Depths No. 8

The width of the image above is 6.4e-6 or 0.0000064 which gives a magnification of 390,625,000,000 or approx. 3.9e11.

There 26 images in total before we get to the start of the original set of images. As the depth of zoom increases the number of iterations required and the number of bits used in the calculations increases so the time required to generate each images gets progressively longer. So far from the start of the original sequence there are 58 images, all of which are currently 750×600 seed files which will be expanded to 6000×4000 and then scaled to 1200×800 when they will eventually be posted.

From the next post, in this series, there will be only one image per post.

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This is the fifth part of my guide to a fractal type I’ve called Cczcpaczp for want of a better name. The formula is:

z_n+1 = c(alpha*z_n^beta + gamma*z_n^delta)

So far I’ve dealt with the images that can be generated when:

• the initial values is the location in the complex plane.
• the initial value is the critical value and how to calculate it.
• the powers have opposite signs and the absolute powers are 2 or greater and the initial value is the critical value.
• one power is 1 and the other is an inverse power of 1 or greater, i.e. -1, -2, -3 …
• one power is -1 and the other power is 2 or greater.
• when one power is 1 and the other is an inverse power of 1 or greater and how they are affected by the bailout value.

In this part I’m dealing with positive values for both powers where one power is always 1, to maintain a critical value of 1 the values of alpha and gamma should have opposite signs and that the absolute products of alpha and beta, and gamma and delta are equal. The values of beta and delta should never be equal as the terms will cancel out and the result of each iteration will be zero regardless of location. I’m skipping the first pair of values 1 and 2 because the resulting image does not match the pattern starting with 1 and 3. The image is centred on the origin for all these fractals and the number of main buds or bulbs starts at 2 and increases by one every time the power is increased by one.

So for

alpha = -3
beta = 1
gamma = 1
delta = 3

the result is a circular central section with radial buds, there are only two of the largest buds:

For

alpha = -4
beta = 1
gamma = 1
delta = 4

there are three of the largest buds:

and so on …

Now back to the case where one power is 1 and the other is 2, the position of the centre of this fractal is at -0.5 + 0i and not 0 + 0i and there is no central circle with radial buds. This particular fractal has a name all of its own, it is called the Lambda fractal.

alpha = -2
beta = 1
gamma = 1
delta = 2

and z₀ = 1 as it is for all the pictures in this post.

Part 6 will deal with the fractals generated when both powers are positive or negative and their absolute values are all 2 or more.

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This part deals with the fractals generated where the powers have opposite signs and, ignoring signs, one of the powers is one and the other is two or greater. Before I continue, I’ll mention the case where the powers are 1 and -1, which is the start of the sequence of a family of fractals where one of the powers is 1, the sequence where one of the powers is -1 produces a completely different family of fractals.

A reminder of the fractal formula:

z_n+1 = c(alpha*z_n^beta + gamma*z_n^delta)

For the fractals in this part the initial value is set to the “critical value”, for a description of how the critical value is calculated see part 2.

So starting with

alpha = 1
beta = 1
gamma = 1
delta = -1

the critical value or z₀ = 1

The resulting fractal is enclosed in a circle with the two main buds merging into each other. I’ve been using a very high bailout condition of norm(z) > 16000000, so the image is made up of a multitude of dots and a high number of iterations is required, in this case 12000.

An even higher higher number of iterations is required to reveal the structure of the fractal in greater detail, the form will become apparent as the negative power is decreased, the value of alpha is adjusted to maintain the critical value at 1, for delta = -2 alpha is 2, for -3 alpha is 3 and so on.

The following sequence of pictures delta is set to -2, -3, -4 and finally -5.

While preparing this part I discovered that these particular forms of Cczcpaczcp are sensitive to the bailout condition, the results are more pleasing, but that will have to be the subject of the next part of the guide, pushing back what would’ve been in part 4 to part 5.

At the start of this part of the guide I mentioned that a completely different sequence of fractals is produced where one of the powers is set to -1 instead of 1 and that the other power was positive and greater than 2.

So using

alpha = 1
beta = 2
gamma = 2
delta = -1

z₀ remains at 1. The critical value can be kept at 1 by setting gamma to be the same value as beta and keeping alpha and delta to 1 and -1, you can of course use different values which will require z₀ to be calculated so that the images produced match the following examples starting with beta equal to 2 and increased by one for each following image.

Like the rings of Mandelbrots in part 2 these forms of Cczcpaczcp, the number of their main features, be it buds or clover like leaves, matches the sum of the powers, ignoring signs. In what would’ve been the next part I’ll show a form of this fractal where the number of main features does not match the sum of the unsigned powers. That subject will have to wait to part 5.

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In part 1 I introduced the Cczcpaczcp fractal as produced with the default values and the initial z value set to the location in the complex plane. The final picture showed four Mandelbrots arranged in a ring, changing the powers so that both values are 2 and above (ignoring signs) leads to five, six, seven and more Mandelbrots in a ring. Using the location in the complex plane as the initial value makes it very difficult to determine the values of alpha and gamma that will produce well formed Mandelbrots, the solution is to use the same initial value for all calculated positions. The initial value used for the standard Mandelbrot is zero which happens to be be its critical value, using any other value will distort and degrade the shape of the Mandelbrot. It isn’t a bad thing in itself as the resulting “perturbed” Mandelbrot can have some very pretty or striking structures.

So after mentioning the critical value an explanation of how it is derived is in order. As is the nature of fractals Mathematics is now going to intrude. The critical value is found by differentiating the fractal function.

For fractals the following general formula is iterated:

z_n+1 = f(z_n)

The critical value is found by solving the following general equation:

f'(z) = 0

NOTE: in the following formulae c is a constant representing the location in the complex plane, and * indicates multiplication.

For the Mandelbrot:

f(z) = z² + c
f'(z) = 2z

so

f'(z) = 0

is

2z = 0

So the critical values is 0.

For Cczcpaczcp:

f(z) = c(alpha*z^beta + gamma*z^delta)
f'(z) = c(beta*alpha*z^beta-1 + delta*gamma*z^delta-1)

to find the critical value:

c(beta*alpha*z^beta-1 + delta*gamma*z^delta-1) = 0
c*beta*alpha*z^beta-1 = -c*delta*gamma*z^delta-1

c cancels out so, for Cczcpaczcp, a value of z that satisfies the following equation is the critical value:

c*beta*alpha*z^beta-1 = -c*delta*gamma*z^delta-1

For

alpha = 1
beta = 2
gamma = 1
delta = -2

we get

2*1*z = -(-2)*1*z^-3

which boils down to

z⁴ = 1

So the critical value is 1.

For

alpha = 1
beta = 3
gamma = 1.5
delta = -2

The critical value is again 1, different values could be used for alpha and gamma but you’ll end up a power of z equaling some value other than one and hence a different critical value. The size of the image will also be different but the structure of the fractal will be identical. The resulting picture is a ring of 5 Mandelbrots:

Swaping the signs for beta and delta results in a different 5 Mandelbrot ring:

Maintaining a critical value of 1 the following:

alpha = 1
beta = 3
gamma = 1
delta = -3

will result in a 6 Mandelbrot ring.

A different 6 Mandelbrot ring can be produced using

alpha = 2
beta = 2
gamma = 1
delta = -4

maintaining the critical value at 1 and this ring:

Swapping the signs for beta and delta produces a third version of the ring:

Where beta and delta have opposite signs then, ignoring signs, the sum of beta and delta will determine the number of Mandelbrots in the ring provided beta and delta are 2 and above. This is why this fractal will not produce rings of this type containing two or three Mandelbrots.

If one of beta and delta is 1 or -1 the resulting fractals are completely different. Those fractals will be the subject of part 3.