Sequence Fractals Part I #22


si = -1, 1, -1, 1, …
Center: C2.00: 0.053427, 0.781326, Zoom = 16

First of three pairs of two-cycles, C2.00. Below and to the left of C1.00 in Sequence Fractals Part I #21

  6 two cycles
C2.00:   0.053427,  0.781326
C2.01:   0.053427, -0.781326
C2.02:   0.633096,  1.288065
C2.03:   0.633096, -1.288065
C2.04:   0.813477,  1.815442
C2.05:   0.813477, -1.815442

Everything is reflected across the real axis. C2.01 is exactly the same, just flipped vertically. So I will ignore everything going on in the bottom half of the complex plane.

Sequence Fractals Part I #21


si = -1, 1, -1, 1, …
Center: C1.00: 0.500000, 1.322876, Zoom = 0.8

The helper functions will be redefined for the new sequence. (Notice the sign flip in key places.)
f0(z,c) = z2+c-1
f1(z,c) = z2+c+1
fc(z) = f(z,c) =f1(f0(z,c),c) = (z2+c-1)2+c+1.
Derivative: fc‘(z) = 4z(z2+c-1)

z0 = 0 is a critical point and will again be the default starting point. The other two critical points will be ignored without guilt. They just bring us back to the sequence starting 1,-1…, about which we already know more than we ever wanted to know.

Fixed points are found when f(0,c) = 0. so c = \frac{1\pm\sqrt{-7}}{2} = 0.5 ± 1.322876i.

  2 fixed points
C1.00:   0.500000,  1.322876
C1.01:   0.500000, -1.322876

I will start naming named points with a leading letter to avoid confusion with decimal numbers.

Sequence Fractals Part I #20


si = -1, 1, -1, ,1, …
Center:0+0.i, Zoom = 0.25,
Variable plane (Julia), c0 = 0

Here is a picture of the Julia set for the new sequence (starting with -1). It is disconnected, has interior, yet is not dust.

If you are reading these posts you probably know Julia sets. But just in case, here is the wiki definition: Julia set. The picture is the Filled Julia Set. The actual Julia set is the boundary around the white areas.

Stepping way back to the basics.(I should have explained this a long time ago, I just assumed it was obvious.) We have a complex function in two variables, f(z,c). I like this view, it has symmetry, calling out both variables avoids confusion. Equivalently, and traditionally, we are looking at a family of complex functions fc(z), indexed by a complex value, c. z is the variable and c is the parameter. Fix two numbers c0, z0, and define a sequence of numbers by iteration, zi+1=fc0(zi), called an orbit. The question is how does this orbit behave in the limit. The simplest dichotomy is the orbit goes to infinity or it does not. In all of the pictures the non-escaping points are colored white (traditionally they are black, I am just changing things up to be different). The escaping points are colored (approximately) by how quickly the orbit rushes off to infinity.

In the previous pictures in this series, z0 = 0 (usually) and c0 depends on the pixel. A linear map maps pixel coordinates to complex numbers. Such pictures could (and probably should) be called “Non-escaping parameter plane” pictures. For the function family z2+c, this is the Mandelbrot set.

In today’s picture the c value is fixed, in this case c0 = 0, and the pixels are mapped to the complex plane for the starting z0 value. z is the variable so these are “Non-escaping variable plane” pictures.

Sequence Fractals Part I #19


si = 1, -1, 1, -1, …
Center:0+0i , Zoom = 0.25

Way back at the start Sequence Fractals Part I #1 we had these definitions:
f0(z,c) = z2+c+1
f1(z,c) = z2+c-1
fc(z) = f(z,c) =f1(f0(z,c),c) = (z2+c+1)2+c-1.
Derivative: fc‘(z) = 4z(z2+c+1)

Critical points are where fc‘(z) = 0. One critical point is 0, which we have studied in some detail. The other two are the roots of (z2+c+1), \pm\sqrt{-c-1}

The sign does not matter (it disappears with the first z2), both roots behave the same. So, for today only, let z0 be one of these roots and the starting point for the fractal iteration. Also forget about the two step function for now, and look at the original sequence fractal definition, and write out the first few iterations (notice the alternating function subscript)
z0,f0(z0,c), f1(f0(z0,c),c), f0(f1(f0(z0,c),c),c), f1(f0(f1(f0(z0,c),c),c), c)…

z0 was defined to be a root of (z2+c+1)=0 which is exactly f0(z,c). How convenient! We can rewrite the orbit as
z0, 0, f1(0,c), f0(f1(0,c),c), f1(f0(f1(0,c),c),c)…

After the first step this is exactly the sequence fractal for si = -1, 1, -1, 1, … starting at 0. That sequence will be the subject of the next several posts.

Sequence Fractals Part I #18


si = 1, -1, 1, -1, …
Center:–3.051388+0.210786i , Zoom = 10000

For orientation, this center of this picture is up and to the right of Sequence Fractals Part I #12. Using the same notation, as Sequence Fractals Part I #16, z_3=z_1\ and\ z_2\neq z_0 so for this c, the critical point orbit is preperiodic with pre-period 1 and period 2.

In general if zn+k=zk and n,k are the smallest values for which this is true, then the orbit is preperiodic with pre-period k, period n, and denoted Mk,n. Such points should be called Misiurewicz points. See

The wiki article says that it is not a Misiurewicz point unless you are looking at a polynomial conjugate to zd+c for d >=2. If you are not familiar with the term ‘conjugate’, it is a particular, precise mathematical concept for a type of “similar to”. The key property is that there is only one critical point. Our polynomial has three critical points. (We have been ignoring the other two.) I have been careful to avoid calling something a Mandelbrot set that is not a Mandelbrot set. So I will also avoid calling these Misiurewicz points. However I will stick with the M_ notation.

Sequence Fractals Part I #17


si = 1, -1, 1, -1, …
Center:-3.631717+0i , Zoom = 6000

This is a zoom into the far left side of #1, also see #9, #10, #11, for more views of the neighborhood. The center is another M1,1 point. This point is similar to the point -2, a M2,1 point in the Mandelbrot set. (The Mandelbrot set has no M1,1 points.)

Self-similarity is in full force here. Every zoom looks the same. I offer the following “proof by recursive pictures”: For the parameter space non-escape set, M, for our function, -3.631717… is an accumulation point of M, a boundary point of M, and not connected to any other point of M.

In the Mandelbrot set, -2 has the first two properties, but not the third.

If you liked the bit about derivatives in yesterday’s post, for today’s picture

 f'(z1)  = 24.1509+0i
|f'(z1)| = 24.1509

So there is self-similarity with a 24x zoom and no rotation.

Sequence Fractals Part I #16


si = 1, -1, 1, -1, …
Center:0.034695+0.488109i , Zoom = 40

Take a look back at Sequence Fractals Part I #7, the two-cycle 2.05. Today we are looking at a zoom of the left most subgroup of the group of islands in the top left.

No, the camera did not shake. The center is as intended, the bottom of the small group in the center. You may be sensing a pattern here.

Denote the critical point orbit by zi. More formerly, z0 = 0 our critical point, and zi+1 = f(zi,c).

For c at the center of today’s image, z_2=z_1\ and\ z_1\neq z_0. So z0 is not a fixed point, but after one step it becomes a fixed point. z0 is called a preperiodic point with pre-period 1 and period 1. Often denoted M1,1.

The fixed point, z1 = -0.34122 + 1.43075i. It is a repelling fixed point. The derivative is

 f'(z1)  = 4.11157-4.85809i
|f'(z1)| = 6.364437

When referring to fixed points and cycles, the derivative is sometimes called the eigenvalue or the multiplier, and is denoted λ. |f’| > 1 indicates a repelling fixed point. f’ also gives a clue to the self-similarity behavior. In polar coordinates f'(z1) = re, r=6.3644, ϴ=–0.868 = -50 degrees. Take the image, blow it up by a factor of 6, and rotate it -50 degrees and you get the original image back image.

Sequence Fractals Part I #15


si = 1, -1, 1, -1, …
Center:0.104301+0.479303i , Zoom = 4000

Hosts for three three-cycles, 3.22, 3.24, 3,26 surround the two cycle host 2.04. (3.23, 3.25, 3.27, 2.05 reflections below the axis). See Sequence Fractals Part I #11 and Sequence Fractals Part I #6 for the definition of these points. 3.26 is today’s picture. In Sequence Fractals Part I #7 3.22 is the biggest mini in the lower left. Find 3.24 and 3.26 are up and to the right of the main feature.

Sequence Fractals Part I #14


si = 1, -1, 1, -1, …
Center:0.146654+152514.i , Zoom = 256

There are four three cycles surrounding c=0, see them at the four corners surrounding the main shape in Sequence Fractals Part I #2. They are 3.15, 3.16, 3.28, 3.29 in the Sequence Fractals Part I #11 list. This is a zoom of 3.28 in the NE corner. 3.29 is a reflection, 3.15, 3.16 are similar.