# On a variant of rhodonea curves

Rhodonea curves or rose curves are plots of a polar equation of the form
$$r = \cos(k \theta)$$.

If we specialise to equations with

$$k= \frac{n}{d}$$

for n and d integers (>0), then we have plots of the form below. In the table n runs across and d down

Now, just for fun I considered a slight variant of this given by

$$r = \cos( k \theta) – k$$

The plots are as follows

For another variant I considered

$$r = \cos( k \theta) – k^{-1}$$

I am not sure there is anything mathematically deep here, I just like the images and classify this as some basic mathematical art.

# More experiments with random walks

I have again been playing with some random walks, using the same method as here. This time I used 1000000 iterations and added some colour.

Below are random walks, on the plane (not a lattice) for which step size gets (on average) smaller and smaller with each step. I pick the step size using the Maxwell-Boltzman distribution (with a =1) and a suitable scaling which depends on the iteration parameter. I the add a opacity depending on how many times the points are visited: bright white means a lot, while grey means not many and black never.

Once again, these images are rather for artistic purposes than scientific purposes.

# This is like so random…

Below are random walks on the plane (not a lattice) for which step size gets (on average) smaller and smaller with each step. I pick the step size using the Maxwell-Boltzman distribution (with a =1) and a suitable scaling which depends on the iteration parameter. I the add a opacity depending on how many times the points are visited: bright white means a lot, while grey means not many and black never.

I may play with these further, but they make some interesting pattens. We have approximate self-similarity and so these patterns have fractal-like properties. Anyway, enjoy….

These images were created for artistic rather than scientific reasons. That said, random walks are have been applied to many fields including ecology, economics, psychology, computer science, physics, chemistry, and biology.

Probably the most famous application of a random walk is to Brownian motion, which describes the trajectory of a tiny particle diffusing in a fluid. I have no idea if there is anything scientific in these images, but I would not be surprised if for small step sizes we have approximately Brownian motion. However, I would need to think a lot more about this before making concrete statements.

# Even more fun with IFS

Another IFS that my wife and I created. Not sure what to call this one!

# A fractal fern

The above is an IFS fractal that resembles a fern. Maybe not as good as Barnsley’s fern , but mine was generated using two affine transformations and not the four as used by Barnsley. It is a nice image and I am happy with it.

# On to infinity: a peice of art

This is another piece of mathematical art my wife and I created. I call it “On to infinity”.

# Some (more) IFS fractals

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.

# A very symmetric Julia set

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.

# 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.