Navigating to a Leavitt Embedded Julia Set  

This is an example of how I typically locate a Jonathan Leavitt style image, provided I have a clear idea of its design (perhaps from an understanding of the self-squared and exponential mappings).

The technique uses some of the special features of Power MANDELZOOM. Here I will be finding a simple 3nd-order EJS based on the pattern I call "X chromosomes".

We'll start with this view, which is centered at -1.76863 +0.00266 i, size = 0.00125:

To get the chromosomes we need a free chain of the tip of a pointy 1st-order IJS. This involves zooming in for a while near the tip of this embedded Julia set:

(Coordinates: -1.76863411 + 0.00266330 i, size = 0.0000195)

I zoom in close to its "tip" until I get to here:

Then I just arbitrarily pick a tiny Julia set, which is going to be the first one I actually zoom in to the center of. The one I pick is near the "O" in the above image, just under the "X". Here is that EJS:

(Coordinates: -1.76863411915716 + 0.00266330452172 i, size = 1.9×10-11)

This is a 2nd-order embedded Julia set, because it is influenced by two islands: the large period-3 island, and the period-65 island at the center of the Julia set whose tip we were approaching during the previous few images (that island was unseen because we never had it in the center of the view).

At this point I use the special period-finding command, which uses the algorithm described in the period article, starting with the section titled "Finding the Period of a mu-Atom". It uses the Jordan Curve Theorem to track the vertices of a polygon that starts as the rectangular boundary of the current view, iterating all vertices one step at a time until the resulting polygon contains the origin. The number of iterations performed is thus the period of a mu-atom somewhere inside the original polygon. It uses an inverse bilinear interpolation technique to figure out approximately where in the window the center of the mu-atom is likely to be.

As you can see, it found a period 100, which is labeled with a large circled "1" in the center of the image because it is the lowest-period found. Next to the circled "1" is a tiny "100" indicating the period. My program also scans across the image trying little squares to locate more mu-atoms, and reports the lowest 9 periods it finds. It found some with periods 103, 104 and 105. As seen here, the positions are sometimes wildly inaccurate.

After executing that command I can then zoom in with a "Zoom to 1st Found Period" command (and there are similar commands for 2 through 9).

At periodic intervals I need to repeat the "find period" command because its calculation of the location of the period-100 is not exact. (As it turns out, the closer you zoom in, the more exact it is, because the precision of the linear interpolation depends on how far away the next lower period is located in relation to the size of the polygon, which is the same as the size of the current view. In this case we're zooming in towards a period-100, and the next-lower period is the owner of the nearby Julia set that you'll see if you look at -1.7686341190573 + 0.0026633044939 i, size 2×10-11. That is one of the Julia sets that I passed by on my way to this one).

I continue zooming on the center:

Until I get to this point:

(Coordinates: -1.76863411915716693095 + 0.00266330452171772597 i, size = 7.1×10-17)

I want this large X shape to get distorted into an "X-chromosome", so I need to "go off-axis", which is how I describe the act of picking a new center to focus on. A doubled, distorted version of everything seen here will become part of the exponential map of the new center-point.

To get the desired "X" shape I have to pick a new zoom point near the "O" in the image, which you will find near the right edge (with a small "X" above it). There is a Julia set under that "O" but it's too small to see.

I zoom at that point manually until the Julia set fills the view, and then do "find period" again:

(Coordinates: -1.768634119157166907234 + 0.002663304521717732925 i, size = 1×10-18)

Now I can zoom automatically again. As before, I continue to zoom in using the Find Period command every now and then to make sure it's centered.

At this point we see the two "squared" copies of the X shape. I want four in a symmetrical circular arrangement, so I just continue:

The cell now has four "X" chromosomes. "Y" chromosomes are left as an exercise for the reader.

The coordinates of the final image are:

-1.768 634 119 157 166 907 234 009 209 392 062 +0.002 663 304 521 717 732 927 981 067 912 799 0.000 000 000 000 000 000 000 000 000 002 019 Nmax: 20000

and it required 161-bit (triple-double) precision to draw the final image.

Here is an exponential map of the same zoom sequence:

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This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2017 Feb 02. s.11