Compasses Revisited Part 1   1 comment


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

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

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

so

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

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

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

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.

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