Mandelbrot set and prime numbers

Solution 1:

First off, I don't think it's the case that spirals with a prime number of arms appear any more or less frequently than spirals with a composite number of arms. The fact is that, for every $n$, there are infinitely many spirals with $n$ arms. Here's a spiral with 12 arms:

What truly is amazing and connects the number of arms in a spiral to certain types of numbers is that the distribution of the bulbs on any component of the Mandelbrot set can be determined using Farey fractions. This is beautifully described in this paper by Bob Devaney.

Consider, for example, the following image of the Mandelbrot set:

The fractions $1/2$, $1/3$, and $2/5$ label disk-like bulbs hanging off the main cardioid of the Mandelbrot set. Those fractions give us a lot of information about the structure of the set near there. In particular, the denominator of each fraction tells us how many arms spiral in off of decorations nearby. Given two such disk-like bulbs labeled by fractions $a/b$ and $c/d$, the largest bulb between them should have label $(a+c)/(b+d)$. That is exactly how the $2/5$ bulb arises between the $1/2$ bulb and the $1/3$ bulb. Similarly, there is a $3/8$ bulb between the $2/5$ bulb and the $1/3$ bulb; if you zoom in near there, you're sure to find spirals with 8 arms.

If all this is right, we might be able to use it to help us find where your image lies in the Mandelbrot set. In fact, I was able to come up with the following image:

This image does not lie immediately off of the main cardioid. To find it, I had to use the portion that's boxed in the figure. Near a $c$ parameter in the Mandelbrot set, it tends to "look like" the Julia set with that $c$ parameter. I happen to know that we can generates pictures like the one in the box by choosing a $c$ value near the yellow dot in the picture of the Mandelbrot set above. Zooming in near there and finding a period $11$ bulb, I was able to find your picture.

Solution 2:

Apparently fractals are related to partition numbers, and Ramanujan found a way to relate some prime numbers to partition numbers (5, 7, 11), according to the following articles:

http://www.wired.com/wiredscience/2011/01/partition-numbers-fractals/

http://www.aimath.org/news/partition/

I see your primes are higher, but who knows, maybe you will find these ties in higher partition numbers or different fractals.