Glynn fractals.
Glynn fractals are actually a kind of Julia set. They were discovered by Earl F. Glynn II whose site is linked above. It will help to read the Julia set article before reading this one. The original Glynn fractal was based on the function f(z)=z^1.5+mu, like a quadratic Julia set except for the funny exponent. This type of fractal has a tree-like structure, as can be seen in the picture above, left. That tree-like fractal is one of the bumps from the arrow-like fractal on the right. On this web site we call any Julia set based on z^e+mu, where e is a fraction, a Glynn fractal. As with the transcendental Julia sets some adjustment of the standard Julia set algorithm is required. In this case |z|<2 becomes |z|<4. The revised algorithm is given below.
Glynn fractal membership algorithm. |
Choose the parameter mu=a+bi
Pick a Glynn exponent, e.g. e=1.5.
Pick an iteration cutoff limit.
For every point (x,y) in your view rectangle
Let z=x+yi
Set n=0
While(n less than limit and |z|<4)
Let z=z^e+mu
Increment n
End While
if(|z|<2) then z is a member of the approximate
Glynn fractal, plot (x,y) in the Julia set color
otherwise z is outside the Glynn fractal,
plot (x,y) in the outside color.
End for
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We call the ``full'' Glynn fractals, like the one at the top left Glynn arrows because of their appearance. It's worth mentioning that Glynn fractals mostly look awful when colored with the cosine method. The Glynn fractals in the gallery are all paletted.
Examine the two Glynn fractals below. They are both based on the function f(z)=z^1.75+mu and use the view window with corners (-0.39,-0.7) and (0.19,-0.12). The only difference is that the left image uses mu=-0.375 while the right uses mu=-0.375+0.002i. This addition of a small imaginary perturbation creates the fault lines in the image. These fault lines grow rapidly with the imaginary part of mu. They are related to the branch of the complex fourth-root function which becomes far more visible when the imaginary part of mu is non-zero.
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