Sierpinksi again

I like the Sierpinski-inspired fractals.  They’re (mostly, usually) easy to hand-draw in paint programs like Gimp that have snap-to-grid functions. The classic is the Sierpinski Triangle, in which you take an equilateral triangle and remove the middle quarter, and just keep repeating the process on the remaining portions down infinitely. This gives you something with infinite surface area and zero mass. For some reason, though, Gimp doesn’t have infinite precision on pixel width, so I usually have to stop after a few iterations. On the other hand, because each triangle is a copy of the triangle one-half the length of its side, you can invert the process by printing three triangles and taping them together, then repeating again, to make a huge Sierpinski paper blanket that’s infinitely large (if you have enough paper and your company pays for the ink).

(Sierpinski Triangle)

(Sierpinski Carpet.)

You can do the same thing with a square, which gives you the Sierpinski Carpet, and circles (which I haven’t tried doing by hand). If you have a CAD package and a 3D printer, you can build real three-dimensional versions, too.

(Koch Snowflake, from the wiki page.)

Now, there’s the opposite approach, which is to draw the triangle, and add triangles at the sides, pointed outward, that are 1/3 the length of each side. If you only plot the exterior contour, you get the Koch Snowflake (I haven’t tried drawing this by hand because Gimp’s not really suited for it).

I wanted to know what would happen if I combined the Sierpinski Carpet with the Koch Snowflake, applying the algorithm to all open surfaces. The result isn’t really what I was hoping for, but if I use a coloring scheme that fades to white with each new iteration, it’s kind of interesting.

(Extrusions with seeds of side=8 and 9.)

The next step is to add variations. Instead of changing the length of the side of each extruded layer by 1/3 of the original length, what happens if the sides are prime numbers (not including 2)? A square of length 3 is the smallest possible “dust”, and the recursion process stops there. So, go to a “seed” square of length 5, and the make the first extrusion 3 (you can see the result in the upper-left corner of the below picture). Next, use a seed cube of length 7. The first extrusion will be of length 5, and the second (extruded from all “naked” sides of the 5 cubes) is length 3. Stop. Go to a seed of length 7, and repeat. Very quickly, the extruded sides overlap, and then start curling in towards the center of the seed, making it really hard to keep track of them all by hand by the time you get up to 13. At 19, I decided to not calculate the interior, and just fudged it with shrinking squares with sides of prime number lengths.

(Lower left, going clock-wise: Seeds of 3, 5, 7 and 11. The central object is kind of a 13-19 hybrid.)

It’s something to do when you’re bored.

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