1. A fractal made with circles?

Quite possibly this problem (or one like it) has been discussed before, if so I would be interested in any information about it. I just thought about it tonight, and so I haven't done any work on it yet, except to generate the image I will use to describe the process:

http://nowisforever.me/pics/CircleFractal.jpg

(I'm hosting it on my website to avoid the clutter that would be involved had I uploaded it elsewhere.)

Starting with four circles, each of radius 1, and centered respectively at (0,0), (2,0), (0,2), and (2,2), we have a framework for the addition of other circles of diminishing radii, which are all tangent to prior circles in the construction. Next is the circle centered on (1,1) with radius √2 - 1. Continuing, it is evident that the next set of circles (not shown) are four in number, and are each tangent to two of the original set of four and also tangent to the single circle of the prior generation.

Not only the circles and their construction are of interest, but the region of the domain lying outside all the circles would be, I assume, a fractal, and might have non zero measure and dimension.

Any pointers to "prior art" would be appreciated.

Charles

2. I don't have much to say, I just noticed that starting with four circles gives different convergence and divergence - since the outer bounds would be the added circles, and the inner - the already added circles?

Actually, starting with three circles would be somewhat the same.

3. Originally Posted by ellensius
I don't have much to say, I just noticed that starting with four circles gives different convergence and divergence - since the outer bounds would be the added circles, and the inner - the already added circles?

Actually, starting with three circles would be somewhat the same.
No doubt it would be easier to calculate, and the general results would be the same, methinks.

4. Originally Posted by undefined
That's it, thanks!

5. I hope this is an ok 'circling' around the problem?

This is just a quick sketch and I measured directly in the vector program. (also I'm not sure I know how to calculate this...)

If I divide the two circles diameter, the one that is bound by the four,with the one that is bound by the expansion of other circles, and takes that times 6, it looks like it should be pi?

6x(smaller diameter)/(larger diameter)=pi ?
(I got 3.13 from graphical measuring)