Octics Part 11

I exhausted the Octic formulae that consist of z to the eighth power plus z to another power. Those formulae allow the critical points to be found relatively easily. The next formula is MC 8.7.6 which also allows the critical points to found relatively easily.

The MC 8.7.6 formula is:

z = αz⁸ + βz⁷ + γz⁶ + c

The critical points are found by solving f'(z) = 0:

f(z) = αz⁸ + βz⁷ + + γz⁶ + c
f'(z) = 8αz⁷ + 7βz⁶ + 6γz⁵

so

z⁵(8αz² + 7βz + 6γ) = 0

so the solutions are 0 and the other two can be found using the well known formula for solving quadratic equations.

As there are three critical points there are three component pictures, for α, β and γ all set to 1 the composite picture and its three component parts are as follows:

Composite Picture

Zero critical point

A non-zero critical point

The other non-zero critical point

Changing the values of α, β and γ changes the appearance of the fractal, setting β to -1 produces this composite picture:

α = 1 β = -1 γ = 1

Changing γ to -1 instead of β produces a fractal that looks like a distorted M7 Mandelbrot with a small satellite M6 Mandelbrot to the left and an M2 Mandelbrot to the right:

α = 1 β = 1 γ = -1

Changing γ again to -2.5 has a dramatic affect, the resulting fractal is like an M6 Mandelbrot with an extrusion of M2 Mandelbrot features to the right:

α = 1 β = -1 γ = -2.5

Changing β to -1 and the right hand extrusion disappears and the main body of the fractal looks even more like an M6 Mandelbrot but with a less distinct extrusion to left. The left hand extrusion is seen in more detail in the picture following the composite picture of this fractal:

α = 1 β = -1 γ = -2.5

Detail of left hand extrusion

Altering β again to 0.375 again produces a main body similar to an M6 Mandelbrot with a remote attendant M2 Mandelbrot:

α = 1 β = 0.375 γ = -2.5

So that concludes a brief exploration of the MC 8.7.6 octic formula. Like MC 8.6 it features M6 and M2 Mandelbrots and it too has 3 critical points.