Ummmm.... More Fractals

Twitchy Hilbert

A while ago I read this article about the Hilbert Curve and I totally misunderstood it. I thought it was presenting a 1-1 mapping between the number-line and the Hilbert Curve. I was very confused at the time and if I had looked a little more carefully I would have seen I was wrong.

The mapping that was presented was not 1-1 but it was REALLY COOL. And it did map every point in [0,1] to a point in [0,1]X[0,1]. My misunderstanding lead me to use the mapping presented in a Java program that tried to draw the Hilbert curve using the mapping. Essentially I had a parameter t which would go from 0 to 1 in steps of size S. I drew a line between t_i and t_i+1 for every i. At the end I connected to the starting point and filled it in. I did not make the Hilbert curve, but my creation was very interesting. Below is a GIF of it where I vary S over time.

Some pictures of the shape with various increments:

Koch snoflake

The Koch snowflake is a classic fractal. It, like many other similar fractals, can be made a number of ways (IFS, Sierpinski hexagon, L systems, trees, more?). The usual way to make it is to draw an equilateral triangle, then for every line, draw a new equilateral triangle that has one edge along the middle third of the line. The process of replacing each line with a line that has a spike in the middle is repeated until you have the image below. I found the point of the added triangles by first finding the 2 points that separate the lines into thirds, then rotating the point that lies further counter clockwise and rotating it 60 degrees clockwise around the other point.

The modifications of the normal figure found here were made in several ways. Some where made by rotating the point counterclockwise rather than clockwise at varying times (e.g. every third triangle piece). Others were made by rotating it by different angles. One of the videos was made by varying things in a way I have not analyzed enough to understand; I believe it varied both angle and radius but if so it was done indirectly through rotation matrix shenanigans.

Koch

Anti-Koch

Video time!!!!

Some Newton Fractals and Images Made of Images

I like to pretend some day I will put explanations for the pictures I post. Its strange because I am always eager to explain them to people individually, somehow I feel obligated to formalise things if I explain them here and that feels boring and annoying. Feel free to ask me stuff through comments or email, I will be more than happy to answer.

Newtons Fractal

Not working yet

Still not working

The shape is almost right

The shape is right but it looks very different from what I have seen before

Getting closer to the standard

Very close but there's this weird bit at the bottom

This is what most look like.

It turned out that the program was having difficulties with 0 and 2pi.