# mathr

## Julia morphing symmetry

Wolf Jung pointed me towards an interesting pattern in the angled internal addresses corresponding to the external angles in a previous post. The pattern is:

$$1_{\frac{1}{2}} \to 2_{\frac{1}{2}} \to 3_{\frac{1}{2}} \to 4_{\frac{1}{2}} \to 5_{\frac{5}{11}} \to 54_{\frac{1}{2}} \to 64_{\frac{1}{2}} \to 69$$
$$1_{\frac{1}{2}} \to 2_{\frac{1}{2}} \to 3_{\frac{1}{2}} \to 4_{\frac{1}{2}} \to 5_{\frac{5}{11}} \to 54_{\frac{1}{2}} \to 64_{\frac{1}{2}} \to 69_{\frac{2}{3}} \to 143$$
$$1_{\frac{1}{2}} \to 2_{\frac{1}{2}} \to 3_{\frac{1}{2}} \to 4_{\frac{1}{2}} \to 5_{\frac{5}{11}} \to 54_{\frac{1}{2}} \to 64_{\frac{1}{2}} \to 69_{\frac{2}{3}} \to 143_{\frac{2}{3}} \to 291$$
$$1_{\frac{1}{2}} \to 2_{\frac{1}{2}} \to 3_{\frac{1}{2}} \to 4_{\frac{1}{2}} \to 5_{\frac{5}{11}} \to 54_{\frac{1}{2}} \to 64_{\frac{1}{2}} \to 69_{\frac{2}{3}} \to 143_{\frac{2}{3}} \to 291_{\frac{2}{3}} \to 587$$

where the rotation number (internal angle) $$\frac{2}{3}$$ is the same each time. The alternative automatic zooming method at the end of my earlier post doesn't respect this, it would unpredictably choose $$\frac{1}{3}$$ or $$\frac{2}{3}$$ with no efficient way to check whether you ended up with the desired alternative. I thought I'd check to see if this ambiguity matters in practice, so I rendered 16 images with angled internal addresses corresponding to all possible choices:

$\left\{ 1_{\frac{1}{2}} \to 2_{\frac{1}{2}} \to 3_{\frac{1}{2}} \to 4_{\frac{1}{2}} \to 5_{\frac{5}{11}} \to 54_{\frac{1}{2}} \to 64_{\frac{1}{2}} \to 69_{\frac{a}{3}} \to 143_{\frac{b}{3}} \to 291_{\frac{c}{3}} \to 587_{\frac{d}{3}} \to 1179 \\ : a,b,c,d \in \left\{1,2\right\} \right\}$

The good news is that the central morphed tree-like pattern is almost indistinguishable across all the variants, but the way it aligns with the surrounding ring of features shows some clear differences - comparing the first and last images, one has the longest thin filament to the left of a larger blob, the other to the right. The rotations are different of course, but I expected that - I didn't expect the alignment of the surrounding ring of features to be so consistent across the images.