Fractals/Iterations in the complex plane/pcheckerboard
Contents
Name[edit]
 Parabolic chessboard = parabolic checkerboard
 zeros of qn
Algorithm[edit]
Color depends on:
 sign of imaginary part of Fatou coordinate ( )
 position of under in relation to one arm of critical orbit star: above or below ( geometric interpretation)
Steps[edit]
 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 fragmnents of p components ( sectors). Divide each part of target set into 2 subsectors ( above and below critical orbit ) = 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 precritical 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]
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[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 2016Jan19
 Holomorphic checkerboard by etale_cohomology
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 nearparabolic 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