Kompassbrot uses and transcendental functions?

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Offline hgjf2

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« on: September 14, 2019, 09:47:54 AM »
If for Kompassbrot kind 2 this formula F=((-8*k^3-36*k+-(k^2-18)sqrt(10*k^2+36))/108) is avaible only for border of main "cardioid" in rest for each number k<-C the value K2(k) may be just transcendent number, an example K2(2.5)={-1,3685948042...;-2,5128866773...}. Else F(2,5)={-1,43068...;-2,55080...}
I don't excludes the case when those numbers -1,3685948042... and -2,5128866773... to be simmilar with the Feigenbaum constant: so=4,6692016091... and with a convergence point -1,4011551892 of the classic Mandelbrot set.
My research don't stops here.
« Last Edit: September 21, 2019, 08:10:50 AM by hgjf2 »

Offline marcm200

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« Reply #1 on: September 14, 2019, 12:18:32 PM »
((-8*k^3-36*k+-(k^2-18)sqrt(10*k^2+36))/108) ... K2(k) may be just transcendent number, an example K2(2.5)={-1,3685948042...;-2,5128866773...}.
From my understanding, entering a rational number k into the ((-8*k...) gives you an algebraic number, not a transcendental one.

If k is a rational, everything in the formula is rational except "sqrt(term)". But if "term" is a rational, then sqrt(term) is algebraic as it is a solution of the polynomial x▓-term=0 (and every coefficient there is rational).

Offline hgjf2

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« Reply #2 on: September 15, 2019, 08:37:46 AM »
No! The function K2(k) is not algebrical function. Is position of Kompassbrot kind 2 point.
If K1(k)=(-2k^3-9k)/27 at the Kompassbrot kind 1.
Details are in the pictures:

At Cubic Mandelbrot fc(z)=z^3+2.5z^2+c, the points of Kompassbrot kind 2 at the numers -1,3685948042+0i and -2,5128866773+0i are marked with yellow points.

I keep searching the math formula for K2(z)

Offline marcm200

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« Reply #3 on: September 15, 2019, 08:44:15 AM »
Is K2(k) defined as ((-8*k^3-36*k+-(k^2-18)sqrt(10*k^2+36))/108) as I understood your post?

If so, I disagree. The result of that K2 is algebraic in case k is rational and transcendental for transcendental k's. So the function is neither fully algebraic everywhere nor fully transcendental.

Offline gerrit

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« Reply #4 on: September 15, 2019, 07:17:22 PM »
What is a "Kompassbrot"?

Offline hgjf2

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« Reply #5 on: September 21, 2019, 08:07:32 AM »
Not! The formula F=(-8*k^3-36*k+-(k^2-18)*sqrt(10*k^2-36))/108 is formula for beta of Kompassbrot kind 2, isn't right formula for Kompassbrot kind 2. The points market with yellow in my graphic is for right Kompassbrot kind 2.
For k=2,5; F={-1,43068...;-2,55080...} not {-1,36859...;-2,51288...} . I marked with yellow the point K2={-1,36859...;-2,51288...} at the fractal MC fc(z)=z^3+2,5*z^2+c.

Offline gerrit

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« Reply #6 on: September 21, 2019, 10:37:22 PM »
What is a "Kompassbrot"?

Offline 3DickUlus

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« Reply #7 on: September 22, 2019, 03:25:29 AM »
 ???
Fragmentarium is not a toy, it is a very versatile tool that can be used to make toys ;)

https://en.wikibooks.org/wiki/Fractals/fragmentarium

Offline hgjf2

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« Reply #8 on: September 22, 2019, 07:32:46 AM »
What is a "Kompassbrot"?
I told you that kompassbrot is cubic Mandelbrot fc(z)=z^3+k*z^2+Kn(k) where Kn(k) is Kompassbrot kind (n) as complex function.
K1(k)=(-2*k^3-9*k)/27  [Kompassbrot kind 1], at other kinds I yet don't know .
At each cubic Mandelbrots based by z^3+k*z^2+c, the Kn(k) are the point where bulbs or antenas has simetry, like in my picture with cubic mandelbrot based by z^3+2,5*z^2+c where I marked with yellow the points of Kompassbrot kind 2.

The kind of Kompassbrot is stabilised by the kind of bulbs whick has simetry (complex square collisions).
I yet not found formula for other Kompassbrot than kind1

Kompassbrot kind 2 handmade look so:
« Last Edit: September 22, 2019, 07:43:58 AM by hgjf2 »

Offline gerrit

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« Reply #9 on: September 22, 2019, 08:46:14 AM »
I told you that kompassbrot is cubic Mandelbrot fc(z)=z^3+k*z^2+Kn(k) where Kn(k) is Kompassbrot kind (n) as complex function.
K1(k)=(-2*k^3-9*k)/27  [Kompassbrot kind 1], at other kinds I yet don't know .
At each cubic Mandelbrots based by z^3+k*z^2+c, the Kn(k) are the point where bulbs or antenas has simetry, like in my picture with cubic mandelbrot based by z^3+2,5*z^2+c where I marked with yellow the points of Kompassbrot kind 2.

The kind of Kompassbrot is stabilised by the kind of bulbs whick has simetry (complex square collisions).
I yet not found formula for other Kompassbrot than kind1

Kompassbrot kind 2 handmade look so:
Thanks for explaining.

Do I understand correctly that you mean the parameter space image of \( z^3+kz^2+F(k) \), following some critical orbit (which?) with k the parameter (pixel), and F(k) a function you get somehow from parameter space image of \( z^3 + kz^2+c \) with k now a constant and c (pixel) the parameter?


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