Saturday, December 15, 2007

Merry Yuletide

Here is a movie in which I have created an animation of a festive texture on the surface of a trefoil knot. Merry Yuletide!

video

How did I create this movie?

I have been playing around with the Belousov-Zhabotinsky reaction simulation that I described here.

One thing that you can do is to find limit cycles of the simulation, where the state of the array of cells returns to a state that it visited earlier in the simulation. The whole sequence of states between two such repeats (including one of the end points) is then a limit cycle of the BZ simulation. Such limit cycles must exist because the state space is finite in size, so it it inevitable that the simulation must eventually revisit states that it visited earlier, though starting from a random initial state the likelihood that this occurs in a given timescale decreases rapidly as the size of the array increases.

One example that I particularly like is a cycle of length 10 that I found on a toroidal 13 by 13 array (using the same parameter values and colour scheme as here), and I show 2 of these cycles in the movie below:

video

This cycle is unusual because it has a low symmetry, and because it is pretty to look at despite its short length.

Using this cyclic BZ solution on a torus it is relatively easy to create the movie at the start of this posting, by using it to texture the toroidal surface of a trefoil knot, and choosing a colour scheme that maximises its festive feel.

2 comments:

Anonymous said...

Pretty and clever, but is it art? Were Henry Cow before your time at Cambridge?

Stephen Luttrell said...

I don't think I said it was art - the tag says "animation". I guess you were referring to the passing remark I made in the earlier BZ posting here, where I said "...you could imagine using variants of this type of simulation for generating interesting dynamic art forms".

I certainly think that some animations can be art, but this would need much more effort than a mere BZ simulation to do creatively. At the very least you would need to create a special set of tools (i.e. "paint brushes") for creating and working with these sorts of dynamical object.

I've never heard of Henry Cow.