Archive for the ‘Mathematics’ Category
So far adding critical point support to some of the formulae in Saturn and Titan, for the next release, has been straight forward. The next formula “Mandelbrot Division” is somewhat trickier. The formula features two parts that match a standard Mandelbrot with powers that are not necessarily 2, one part is divided into the other, in addition there are two parameters added to the result. All parameters in the new version of Saturn can be a number or a number modified by multiplying or dividing the location in the complex plane, strictly speaking the “Mandelbrot parts” aren’t Mandelbrot unless the parameter is modified, so Mandelbrot Division is only a loose description but it is convenient as a name.
The current formula is as follows:
z = (zα + β)/(zγ + δ) + ε + ζ
To ensure that the critical points can be found a new version of the formula “Mandelbrot Division IP” will be added where IP indicates Integer Power.
z = (zA + α)/(zB + β) + γ + δ
The critical points are found as usual by solving f'(z) = 0, γ and δ simply disappear leaving the Mandelbrot parts to be differentiated using the quotient rule:
(f(z)/g(z))’ = (f'(z)g(z) – g'(z)f(z))/g(z)2 = 0
Multiplying both side by g(z)2 simplifies things so critical points are the solutions of the following:
f'(z)g(z) – g'(z)f(z) = 0
The Mandelbrot parts form the definitions of f(z) and g(z):
f(z) = zA + α
g(z) = zB + β
Their differentials are:
f'(z) = AzA-1
g'(z) = BzB-1
So the formula to solve is:
(AzA-1)(zB + β) – (BzB-1)(zA + α) = 0
which simplifies to:
(A – B)zA+B-1 + βAzA-1 – αBzB-1 = 0
When the values of A and B are known the smallest power can be factored out and the other powers reduced accordingly, as a power of z has been factored out it follows that Mandelbrot Division IP always has a critical point of zero. The other critical points are found by solving the remaining polynomial which will be handled by Saturn and Titan.
This formula hasn’t yet been implemented so I can only show some pictures using the current Mandelbrot Division formula where the initial value is set to zero (a critical point).
Example 1 – Mandelbrot Features
Example 1 – parameter summary
Example 2 – Multibrots
Example 2 parameter summary
The two Titan summaries show changes from the currently released software the most noticeable of which is the display of colour maps at the bottom.
The focus of this part on Octic fractals is the MC 8.6 formula:
z = αz8 + βz6 + c
The critical points are found by solving f'(z) = 0:
f(z) = αz8 + βz6 + c
f'(z) = 8αz7 + 6βz5 + c
z5(8αz2 + 6α) = 0
so the solutions are 0 and the two square roots of -6β/8α.
Again the number of critical points has reduced by one, compare with MC 8.5. This time the pictures for the square root values are identical, so there are only two component part pictures. Using α = 1 and β = 1 the pictures are:
Critical point of zero
Critical points of the two square roots of -0.75
The component parts combined, ignoring overlapping outer colouring produce this:
Composite of both component pictures
The composite image shows regions of M2 Mandelbrot features and regions of M6 Mandelbrot features with the latter dominating. The three following pictures show the same area, one for each component and one for a composite using inner colouring other than black and averaging the colours of the two components:
Critical point: zero
Critical point: a square root of -0.75
Composite of the two component parts
As with the other formulae in the family of Octic fractals changing the values of α and β results in different pictures:
α = 1 β = -1
α = 1 β = -1.5
Detail using α = 1 and β = -1.2
The previous set of posts on the Compasses fractal back in 2011 did not cover positive integer powers greater than 2. The Compasses formula has changed slightly, there are now three parameters instead of two, the location in the complex plane, c, was incorporated directly in the original formula. The current version of Saturn allows any parameter to be set to c or -c so the original formula can be reproduced.
The formula is:
z = zα – αβα – 1z + γ
With α = 2, β = c and γ = 0 the formula becomes:
z = z2 – 2cz
To find the critical point f'(z) = 0 needs to be solved:
f(z) = z2 + 2cz
f'(z) = 2z – 2c = 0
so there is one critical point c, the location in the complex plane. The initial value of z is set to the location in the complex plane, so a different value is used for each point in the fractal’s picture. When I’ve shown how to derive the critical value in other posts the value has been a number which is used for all points in the fractal picture. Currently, Saturn allows the initial value of z to be set to a number, the transformed or untransformed location in the complex plane which is fine in this case but will prove to be inadequate.
The picture produced for α = 2, β = c and γ = 0 is as follows:
α = 2 β = c γ = 0
Changing the value of γ alters the appearance of the fractal but as γ is not present in f'(z) the critical point is not affected.
α = 2 β = c γ = -0.8
Increasing α to 3 produces a different fractal, again using β = c and γ = 0 the critical point can be found:
f(z) = z3 + 3c2z
f'(z) = 3z3 – 3c2 = 0
3z2 – 3c2
obviously the critical point is again c, the location in the complex plane and the fractal produced looks like this:
α = 3 β = c γ = 0
All looks fine, M2 Mandelbrot type buds and no barren areas, however, change the value of γ and all is not so fine:
α = 3 β = c γ = -0.25
A picture of Saturn’s main window shows a barren area highlighted within the rectangle:
Highlighting a barren area
Why has this happened? There are two critical points, not one. I have encountered a phenomenon with Neptune where more than one critical point can produce the same image which is the case when γ = 0. Neptune is a program for generating “multiple critical point” fractals, it combines the component parts, one for each critical point, of the fractal into a final image.
To get more than one critical point where 3z2 – 3c2 treat c2 as being 1 multiplied by c2, the square root of 1 is 1 and -1 hence two critical points c and -c. Saturn can’t set z to -c and Neptune can’t calculate critical points dependent on the location in the complex plane which is why Neptune does not include any Compasses formulae.
For higher values of α the critical points are the roots of 1 multiplied by c, so for &alpha = 4 the roots are 1 and two complex values, for α = 5 the roots are 1, -1, i and -i. The roots of 1 are always equally spaced on a unit circle centred at the origin of the complex plane.
The next versions of Saturn & Titan and Neptune & Triton will support critical points dependent on the location in the complex plane. The next part will focus on setting γ to c.
On to MC 8.2. These fractal have the following formula:
z = αz8 + βz2 + c
To determine the critical points f′(z) = 0 needs to be solved:
f(z) = αz8 + βz2 + c
f′(z) = 8αz7 + 2βz = 0
z(8αz6 + 2β) = 0
So the solutions are z = 0 and the six sixth roots of 2α/7β.
Critical points for α = 1 and β = 1 shown in Neptune’s settings window.
Here is an MC 8.2 Octic fractal:
MC 8.2 Octic α = 1 and β = 1
Ignoring outer colouring where inner and outer areas overlap the same fractal looks like this:
Ignoring outer colouring for inner/outer overlap
This fractal looks odd when compared against the standard Octic or M8 Mandelbrot (see part1). I have a tendency use Heptabrot (based on the number of primary buds) to refer to M8 Mandelbrots. The following pictures show each of the component pieces, one for each critical point.
Critical point 1
Critical point 2
Critical point 3
Critical point 4
Critical point 5
Critical point 6
Critical point 7
From the above pictures it is clear that there are only four distinct components: critical point 1 is unique but the points 2, 3 and 4 are duplicated as 5, 6 and 7. I’ve come across cases where there are identical components before which is why I added a facility to Neptune to ignore critical point(s) as unnecessary calculation can be avoided.
There is a clear difference between MC 8.1 Octics and MC 8.2 Octics. MC 8.1 Octics have just one component in regard to its shape but has seven different locations. MC 8.2 Octics have four distinct components in regard to their shapes and three locations are shared. I’ve checked MC 8.2 with different values for β and they all have four distinct shapes and three locations are indeed shared. I suspect this is always the case, I do not have a mathematical proof and if I had it would be too technical for these observations on Octics.
Changing the value of β changes the appearance of the fractal. All the values of β used for the following pictures are on a unit circle with its centre at the origin of the complex plane (0 + 0i). Outer colouring is ignored where inner and outer areas overlap.
β = -1
β = 0 + 1i
β = 0 – 1i
β = 1/√2 + 1i/√2
β = -1/√2 + 1i/√2
β = 1/√2 – 1i/√2
-1/√2 – 1i/√2
The next part will continue with MC 8.2 with values of β that have absolute values less than and greater than 1. There will also be detail showing how the component parts overlap and the presence of M2 Mandelbrot islands (copies of the standard Mandelbrot, often distorted).
Continuing on from part 2. More MC 8.1 fractals mostly with imaginary values for β. As a reminder the MC 8.1 formula is:
z = αz8 + βz + c
α = 1 β = 0 + 1i
α = 1 β 0 +0.5i
α = 1 β = 0 + 0.5i One critical point
α = 1 β = 0 +0.25i
While β is less than 1i most of the areas of the component parts (one for each critical point) overlap. Above 1i the component parts look more like standard Mandelbrots and the overlapping areas are reduced in size until there is virtually no overlap at all.
α = 1 β = 0 + 1.25i
Once the fractal has split apart the colouring of the image is affected: inner areas are entirely overlapped by multiple outer areas so that inner colouring has a greatly diminished effect on the final picture. Inner colouring only can be used for areas where at least one critical point produces inner colouring, any overlapping outer areas are ignored.
Inner colouring ignoring outer overlaps
Using negative imaginary values produce the same images as for the positive values except that they are mirror images of the positive value when rotation is taken into account. This behaviour is markedly different to real values for β where the negative value produces a completely different picture to the corresponding positive value. All the pictures above have the origin (0 + 0i) of the complex plane at theirs centres.
The following two pictures show how unlike a standard Mandelbrot a component picture (1 critical point) can be.
α = 1 β = -1
α = 1 β = -1.125
Finally detail from the centre of an MC 8.1 fractal.
α = 1 β = -1.125
The next part will focus on MC 8.2.
This post refers to Saturn and Neptune, they are programs for exploring fractals and are available from the download page.
Octic fractals are generated using an eighth order polynomial, the eighth power Mandelbrot is a special case where all coefficients except that for z8 are zero. It has a single critical point at zero in common with all Mandelbrots defined using the following general formula:
z = zn + c
where n is an integer greater than or equal to 2.
The eighth power Mandelbrot can be found in the list of Saturn fractal types under the name M08 Mandelbrot. It looks like this:
Eighth power Mandelbrot
Saturn also has a fractal type called Octic which is a full polynomial, its formula is:
z = αz8 + βz7 + γz6 + δz5 + εz4 + ζz3 + ηz2 + θz + ι
For use with Mandelbrot algorithm at least one of the parameters (or coefficients) must be the location in the complex plane (usually denoted as c) for convenience ι is set to c. When using the Mandelbrot algorithm the initial value of z is set to a critical point, other values can be used and resulting pictures are “perturbed”. So what are the critical points of an “Octic”? Critical points are the roots of the first derivative of the fractal formula.
So, for Octic:
f(z) = αz8 + βz7 + γz6 + δz5 + εz4 + ζz3 + ηz2 + θz + c
the first derivative is:
f'(z) = 8αz7 + 7βz6 + 6γz5 + 5δz4 + 4εz3 + 3ζz2 + 2ηz + θ
The critical points are the roots of:
f'(z) = 0
I mentioned earlier that any of the coefficients of the polynomial can be the location in the complex plane, in which case, the critical points have to be calculated for every location. Neither Saturn nor Neptune can do this which is why ι was set to c. Saturn only allows a number or the location in the complex plane to be used as the initial value for z. So what value should be used? It is possible to zero out some of the terms so that some critical points can be found relatively easily but that precludes the use of all the terms, solving f'(z) = 0 where all the terms of the polynomial are present isn’t easy, fortunately you don’t have to work them out as Neptune can do it for you. Neptune was specifically designed to produce pictures of fractals with multiple critical points, its version of Octic is MC 22.214.171.124.126.96.36.199.
To start the exploration of Octic fractals all parameters except ι are set to 1, i.e.:
f(z) = z8 + z7 + z6 + z5 + z4 + z3 + z2 + z + c
Neptune produces this picture:
Octic f(z) = z8 + z7 + z6 + z5 + z4 + z3 + z2 + z + c
Saturn can only produce pictures using one critical value at a time, the z0 value to be used can be found on Neptune’s settings window:
Neptune’s settings window
Using the first and sixth critical points Saturn produces:
First critical point
Sixth critical point
Zooming into an area at the left hand size of the sixth critical point picture standard second power Mandelbrot islands can be found.
There are no eighth power Mandelbrot islands (multibrots) as found in the first picture of this post.
Saturn can also generate Octic Julias, the Julia algorithm uses c as the initial value where all parameters have fixed values. Here is an example Julia using the Julia form of the formula used for the pictures above:
Octic Julia ι = 0.17 – 0.33i
There is an enormous variety of Octic “Mandelbrot sets” here are two examples using different coefficients:
Octic f(z) = z8 + z7 + 3z6 + z5 – 0.5z4 + z3 – 2z2 – 0.6z + c
Octic f(z) = z8 + z7 + z6 + 2z5 + z4 + z3 + z2 – z + c
Where all terms of the polynomial are present only Octic Mandelbrot sets will contain second power Mandelbrot islands. Zeroing out various terms other than z8 and c results in sets that contain both Mandelbrot and cubic Mandelbrot islands. Higher order Mandelbrot islands are also found, I don’t know whether more than two types of Mandelbrot islands can be present in the same set, over the course of the next parts I intend to find out.
Neptune will be exclusively used from now on, so no more Julias. Neptune has 9 version of the Octic formula with the number of critical points varying from 7 to 2:
- MC 8.1 z = αz8 + βz + c
- MC 8.2 z = αz8 + βz2 + c
- MC 8.3 z = αz8 + βz3 + c
- MC 8.4 z = αz8 + βz4 + c
- MC 8.5 z = αz8 + βz5 + c
- MC 8.6 z = αz8 + βz6 + c
- MC 8.7 z = αz8 + βz7 + c
- MC 8.7.6 z = αz8 + βz7 + γz6 + c
- MC 188.8.131.52.184.108.40.206 z = αz8 + βz7 + γz6 + δz5 + εz4 + ζz3 + ηz2 + θz + c
Part 2 will focus on MC 8.1.
Zoom into Somewhere Else No. 59
Image centre: -1.36948684035838934726941994120263305553285546666666666666 + 0.00696522706387073028507583308790115344792666666666666666667i
Image width: 1.28e-39
Precision: 160 bits
Maximum iteration: 28824
Outer colouring: absolute log of exponential sum of magnitude, scale = 8800
Inner colouring: fixed colour