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 .
The Tricorn set is generated by . 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, and use Taylor series to show that
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 . 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 .
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.