It has been a while since I have produced any iterated function system fractals, so here is my latest effort. I have not coloured them, rather the opacity describes the number of times a point has been visited. This gives a nice effect for some, but not all of these fractals.
This is just a Julia set that I created. I was surprised at just how symmetric it is as well as showing clearly the fractal property of self-similarity.
My wife and I have created a few more iterated function system fractals for you to enjoy.
You can find other examples of our collaboration here, here and here.
Above is the Julia set of \(F_{c} = e^{z^{3}}\) at the point \(– 0.621\).
Above is the Julia set of \(F_{c} = (1+z+ \frac{z^{2}}{2})Exp[z^{-3} -z]\) at the point \(-0.6 -i\).
Above is the Julia set of $latex F_{c} =-\frac{\cosh \left(-z+1+\frac{1}{z}-\frac{1}{z^2}+\frac{1}{z^3}\right)}{|z|}+\sinh
\left(-z+1+\frac{1}{z}-\frac{1}{z^2}+\frac{1}{z^3}\right)$ at the point \(-(0.62-0.4 i)\).
I have posted other Julia sets here.
You can find out more about Juila sets here.
Above is another iterated function system fractal my wife and I created. You can see the self-similarity of the “swirls” clearly. We are both pleased with this one.
This one has a rather organic shape. You can see that this one consists of overlapping spirals and almost a ghostly appearance.
The above is similar to the first IFS system fractal we created together. It has a spongy-organic look to it.
You can find other IFS fractals we have created here and here. We may post more in the future.
This is the second collaboration with my wife, she’s quite pleased with it and told me to put it up!
The first attempt can be found here.
Above is an iterated function system fractal generated from two affine transformation of the plane and Barnsley’s chaos game.
This image represents the first proper artistic collaboration between myself and my wife. Hopefully more will follow.
Nathan Shields has been very creative with his breakfast, making pancakes that resemble fractals.
Above is the tastiest I have ever seen. Compare it will the image below.
Other examples can be found via the American Mathematical Society website here.
Links
Above is a discrete fractal generated by creating a table of zeros and ones by deciding if the binomial coefficients are even or odd. The “key” here is paint black if odd, otherwise leave light blue.
The pattern is closely related to Pascal’s triangle.
The pattern clearly shows self-similarity as all fractals do.
As far as I know, this pattern was first noticed in [1]. Also note that we have a structure very similar to the Sierpinski Sieve. In the limit of infinite rows we recover the Sierpinski Sieve, up to a shift in the positions of the zeros and ones.
A slight variant
Just for fun I used the same algorithm to study the pattern associated with modified binomial coefficients of the form
\(\left( \begin{array}{c} (-1)^{k}n\\ k \end{array} \right)\)
Again the pattern shows lots of self-similarity.
References
[1] S. Wolfram: American Mathematical Monthly, 91 (November 1984) 566-571
Random walks can be found throughout nature in many different contexts. For example they been used in ecology, economics, psychology, computer science, physics, chemistry, and biology. Above is an example of a (simple) random walk I created. There is 8 directions to this walk and 1000000 points.
The random walk above is an example of a Markov process, that means that the next step only depends on the present step. Such processes have “no memory”.
Random walks are closely related to Brownian motion, which is the physical phenomenon of minute particles diffusing in a fluid.
Random walks are examples of discrete fractals. They show self-similarity on large scales (such as in the picture above), but on the smaller scales the discrete nature of the grid becomes apparent. See the picture below.
Here we have another random walk generated in exactly the same way as the previous one, but not just 1000 points. One can consider this as a “zoom in” on the random walk with a million points. The finite step size is apparent and the resemblance to genuine fractals is far less clear.
Links