A world in dk(decay/denmark)

I stumbled upon a toyshop that is quite fascinating, Grand illusions

Thay have a nice collection of different, and perhaps a little geeky toys. But the best part is there are several articles and videos documenting the toys, not only for the stuff they have in there shop but also from Tim’s huge collection of toys. I don’t know who Tim is, but he does a very nice and distinctly British job of presenting those nifty gadgets, and i am left thinking that it would be a great joy to be his grandchild.

And as can bee seen be the heading I have read Jacob Nielsen: “Passive Voice Is Redeemed For Web Headings”

I the moment I’m taking a course in Numerical Analysis, and here we were of course introduced to the Newton-Raphson method for finding roots. That is the method where, geometrical speaking, we take at tangent line to a function and use that to approximate the root, and keep on doing that until we reach satisfying precision.

Depending on where you start the iteration you get a a different root, and requiring a different amount of iterations. That’s self-evident. The “funny” part is that when searching for complex roots there are areas in the complex plane, where a very small difference in starting point gives different roots, and with remarkable difference in the number of iterations.

This is a plot where each root in the polynomial x^4+2 have been assigned a different color. And each point is colored according to which root a iteration starting here finds.

Now thats the mathematical part of it. Now to the beautiful part or at least more fascinating part. The pattern reapeats it self, as a fractal, when you zoom in.

So i modified my code to zoom in, and i used a diffenrent coloring scheme where i only colored according to the number of iterations taken. This made it possible to generate a video zooming in og showing the rapeating patterns in “nice” colors.

Here a some videos :

4 roots (14.6 mb)
7 roots (50 mb)

Yes I know that its very large files, and maybe i will upload them to google video at a later time for easy viewing and less quality.

As mentioned before I’ve made a one dimensional Cellular Automata(CA). And although it was intriguing they quickly became very predictive. Actually it was in two dimensions. As the a y-coordinate represented time, but since the states were only depending an the adjacent neighbor’s to the right or left its considered to be one-dimensional.

But they are not as fascinating as CA’s where you expand the neighborhood to include all the eight adjacent cells. That’s called the Moore Neighborhood.

Representing the time dimension. As the intersting part is the evolution of the system, representing time is essential. With a two-dimensional CA it has to be done using animations, and who doesn’t like blinking pixels. This makes CA’s makes even more fun. Continue reading ‘Cellular Automata II’ »