Four years ago, I wrote a blog post (What to do with 1 million iterations?) about zooms into the Mandelbrot set that require a lot of iterations (1 million, in that case), but that weren’t deep zooms.  I also featured the Color of the Year for 2009, mimosa.  Now, in honor of 2013, I return with a different zoom, featuring 2013’s Color of the Year, emerald.

This image, “Thirteen,” pays homage to its namesake number through a progression of disks with 13-armed spirals.

Each disk in the Mandelbrot set has associated with it, a fraction in lowest terms.  Around the main cardioid, the numbering starts with 0/1 at the cusp, around the top (1/3 represents the largest disk on top), to the left edge (disk 1/2), around the bottom (2/3 for the largest on the bottom), and finishing up with 1/1 back at the cusp.  The largest disk between any two disks has a fraction that is between the two disks' fractions, in a special way.  If the two surrounding disks have fractions a/b and c/d, then the largest between them has the fraction (a+c)/(b+d).  For example, between both extremes of 0/1 and 1/1, lies the largest disk, 1/2, or (0+1)/(1+1).  Between 1/2 and 0/1 lies the 1/3 disk on top, and so on.  Working clockwise down to the cusp, the largest disks are 1/4, 1/5, 1/6, etc.  Way down there is 1/13:

Its “headdress,” or the system of arms emanating from the top of the disk, reflects the fraction 1/13, in that there are 13 arms.  Label the one coming out of the disk as arm 0 and count clockwise.  The first arm, arm 1, is the most prominent of the bunch.  Also, the lengths of the arms vary smoothly from one to the next.  This will be important later.

Back up the cardioid is the 2/13 disk.  It is the largest disk between those for 1/6 and 1/7.

This disk also has 13 arms in its headdress.  Again, counting clockwise from 0 (the arm coming out of the disk), arm 2 is the most prominent.  This time, instead of the arms lengths varying smoothly, they oscillate, with the even-numbered arms being longer than the odd-numbered arms.

Moving back up the cardioid to the 1/3 disk, we encounter the 3/13 disk.

This one is a bit trickier to find:  first, find the 2/9 disk, between 1/4 and 1/5.  Then, between the 2/9 and 1/3 disks is the 3/13 disk.  It also has 13 arms and the third one is the most prominent.  Notice here how the arms get longer in groups of three.

It seems then, that every disk with a fraction of p/q has q arms emanating from its headdress, the pth arm is the most prominent, and arms whose numbers are multiples of p are longer than their neighbors.

Each disk has disks of its own, and they have disks, and so on, down to infinity.  These sub-disks carry with them their heritage of the disks that came before them.

Here is the 2/13 disk of the 1/13 disk (for sub-disks, 0/1 and 1/1 are at the base where it connects to its parent disk).  Immediately above the disk, and darker in this image, are the arms reminiscent of the 2/13 disk, 13 in number with arm 2 being the most prominent.  Following this arm outward leads to the spiral remnants of the 1/13 disk.  Since this disk is a child of the 1/13 disk, as are all in this image, all have the same 1/13 spiral structure at the ends of their arms.  However, their inner structure depends on their own fractions.

This final example shows how the structure in “Thirteen” came to be.  The largest disk at the bottom-right of this image is the 3/13 disk of the 2/13 disk of the 1/13 disk of the main cardioid.  The 3/13 structure is clear in gray immediately above the disk.  Arm 3 extends toward the top of the image, bearing a bright white spiral center of the 2/13 arms.  Arm 2 of this spiral reaches almost to the top of the image.  Very small is the 1/13 spiral, culminating in the tip of arm 1, just barely visible.

As the chain of disks gets longer, the structures become increasingly compressed and tightly wound, in order to incorporate the lengthening train of parental arms.  Ultimately, the image becomes filled with spiral structure.  This reflects the fact that the boundary of the Mandelbrot set has a fractal dimension of 2 and is an area-filling curve.