# Fractals/Iterations in the complex plane/pcheckerboard

# Name[edit]

- Parabolic chessboard = parabolic checkerboard
- zeros of qn - use only interior of Filled Julia set

# Algorithm[edit]

Color depends on:

- sign of imaginary part of numericaly aproximated Fatou coordinate (numerical explanation )
- position of under in relation to one arm of critical orbit star: above or below ( geometric explanation)

## Steps[edit]

First choose:

- choose target set, which is a circle with :
- center in parabolic fixed point
- radius such small that width of of exterior between components is smaller then pixel width

- Target set consist of fragments of p components ( sectors). Divide each part of target set into 2 subsectors ( above and below critical orbit ) = binary decomposition

Steps:

- take = initial point of the orbit ( pixel)
- make forward iterations
- if points escapes = exterior
- if point do not escapes then check if point is near fixed point ( in the target set)
- if no then make some extra iterations
- if is then check in what half of target set it ( binary decomposition)

## Target set[edit]

- circle ( for periods 1 and 2 in case of complex quadratic polynomial)
- 2 triangles ( for periods >=3) described by :
- parabolic periodic point for period p , find
- critical point
- one of 2 critical point preimages ( a or b )

# How to compute preimages of critical point ?[edit]

- (a,b)
- (aa, ab)
- (aaa,aab )
- (aaaa, aaab )
- (aaaaa, aaaab )

# dictionary[edit]

- The chessboard is the name of this decomposition of A into a graph and boxes
- the chessboard graph
- the chessboard boxes :" The connected components of its complement in A are called the chessboard boxes (in an actual chessboard they are called squares but here they have infinitely many corners and not just four). "
^{[1]} - the two principal or main chessboard boxes

# description[edit]

"A nice way to visualize the extended Fatou coordinates is to make use of the parabolic graph and chessboard." ^{[2]}

Color points according to :^{[3]}

- the integer part of Fatou coordinate
- the sign of imaginary part

Corners of the chessboard ( where four tiles meet ) are precritical points ^{[4]}

or

"each yellow tile biholomorphically maps to the upper half plane, and each blue tile biholomorphically maps to the lower half plane under . The pre-critical points of or equivalently the critical points of are located where four tiles meet"^{[5]}

# Images[edit]

Click on the images to see the code and descriptions on the Commons !

ray 0

Examples :

- Tiles: Tessellation of the Interior of Filled Julia Sets by T Kawahira
^{[6]} - coloured califlower by A Cheritat
^{[7]}

# code[edit]

For the internal angle 0/1 and 1/2 critical orbit is on the real line ( Im(z) = 0). It is easy to compute parabolic chesboard because one have to check only imaginary part of z. For other cases it is not so easy

## 0/1[edit]

Cpp code by Wolf Jung see function parabolic from file mndlbrot.cpp ( program mandel ) ^{[8]}^{[9]}

To see effect :

- run Mandel
- (on parameter plane ) find parabolic point for angle 0, which is c=0.25. To do it use key c, in window input 0 and return.

C code :

```
// in function uint mndlbrot::esctime(double x, double y)
if (b == 0.0 && !drawmode && sign < 0
&& (a == 0.25 || a == -0.75)) return parabolic(x, y);
// uint mndlbrot::parabolic(double x, double y)
if (Zx>=0 && Zx <= 0.5 && (Zy > 0 ? Zy : -Zy)<= 0.5 - Zx)
{ if (Zy>0) data[i]=200; // show petal
else data[i]=150;}
```

Gnuplot code :

```
reset
f(x,y)= x>=0 && x<=0.5 && (y > 0 ? y : -y) <= 0.5 - x
unset colorbox
set isosample 300, 300
set xlabel 'x'
set ylabel 'y'
set sample 300
set pm3d map
splot [-2:2] [-2:2] f(x,y)
```

## 1/2 or fat basilica[edit]

Cpp code by Wolf Jung see function parabolic from file mndlbrot.cpp ( program mandel ) ^{[10]}
To see effect :

- run Mandel
- (on parameter plane ) find parabolic point for angle 1/2, which is c=-0.75. To do it use key c, in window input 0 and return.

C code :

```
// in function uint mndlbrot::esctime(double x, double y)
if (b == 0.0 && !drawmode && sign < 0
&& (a == 0.25 || a == -0.75)) return parabolic(x, y);
// uint mndlbrot::parabolic(double x, double y)
if (A < 0 && x >= -0.5 && x <= 0 && (y > 0 ? y : -y) <= 0.3 + 0.6*x)
{ if (j & 1) return (y > 0 ? 65282u : 65290u);
else return (y > 0 ? 65281u : 65289u);
}
```

# See also[edit]

- Checkerboard in Hyperbolic tilings by User:Tamfang : images and Python code
- https://plus.google.com/110803890168343196795/posts/Eun6pZVkkmA
- shadertoy: Orbit trapped julia Created by maeln in 2016-Jan-19
- Holomorphic checkerboard by etale_cohomology
- Sepals of cauliflower
- wikipedia :Zebra striping in computer graphics

# references[edit]

- ↑ Near parabolic renormalization for unisingular holomorphic maps by Arnaud Cheritat
- ↑ About Inou and Shishikura’s near parabolic renormalization by Arnaud Cheritat
- ↑ Applications of near-parabolic renormalization by Mitsuhiro Shishikura
- ↑ Complex Dynamical Systems by Robert L. Devaney, page
- ↑ Antiholomorphic Dynamics: Topology of Parameter Spaces and Discontinuity of Straightening by Sabyasachi Mukherjee
- ↑ tiles by T Kawahira
- ↑ checkerboards by A Cheritat
- ↑ commons:Category:Fractals created with Mandel
- ↑ Program Mandel by Wolf Jung
- ↑ Program Mandel by Wolf Jung