Anyone know a formula for this?

[symbioidlj]

I have this pattern in my head that i've drawn out a few times, and it should be easy to make into a function. However, I don't math that well, nor do I know how I would render it, but I think I should be able to plug it into a computer.

Here's the idea.

One circle(ideally, it would be a sphere, but to simplify in this case, I'll use a circle) Sprout lines out of the four directions(N,E,S,W), perhaps about the same diameter as the original circle, then, draw a smaller circle on the end of each of those lines(so you have what looks like a diamond shape, with a large circle in the middle) These circles may be 1/2 to 3/4 the size of the original circle. Now, since one side of those circles would be taken by the attaching lines to the central circle, there are three sides left on each of those circles. Example: The top circle, has the left, top and right side available. Sprout branches in the same ratio to these as you did with the original circle. So now you have 12 nodes. Then sprout more... ad infinitum. Somehow, you would have to get the right ratio's between distance of line and size of circle to make it attractive. I'd suspect the golden mean would be involved somehow.

Now you can easily extend this into the z-axis as well, so you have a cube of this fractal spheres.

Can you imagine this? If so, can you get a function for it? Do you know someone who may be able to? Any help from anyone would be greatly appreciated....

Comments

[ziggurat.livejournal.com]

1. well, i don't know if you care about technicalities, this thing you describe isn't a real fractal (http://en.wikipedia.org/wiki/Fractal), a real fractal should be self-affine (http://en.wikipedia.org/wiki/Affine_transformation) at every point, the thing you've described won't be, because of the circle (which is not affine with the line segments)


2. it does show some characteristics of a fractal. for example, like any fractal, you won't find an elementary function for describing it. usually fractals are described by iterative algorithms, just like the one you've described above. in other words, the description you give above is more or less the best description possible for these sorts of things


3. so fractals can't be graphed as functions on your TI-83, but they're still easy enough to make. the iterative algorithms used to describe them lend themselves to direct translation into some computer language. i would recommend doing it in logo (http://en.wikipedia.org/wiki/Logo_programming_language), it's very well suited to making drawings algorithmically. i remember making the Koch snowflake (http://en.wikipedia.org/wiki/Koch_snowflake) fractal in logo quite easily when i was 16. for mac, windows, or linux (http://www.cs.berkeley.edu/~bh/logo.html). or else any language that you're comfortable doing graphics in would suffice


4. the thing you describe reminds me of the universal cover (http://en.wikipedia.org/wiki/Universal_cover) of the figure eight (http://en.wikipedia.org/wiki/Figure-eight_knot_%28mathematics%29). You can see what i'm talking about on page 59 of everyone's favorite graduate algebraic topology text (http://www.math.cornell.edu/~hatcher/AT/ATpage.html) by allen hatcher (http://www.math.cornell.edu/~hatcher/) (or just look below)

[symbioid.livejournal.com]

Dude! That is exactly it! Imagine that in 3 dimensions with spheres as nodes. I'd love to see it rendered as a cube.

I'll look into it more later. Thanks much...