The "rabbit" quadratic Julia set with the finite attractor a 3-cycle. Its basin has been colored using a different method for each "phase" of the attractor: a pink to red to black smoothed-iterations gradient in the component containing the critical point, a yellow/orange/brown/cream/tan
phi gradient in another, and closest approach of later iterates to point of origin in browns in the third.
Technical note: The large red, shiny, and brown components sharing a common large triple-spiral at the top contain the attractor points, one in each component. All of the other components are carried by
f(
z) =
z2 +
c onto a chain of components that leads eventually to these three. Which one of the three it is in after a multiple of three iterations determines which coloring method and gradient get used. (The three components containing points of the attractor are cyclically permuted by
f(
z).)
Three of the image layers are responsible for this, one for each of the three phases. The fourth gives the grey to white to blue to black outside coloring, which is by smoothed iterations.
More technical note: the actual computation of the phase is especially simple. First, when the image calculation begins, the attractor is discovered by forward-iterating the critical point 0 and looking for periodicity. When this is detected, the point is iterated further to discover the period (in this case 3) and then further until the total number of iterations elapsed is congruent to zero mod the period. Then the target disk is selected: a very small (1.0x10
-26 in radius) one about the attractor point in the central red component. (Since the components are cyclically permuted by
f(
z), the critical point is in that particular component, and the number of iterations was a multiple of three, the attractor point selected as the target must be in that particular component.)
Subsequently, the phase is simply the number of iterations mod 3 when the target disk is hit. If the point started out in the central red component, this will be zero for the reasons outlined above. Similarly, if it starts out in any component that maps to this component first, it will be zero.
To see this, consider the effect of iterating
f(
z) three times at once. The attractor members must behave as fixed points, and as attracting ones, under this composited map
f3(
z). Each has a separate basin. This defines a partition of the components into three disjoint subsets. Furthermore, a single iteration of
f(
z) must take a basin of
f3(
z) to another basin of
f3(
z); consider a point in a component in a particular
f3(
z) basin and its forward orbit for some large number
N divisible by three of iterations, sufficient to land it in one of the three attractor-components. Consider this point's first two forward-iterates, and the components they are in, and then all three points (and components) iterated by
f3(
z). Since the
Nth iterate was in one attractor-component, and the attractor is a 3-cycle, the
N+1st iterate must be in a second and the
N+2nd in the third. Since those three points were iterated by
f3(
z)
N/3 times in tandem and landed in separate basins of
f3(
z), they must have started out in separate basins of
f3(
z). Since we started with an arbitrary non-escaping, not-in-the-Julia-set point and iterated twice with
f(
z), we see that
f(
z) must take one of
f3(
z)'s basins to a second, and that to the third, and that to the first again, i.e. it cyclically permutes them. So which one it's in after a multiple of three iterations determines which basin of
f3(
z) it started in. The same thing determines how many additional iterations (zero, one, or two) it takes to land in the one with the target disk, and therefore the value of the number of iterations mod 3 when the target is hit.
For what it's worth, besides the red components being zero-mod-3, the yellow ones are one-mod-3 and the brown ones are two-mod-3.
Freely redistributable and usable subject to the Creative Commons Attribution license, version 3.0.
Detailed statistics:
Name: Dragon III
Date: February 11, 2009
Fractal: Julia set
Location:
c = -0.15273437498 + 0.67526041667
iDepth: Very Shallow
Min Iterations: 1
Max Iterations: 338
Layers: 4
Anti-aliasing: 3x3, threshold 0.1, depth 1
Preparation time: 10 minutes
Calculation time: 2 minutes (2GHz dual-core Athlon XP)