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.

Here is a closer look zoom at the ropes/webbing. This is a magnification of the top left of Bugs #3

View port center (-1.65, 0.49), width 0.25

The main characteristic of the bugs fractal series is the "web" network surrounding and connecting the areas of stability. That will be the focus for this series. However, they also often contain regions of chaotic behavior. So today we take a digression to look at the chaos.

First the TL;DR version. Fractal images based on the ubiquitous $ z^2+c$, are clean and orderly. Complex yes, but also clean. Fractal based on non-holomorphic functions, (and many rational functions) are messy with dust and brush strokes, like today's image.

In a dynamical systems, orbits either (see Stability of orbits.)

• escape to infinity
• converge to single point or a finite periodic cycle
• are non-escaping, non-periodic (chaotic)

For a dynamical system based on polynomials, almost all orbits fall into the first two buckets. In the usual $ z^2+c$ fractals, the orbits of the third type lie on the thin one dimension boundary between escaping and periodic regions, and the one dimensional filaments. Much too small to see.

But when you get away from well behaved polynomials, two dimensional regions with chaotic orbits of the third type appear.

While the bug formula, $ bug(x+iy) = x + i* sin(y)$ seems simple, it is not differentiable as a complex function. So adding this to the fractal calculation produces a non-holomorphic formula with potential for two dimensional chaos.

In general, fractals with two dimension chaos is a worthy topic to study in its own right. I am sure I will return to it from time to time. But tomorrow, focus returns to the webs in the cleaner areas of the bugs fractals.

Here is the same image as yesterday, but with "black inside" turned off.

Traditionally for fractals only the escaping points are colored, and the color indicates how quickly the point escapes. The non-escaping, or "inside" points are colored black. For traditional fractals, the insides are boring. The orbits of the points in the non-escaping region flow towards a single fixed point, or an attracting cycle. A solid inside color helps shifts the viewer's focus to the swirls surrounding the fractal body.

With non-traditional fractals, especially fractals based on non-holomorphic functions, the inside area is more interesting. The orbits can bounce around in chaotic fashion, never settling into a fixed point or cycle. It is not easy to find a meaningful way to color such regions. But that makes it all the more rewarding when you discover a way to visualize the area.

Time for some nerd talk.

First, let's meet the bug formula: $ bug(x+iy) = x + i* sin(y)$.

Typical fractals iterate the formula $ z \to z^2 + c$. Here the basic fractal formula is combined with the bug formula. There are two ways to combine them.

bug first: $ z \to bug(z)^2 + c$
bug last: $ z \to bug(z^2+c)$

Mathematically the order matters, but for pictures the order makes very little difference. I am not consistent in choosing one or the other. This image happens to use the "bug first" variation.

The image is the "Mandelbrot set" for the combined formula. Technically speaking, if you are not using z^2+c, then it is not the Mandelbrot set. A proper description would be the "parameter space" view. But just this once I will indulge in sloppy language since the process is exactly like creating a Mandelbrot set image, but with the slight change in the formula. Compare to basic fractal.

The image is centered at (-1,0) with width 3.0 and height set to maintain a 1:1 mathematical aspect aspect ratio. (Meaning each pixel represents a square in the complex plane.)

I played with these bug fractals a lot in the 90s. Many images on the old site were bug fractals. I return to the formula often, especially when nothing else seems to be working. Two months ago I started playing with them again. There are several images on my hard drive, that need to be sorted and organized. I plan on doing that over the next few days. The first two are presented without explanation as a visual introduction to the series.

This one starts the Bugs series.

I will provide all the nerdy details on how I create these in a later post. For now, just enjoy the image without thinking about how or why.

Here is another one similar to something on my todo list. "curved line cubism". This one has a vague cubist feel. There are lines that run through the image. There are similar colors on each side, but no continuity across the them. The lines are curved, suggesting a flow, but not the usual straight lines, squares, circles you would expect in a cubist painting.

I have a long list of art ideas. Well, it is more like a large collections of scattered notes. The notes take many forms, a sketch or line drawing, written descriptions, mathematical formulas, algorithm flowchart, images classified as "needs more work", photos or other art from the internet that I find interesting. If I were more organized, several of the ideas that could be grouped under "simulated paint flow", and this image is one, "non gravity simulated paint flow".

OK, there is some gravity here, the flow appears to be from the upper left. But for the most part there is no uniform flow direction.

So, at some future date, I planned to start with a much less detailed mental picture of something like this image, and work on creating an algorithm. I have some ideas where to start, but the confetti algorithm is certainly not one of them. Chopping and shuffling is the opposite of the goal, in this case I want smooth mixing.

When I completed this one Confetti #24, the detail in the diagonal boundary separating the top left corner is very close to the flow idea. So a little exploratory side trip seemed justified. The low iteration fractal stretches and distorts the confetti squares, turning them into little rivulets. It is a delicate balance. Too few iterations and they are still rectangles, and in some cases parabolic sections. Too many iterations and they turn into fractals.

I am going to revisit this in the future, and it will have a different and cooler name. "Non gravity simulated paint flow" is just for my internal notes. Although, maybe if I say it often enough…

Here I mix in a low iteration fractal.

What does that mean? A fractal image requires a large number of iterations to get the stars, spirals, tendrils and fireworks to appear. When a fractal is not iterated enough, you may see a vague start of those features, but also flat areas that just scream "this is incomplete". So no one would post a low iteration fractal as a finished product, and if they did, they would not call it a fractal. However, these low iteration fractals do work well with the confetti algorithm. The incomplete fractal is more complex that a simple geometric rendering of circles, lines, and waves. They provide a rich new collection or possible starting points for the confetti algorithm.