# Name

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

# Algorithm

Color depends on:

• sign of imaginary part of numericaly aproximated Fatou coordinate (numerical explanation )
• position of $z_{\infty }$  under $f^{p}$  in relation to one arm of critical orbit star: above or below ( geometric explanation)

## Steps

First choose:

• choose target set, which is a circle with :
• center in parabolic fixed point
• radius such small that width 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 $z_{0}$  = 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

• 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 $\{z:z=f^{p}(z)\}$
• critical point $z_{cr}$
• one of 2 critical point preimages ( a or b ) $z=f^{-p}(z_{cr})$

# How to compute preimages of critical point ?

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

# dictionary

• 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). " 
• the two principal or main chessboard boxes

# description

"A nice way to visualize the extended Fatou coordinates is to make use of the parabolic graph and chessboard." 

Color points according to :

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

Corners of the chessboard ( where four tiles meet ) are precritical points 

$\bigcup _{n=0}^{n\geq 0}f^{-n}(z_{cr})$

or

$\{z:f^{n}(z)=z_{cr}\}$

## 1/1

The parabolic chessboard for the polynomial  z + z^2  normalizing  $\psi _{att}(-1/2)=0$ * each yellow tile biholomorphically maps to the upper half plane
*  each blue tile biholomorphically maps to the lower half plane under $\psi _{att}$ * The pre-critical points of $z+z^{2}$ or equivalently the critical points of $\psi _{att}$ are located where four tiles meet"

# Images

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

Examples :

• Tiles: Tessellation of the Interior of Filled Julia Sets by T Kawahira
• coloured califlower by A Cheritat 

# code

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

Play media
How the target set is changing along an internal ray 0
Play media

Cpp code by Wolf Jung see function parabolic from file mndlbrot.cpp ( program mandel ) 

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

Cpp code by Wolf Jung see function parabolic from file mndlbrot.cpp ( program mandel )  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);
}