Golden Ratio Tiling

[Dinkydau]

Mandel Machine, Mandelbrot set

Magnification:
2^732.365
2.9095965591197057426879587427262 E220

Coordinates:

Code:

Re = -1.756882996047415507177544263880044205560240816681939504337701122205524205232602173448341712703733871053600384797384909496400768301564935457122130644767462662664858651590631630737324402579324531143018225431629992886257332217126295
Im = 0.012178698364084925222456470314532744312959953422670766200219117572544897165197364251729676017200851973935584329711981645307984466766116936909884995575928407038962826724678263924301445881502867259057819318295270283982300902903242

[TheRedshiftRider]

Very cool.

[Dinkydau]

Thanks

[jwm-art]

Interested in how you found this. Was it something that you randomly found, or something you looked for with a rough idea of where to find it based on past exploration, or, knowing precisely where to look based on maths, or located using mathematical algorithm/process

[Dinkydau]

In december I decided I wanted to make more things similar to this image from 2013: http://dinkydauset.deviantart.com/art/Cosmic-Mandelbrot-zoom-383061324

While I was exploring I found that morphing a julia set in one of its spirals close enough to the center can result in such patterns when it is done repeatedly. I went on to try many of the spirals to see what the result looks like, varying how deep to zoom into the spirals and more.

Then the idea came to my mind that once inside a spiral arm, I could do the same thing again: choose a (smaller) spiral to zoom in on. I noticed it lead to a pattern when I chose about 6 subsequent smaller spirals. When I zoomed further the rotational symmetry increased to 8-fold and more and it looked like the image posted here.

I liked what I saw, except the number of squares in an "arm" was only 6 at the time. Imagine the spiral in the image to have only 6 squares in each arm. I thought it would be cool if it looked like an infinite spiral. I tried it again with 30 subsequent smaller spirals (real spirals from the julia set), then some more. This is the result. The beauty of the spiral is completely natural.

The zoom path is very simple. You can see what happens if you visit the coordinates and then zoom out.