Here's a bunch of fractals :)

[wes]

Hello, I've been lurking here for a little while and I want to share some OC. I've got some code I'd like to share (soon) for making M sets as well as any of their Julia sets. In the meantime, here's some images I've made with it.

In reading order:
Mandelbrot 2
Mandelbrot 3
Mandelbrot 4
Burning Ship 2
Burning Ship 3
Burning Ship 4
Perpendicular Burning Ship 2
Perpendicular Burning Ship 3
Perpendicular Burning Ship 4
Mandelbar 2
Mandelbar 3
Mandelbar 4
Celtic 2
Celtic 3
Celtic 4
Mandelbrot 2 Julia of -0.4 + 0.6i
Mandelbrot 2 Julia of 0.285 + 0.01i
Mandelbrot 2 Julia of -0.8 + 0.156i
Burning Ship 2 Julia of 0.29 - 0.29i
Burning Ship 2 Julia of -1 - 1i
Burning Ship 2 Julia of -0.8 + 0.1i
Celtic 2 Julia of -1 - 1i
Celtic 2 Julia of -2
Mandelbrot 3 Julia of -0.65 + 0.5i

The coordinates of the Mandelbrot 2 Julia's are from Wikipedia and the coordinates of the Burning Ship 2 Julia's are from https://theory.org/fracdyn/burningship/julias.html The rest of the Julia coordinates I found myself. I got the color palette from http://stackoverflow.com/questions/16500656/which-color-gradient-is-used-to-color-mandelbrot-in-wikipedia which was originally from Ultra Fractal I think

[SamTiba]

Nice collection of sets you have there! Play around and try to find more.

Did you just translate the formulas or have you already understood them?
Because understanding the underlying math can be quite tough in the beginning but the real creative process starts afterwards. Before that it's a lot copy & paste from things that have already been found before.

In what language did you code?

[wes]

Nice collection of sets you have there! Play around and try to find more.

Did you just translate the formulas or have you already understood them?
Because understanding the underlying math can be quite tough in the beginning but the real creative process starts afterwards. Before that it's a lot copy & paste from things that have already been found before.

In what language did you code?

Thanks I created these with c++. At first I learned by copying stuff off of wikipedia, but then I discovered std::complex from the c++ standard library which encouraged me to read more about complex numbers. I think I understand how complex numbers operate now but I'm still not entirely sure why they make the shapes they do. The cardioid makes sense I guess, the seahorses and elephants are just magical lol.

[SamTiba]

I'm still not entirely sure why they make the shapes they do. The cardioid makes sense I guess, the seahorses and elephants are just magical lol.

Why formulas create which shape ... this question indeed is a tough one
Answer it and i would be stunned
Especially for more complex Julia-Sets I would really love to be able to tell what I get from changing values but it's a lot of trial and error

I'm programming in Java - in my opinion things just really work out better with object oriented programming.

[youhn]

What question would be easier to answer?

1. Why does the Mandelbrot set create the shapes it does?
2. How does the Mandelbrot set create the shapes it does?

I think forum admin already did a pretty good job at question 2. See the following videos:

<a href="https://www.youtube.com/v/ce0lms78nt4&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/ce0lms78nt4&rel=1&fs=1&hd=1</a>

<a href="https://www.youtube.com/v/1W5daRoE7Qs&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/1W5daRoE7Qs&rel=1&fs=1&hd=1</a>

[wes]

I think forum admin already did a pretty good job at question 2. See the following videos:

Thanks for the links. I watched them both. That visualization is amazing. It helps to see all the iterations broken up into their rotation and translation stage. For example the points in the main antenna rotate 180 degrees to a point which would obviously escape, but they are translated back into a stable spot because of their starting position. I think I have to rewatch these 100 more times lol.

[wes]

Why formulas create which shape ... this question indeed is a tough one
Answer it and i would be stunned
Especially for more complex Julia-Sets I would really love to be able to tell what I get from changing values but it's a lot of trial and error

I'm programming in Java - in my opinion things just really work out better with object oriented programming.

I know what you mean. When I play with Julias I just start with a nice round-number and explore randomly. I'd like to know more about what the golden ratio has to do with Julias. Why is the golden-ratio Julia (φ−2)+(φ−1)i so beautiful? Just a coincidence or awsome math? Idk

[SamTiba]

Imagine you have a formula with a lot part-functions and polynoms everywhere and every part of the polynom has a factor with a real and imaginary value - you quickly get to formulas with 10-100 variables.
Telling the outcome for the Mandelbrot-shape might be possible, but imagine to tell what changes when you play with variables, especially if you have a lot of them.
And when is a shape 'interesting' and when not? What does it take to produce a 'good' formula?

[youhn]

Good formulas emulate the imaginary boundaries of nature itself.
They consist of two parts which constantly fight each other,
which of course must be iterative (take a hit, then attack, repeat).
Or maybe dance with each other, like the boundary between air
and water driven by the energy of the wind. Or even the balance
between populations of hunter and prey, or the upward forces
of earth plates versus the erosion due to flows of water and air.
Politics and currencies, open source software communities, etc.
These examples almost always lead to fractal shapes.

Only at these boundaries of balance, things get interesting.
Formulas should reflect this to emulate fractal phenomena.

[greentexas]

Why formulas create which shape ... this question indeed is a tough one
Answer it and i would be stunned
Especially for more complex Julia-Sets I would really love to be able to tell what I get from changing values but it's a lot of trial and error

I'm programming in Java - in my opinion things just really work out better with object oriented programming.

I can venture a guess on why certain formulas produce the shapes they do (mostly for the Celtics and Perpendicular Burning Ships):

The formula for the Perpendicular Burning Ship can be written longhand as Re(z^2 + z₀) + Im(abs(z^2 + z₀))*i.

The real axis contains the same points as the Mandelbrot, and the imaginary axis contains the same points as the Burning Ship. This has to do with the fact that points on the axes of hybrid fractals (such as the Mandelbrot / Burning Ship hybrid, the Perpendicular Burning Ship) will only work with the fractal they share the axis with. I'm not certain of this, but it's a good guess.

The real axis of the Perpendicular Burning Ship contains the same point as the Mandelbrot (real fractal), and the imaginary axis shares that with the Burning Ship (fractal on the imaginary side).

The Celtic's left needle resembles a Buffalo's but is symmetrical. That could be because it has the Buffalo on the real (accounting for the Celtic cross) and a Mandelbrot on the imaginary (explaining the height of this fractal [possibly three units or more] and the symmetry of the crosses). Similar reasoning explains the Perpendicular Burning Ship's behavior.

If you examine the limbs of the cubic fractals, the placement of the limbs of the Cubic Burning Ship Partial Imaginary can make more or less sense. The fourth order fractals do this too.

The Celtic's structure may reveal a bit about the Mandelbrot and Buffalo. The Perpendicular Ship's structure resembles the Burning Ship, but rotating the flipping axis shows a Mandelbrot's behavior better.

Sorry if this makes no sense.