Some Julia sets

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.

Some more IFS fractals

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.

Attempt the Second! – IFS Fractal collab

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.

Nathan Shields eats maths for breakfast!

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.

10-Minute Math

Fractal from Binomial Coefficients

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

A Random Walk

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.

Wikipedia

MathWorld

More fun with Julia sets

I have been rather creative and explored an interesting Julia set. I will say that I picked it for the way it looks, rather than anything scientific. I make no claims that this Julia set is of any real mathematical interest, nor that it is related to any interesting dynamical system or anything like that.

Here is part of the Julia set for $$F_{c}=(1 + \sin(z) ) \log(|z|)$$ and with $$c = – 0.5 i + 2$$. I have included grid lines to help us navigate.

So, let us have a closer look.

The self-similarity in this Julia set is quite striking. the generic features here are also quite generic; the branching off and swirls.

Barnsley fern

Above is my attempt at creating the famous Barnsley fern [1]. It is one of the better known examples of a self-similar set, the other key example is the Sierpinski gasket.

The Barnsley fern resembles real species of Asplenium ferns, see below.

References

[1] Michael Barnsley, Fractals Everywhere, Boston, MA: Academic Press, 1993