Talk:Iterated function system

Equivalent constructions

There might be something unclear in the construction paragraph:

The most common algorithm to compute IFS fractals is called the chaos game. It consists of picking a random point in the plane, then iteratively applying one of the functions chosen at random from the function system and drawing the point. An alternative algorithm is to generate each possible sequence of functions up to a given maximum length, and then to plot the results of applying each of these sequences of functions to an initial point or shape.

Is this a correct rendition of the two methods? Method 1 as described here takes an initial point, applies a randomly chosen transformation, plots the resulting point and applies a randomly chosen transformation on that point again. Method 2 as described here do effectively the same sequence of transformations but only plots the last point. Method 2 is then just Method 1 throwing away most of the plot.

I might be showcasing my ignorance here, but in the chaos game (method 1), we have, intuitively, a point that is jumping around and never settles. Perhaps, we're trying to say that method 2 relies on function sequences such as f1(f2(f1(f3(f2(...(X)....))) where f1,f2,f3 are contractive mappings and that each such sequence have any point X converging on a specific fixed point. This sounds like it would rely on the Banach fixed point theorem but the chaos game seems to show that even though contractive mappings converge to a point, iterative application of different contractive mappings does not necessarily do the same.

So in short, I'm not sure what method 2 is supposed to be :-) EverGreg (talk) 13:54, 23 October 2009 (UTC)

The paper "fractals, graphs and fields" by Franklin Mendivil [1] explained the two methods and their equivalence very well. Recommended if there's others out there getting confused. :-) The construction paragraph seems correct, though it leaves out a lot of detail. EverGreg (talk) 19:40, 26 December 2009 (UTC)

to EverGreg: "Method 2 as described here do effectively the same sequence of transformations but only plots the last point."

There's a subtlety in method 2. Note that it doesn't say "generate a sequence of transformations". It says "generate EACH POSSIBLE sequence of transformations" (implication: from a known starting point in space.)

If you have 3 transformations and you want sequences of length 10, you have 3^10 possible sequences of transformations. Given a common (arbitrary) starting point, this will yield 3^10 end points. Those 3^10 end points are plotted and make your fractal. 46.64.181.38 (talk) 14:11, 19 April 2014 (UTC)

Right. Another way of looking at it is that the chaos-game algorithm is a random depth-first "search" that never backtracks, while Method 2 is an exhaustive search up to a given depth. But yes, this paragraph is super unclear. I just improved it slightly, but a lot more work is needed. Kragen Javier Sitaker (talk) 05:49, 19 July 2014 (UTC)

Example

Just not an encyclopedic, throught with ferns and triangles IFS animation 1 and IFS animation 2 —Preceding unsigned comment added by Edo 555 (talk • contribs) 14:05, 8 July 2010 (UTC)