n-th Order Evolution on a straight line Mandelbrot zoom method
More information about evolution:
https://dinkydauset.deviantart.com/jo...
Software used in this video is my own, see: https://github.com/DinkydauSet/Explor...
The early steps of building a shape by Julia morphing can be difficult because there isn't anything special yet. What if we can use calculations that tell us exactly where to go, when it gets so confusing that using visual references just doesn't work anymore?
4th order evolution looks like this:
https://dinkydauset.deviantart.com/ar...
There are 2 ways that I think of this pattern:
1. 4th order evolution, where all the shapes lie on a straight line, and all the shapes are the same (spirals with increasingly many arms)
2. A visual representation of the solution of a maths problem involving sequences of letters, which probably has a recursive solution.
I use my own (slow) program to create models of what the resulting julia morphings would roughly look like if I zoomed in on similar embedded julia sets in the Mandelbrot set.
The problem can be described for any number of letters that are in a particular order. The ordering (for example: (A B)) is a cycle. We go through all the letters (only A and B with the cycle (A B)) in this order and when we reach the end (B) we return to the first (A). With only the letter A, the sequence is
A
A²
A³
...
This represents how morphing one shape (A) repeatedly leads to one extra symmetry increase (its exponent += 1) every time. A could be a tree, or you could zoom in the center of something, ultimately leading to a minibrot as the exponent goes to infinity.
With the two letters A and B, ordered as A, B (B after A) the sequence is:
A
ABA
AB A² BA
ABA² B² A²BA
ABA²B² A³ B²A²BA
...
This represents what a (second order) evolution set looks like. It's the result of alternating between morphing shape A and morphing shape B. We start with a first iteration of our first shape A. Then we use some of the pattern around it to make the first iteration of shape B. Then we continue with A again which turns it into A², then B again etc.
Alternating between morphing 3 shapes leads to the sequence
A
ABA
ABA C ABA
ABAC A² CABA
ABACA²CA B² ACA²CABA
ABACA²CAB²A C² AB²ACA²CABA
ABACA²CAB²AC²AB²AC A³ CAB²AC²AB²ACA²CABA
...
What is this pattern?
Notice that all of these words are palindromes. In fact that's how they are created. Morphing a shape wraps everything around it around itself another time, achieving 2-fold ratational symmetry compared to before the morphing.
The B² at the center of the 5th palindrome comes from one of the Bs in ABAC A² CABA, which are at a distance of 3 from A², the letter at the center. This problem in general is: how far from the middle letter is the next letter? Or: how many shapes (letters) do we have to skip, couting from the center, to arrive at the next shape to morph, to get 3rd (or more general: nth) order evolution?
I calculated those numbers with a program. A friend also created a program independently and arrived at the same results.
1 letter:
0, 0, 0, ....
2 letters:
1, 1, 1, ....
3 letters:
1,2,1,3,2,5,3,8,5,13,8,21,13,34,21,55,34,89,55,144,89,233,144,377,233,610,377,987,610, ...
4 letters:
1,2,4,1,3,8,4,9,21,8,20,49,21,50,120,49,119,288,120,289,697,288,696,1681,697,1682,4060,1681,4059, ...
5 letters:
1,2,4,8,1,3,8,20,8,17,37,82,20,48,113,263,82,184,416,945,263,608,1400,3216,945,2153,4914,11228,3216, ...
The first two sequences are trivial. The third one is wilder. It appears in the encyclopedia of integer sequences: oeis.org/A053602
It involves the fibonacci sequence and it's also related there to palindromes. That's gotta be it. (I know you may be thinking "SHOW ME PROOF!" but when I do mathematics it actually doesn't get more formal and precise than this.) So we have a formula for it.
The other sequences, however, don't appear and I haven't been able to find a formula for them.