Picture Post. Minimal nerd talk.

a = 1.3, viewport center = -0.0981 + 1.239i. Formula information at Bugs #14.

For the next few days, I am going to post images from the bug formula without the nerd talk.

a = 1.5, center at -0.153+2.163i

Here is a look at the vertical spoke rising above the origin. Msets and Tricorns alternate here as well. The picture back in Bugs #18 show a mini-tricorn. The text describes an infinite grid of points covering the whole plane where the iteration of 0 falls into a 2-cycle. The hot spots in this picture are some of the points on that grid.

Zoom out by another factor of two. The two shapes alternate over the entire horizontal axis at a spacing of $ a*\pi$.

Recall the formula, $ bugR(x+yi) = a*sin(x/a) + yi$, and whatever you may remember about the sin() function. bugR() is periodic with period $ 2*a*\pi$. bugR(z) = 0 for all $ z = a*n*\pi$, n an integer. For the even multiples, $ bugR(z+2n*a\pi) \approx z$ and for old multiples, $ bugR(z+(2n+1)*a\pi) \approx -\overline{z}$. While these observations do not constitute a full proof, it does strongly suggest why the two shapes alternate along the real axis.

This one has the same zoom factor as the previous, the center has been moved to $ a*\pi/2$. As with the bugI images, m-sets and tricorns alternate, this time along the real (horizontal) access. Compare with Bugs #10.

Zoom out by another factor of 2.

No posts for the last two days. I have no excuses, I simply did not make the time. I will try to make up with multiple posts today.

The next few pictures are success out-zooms of Bugs #16 to get an idea of what the neighborhood looks like.

Today's image is a zoom out by a factor of two.

For a quick recap, this formula is bugR, defined back in Bugs #14 , with parameter a = 2.5.

At the next intersection to the east is a mini-tricorn. The minibrots and tricorns alternate.

Yesterday's image was a deep zoom into a "quiet" area. It appears that an infinite grid with ever finer lines is everywhere. At every intersection, a deep zoom holds something interesting. Here is a distorted minibrot.

With z^2+c and a little algebra, you can show that if |z| > 2 + |c| then z escapes. Yesterday I showed a similar argument for bugI() which concludes that if |x| > a + 1 + |c| then z =x + yi escapes. So for both of these formulas, there are easily identified large areas where the orbit is guaranteed to escape. Also we saw that this type of simple algebra argument breaks down for bugR(). That does not prove that no such area exists, only that if it does, it will be difficult to find. Let's dig deeper

Since this is a blog, not a book, I should recap frequently. We are iterating

$ bugR(z^2+c) \quad where \\ bugR(x+yi) = u(x)+yi, \\ u=a*sin(x/a), a>0$

Recall that $ |u(x)|<= a \text{ and } u(a*n\pi)=0$

If a>.5 (an easy assumption, it has been true for every picture so far), u(p)=-.5 has a solution. In fact many solutions when you find one, $ p+a*n\pi$ is also a solution for any integer n.

Look at $ c=c_r+c_i i = p+a*m\pi + \pm \sqrt{p+.25+a*n\pi} i $. where n and m are arbitrary integers.

These points lie on a grid. Equal space horizontally, decreasing spacing vertically because of the square root. But extending to infinity in all directions. Where ever you are in the plane you are surrounded by these points. The following will show that none of these points escape, in fact every one is a 2-cycle.

Lets iterate bugR(z^2+c)

$ \begin{matrix} z_0 & = & 0 \\ z_1 & = & bugR(c)\\ & = & -.5 & + & c_i i \\ z_2 & = & u(.25 - c_i^2+c_r) & + & (2*(-.5)*c_i + c_i)i \\ & = & u(.25 - (p +.25+an\pi)+p+am\pi) & + & (-c_i+c_i)i \\ & = & u(a*(m-n)\pi) & + & 0i \\ & = & 0 \end{matrix}$

So every one of these points generates a two cycle.

It is embarrassing to admit how much time I spent on this. My math skills are rusty.

I wanted visual confirmation that the webbing is everywhere. Here is a deep zoom into the nearly empty area in the top right of yesterday's image. The color has been changed from previous image to enhance the features. The lines get thin very quickly. You see a horizontal line, intersected by many vertical lines, and each of them connected by almost invisible horizontal lines. Because the lines in this third set are thinner than the pixels in the image, Moiré patterns appear.

To save typing and simplify some calculations, define $ u(x) = a * sin(x/a)$. Note that $ u(a*n*\pi) = 0, \text{ and } |u(x)| < a$.

With this notation, $ bugI(x+yi) = x + u(y)i$. Set $ c = c_r+c_i i$ and expand the iteration $ z' = x' + y'i =bugI(z^2+c)$ .

$ x' = x^2-y^2+c_r\\y' = u(2*x*y+c_i)$

After the first step, we always have $ |y'| < a$. With a little algebra, it is seen that if $ x > a + 1 + |c_r| \text{ then } x' > x$. So the real part of the iteration keeps getting larger, and we have many orbits escaping to infinity.

A similar analysis fails with $ bugR(x+yi) = u(x)+yi$. On each iteration step,

$ x' = u(x^2-y^2+c_r)\\y' = 2 * u(x) * y + c_i$

The real part is always bounded, $ |x'| < a$, so the only hope of escape lies with the imaginary part. But if x is a multiple of a*pi, then u(x) = 0, and y has to start over.

As was done earlier with bugI, the next few posts explore bugR from a more mechanical or mathematical perspective.

Compare this image with Bugs #8. Same location, zoom, and color, same parameter a = 2.5,  but with the R version replacing the I version of bug. The webbing appears to be everywhere, although it quickly becomes too thin to see at this image resolution.

Here is another more artistic rendering of bugR. a = 2.0. Compare the web like structures with the bugI images, such as Bugs #12 and Bugs #6. Here the strands are thinner, but are everywhere.

There are several pictures on my hard drive that are queued up to post here. The plan was to introduce the bug formula, provide some nerdy math descriptions and examples, then start posting the more artistic images. I am done with the nerd stuff, and then noticed a problem with that plan; most of the images I have are based on another version of the bug formula. I won't keep you in suspense. You may have guessed the other variation. I added a letter to the name to keep track of which part is gets modified.

$ bugI(x+yi) = x + a * sin(y/a)i \\ bugR(x+yi) = a*sin(x/a) + yi$

As a quick recap, the bug formula is combined with the usual $ z^2+c$, and the images are generated in a typical escape-time fashion. There are two variations of the combined formula, depending on who goes first.

$ (bugX(z))^2+c \\ bugX(z^2+c)$

Most of the pictures here use the second "bug last" variant.

The pictures of the last two weeks have been based on bugI(). Today's picture is based on bugR(), with a = 1.5.

Here is a deep zoom of a bugs fractal. Parameter a = 2.0, viewport center at -0.7871755+0.1839239i, width = 5e-7. Please see the earlier posts, Bugs #3 and Bugs #8 for a description of the formula.

I am short on time today, but still striving for a post a day. So just a picture with very few words. Bugs fractal, parameter a=1.2. Viewport center -1.279+0.371i, width = .008.

This is a zoom into yesterday's image, getting a closer look at the tricorn. The parameter is still a = 2.5. The image center is at $ a*\pi*i$.

Recall the definition of the parameter version of bug: $ bug(x+y i) = x + a * sin(y/a) i$. Then, a little algebra,

$ \begin{matrix} bug(z+a \pi i) & = & bug( x + (y+a \pi)i) \\ & = & x + a *sin(y/a+\pi)i \\& = & x - a * sin(y/a) i \\ & = & \overline{bug(z)} \end{matrix} $

If you are a little rusty the definition of complex conjugate is $ \overline{x+yi} = x-yi$, and a well known trig identity is $ sin(\theta+\pi) = -sin(\theta)$.

Also recall that near z=0 for large a it is almost true that bug(z) = z. More formally, for any $ \epsilon, R > 0$ if a is large enough then $ |bug(z)-z|< \epsilon \text{ whenever } z < R$. Now we can also say $ |bug(z+\pi i)- \overline{z}|< \epsilon \text{ whenever } z < R$

There is a little more to be done, I will skip the details. If you track what is going on in the iteration of $ bug(z)^2+c$ near $ c = a * \pi i$ you will see that it behaves like the tricorn, $ \overline z^2+c$ near $ c = 0$.

Given the periodicity of the sin function, the same is true for c near $ c = n * a * \pi i$ for any integer n. Even n looks like a Mandelbrot set, odd n looks like the Tricorn.

Here is a zoom out by a factor of two from yesterday's picture. The triangular shape at the top is called a Tricorn or Mandelbar. The parameter a is 2.5. (See Bugs #8 for a description of the formula.) The image is centered at $ (a * \pi /2) * i$.

The Tricorn set is generated by $ \bar{z}^2+c$. The standard formula is modified to use the complex conjugate of z. Here is a non-paywall academic paper on Tricorns

Zoom out by a factor of 2 from yesterday's image, we see the rope/web supports extend above and below the main body.

The rest of this post is for the more mathematically inclined readers. Yesterday I stated that the bug function is close to the identity. The real part is already there, so let's focus on the imaginary part, and treat just the imaginary part as a real valued function.

Here is a table of values for three function and their first few derivatives evaluated at 0.

Notice that for the parameterized sin function, a*sin(x/a), with a large value of a, all of the derivatives are close to the derivatives of the identity function, x. Using  Taylor series, the difference can be made arbitrarily small of an arbitrary large neighborhood of the origin.

For example, $ \text{ for } \epsilon, R \text{ where }0 < e < 1 < R, \text{ set }a = R^2/\epsilon$ and use Taylor series to show that

$ |x-a*sin(x/a)| < \epsilon \text{ whenever } |x| < R$

The same result carries over to the complex bug function.

Today's image uses the parameterized version of the bug formula with parameter a = 2.5.

The parameterized version of the bug formula is $ bug_a(x+iy) = x + i*a*sin(y/a)$. When a = 1.0, this is the same as the non-parameterized version.

For large values of a, the bug formula gets close to the identity function, and the images get closer to the Mandelbrot set. By adjusting the parameter a, you can control the degree of distortion / similarity when compared to $ z^2+c$.

In this view, with a = 2.5, the non-cyclic, non-escaping chaotic region in the upper right and lower right of Bugs #3 and Bugs #4 are gone. The body looks pretty normal. Surrounding the body is a bifurcating webbing that is characteristic of the bug formula.

Another image showing the interconnecting webbing. This one surrounding a distorted mini.