Two spirals out...

When zooming into certain parts of the Mandelbrot set, what you have passed before appears again, only this time doubled up and twice the speed - eventually after an infinite number of doublings you reach an island minibrot. Going off-center before reaching the island allows you to sculpt shapes by choosing where the new center of rotational symmetry should be.

The first step is to find an embedded Julia set, here I chose one near the tip of the primary hair of the \(7/15\) child bulb of the western-most period \(5\) island:

2.000e-00
00

1.250e-01
01

3.125e-02
02

8.631e-05
03

3.814e-06
04

3.637e-12
05

I chose to zoom in two spirals out from the middle on the less-pointy side:

2.273e-13
06

2.775e-17
07

Repeating this process builds up a tree structure:

1.734e-18
08

1.462e-25
09

Each level adds another layer of branching:

4.569e-27
10

4.701e-38
11

The off-center point becomes the new center or rotational symmetry:

1.469e-39
12

7.209e-57
13

You have to zoom in more and more between each stage:

2.252e-58
14

2.573e-85
15

Each level seems to be around half as deep again as the previous level:

1.137e-86
16

6.528e-128
17

So with exponent -128 at the previous level, it's expected that the next level will be at expoent -192:

1.442e-129
18

7.013e-192
19

And it is.

3.099e-193
20

6.567e-288
21

Here it's not worth continuing, the next level would be at exponent -432, beyond the current capabilities of mightymandel (other deep zoom perturbation renderers are available, some can go deeper).

Coordinates of final image:

-1.9854703902779482188978906810726942996688299114426679780681272338916743175774089072010938434108465027630679707742753887061853099621177705691946722542084356773160616875969004030169345332729761881544124228888219272043973491806196329841558636640317962690190962083661885189194921623120665639462749e+00 +
 1.7358649082514274346964057196195867573429217546070925891109724640053588221943451089591256131817099960158588688183974122189088110924426704652892751037671575552807282391524257096692515166409544112301864388951397373838634078272155178246562977627152613768821184579962121089988592526297277923837069e-05 i
 6.5672588820774035e-288 @

The mostly-boring manual zooming to the final location took me around 18 minutes, I need to automate (at least some of) the process, and the high resolution render (click the last image) took my computer around 12 minutes.

I'm sure I'm not the first to discover this technique, indeed the shapes of the final image are similar to Old Wood Dish by James W Morris. And I still major work cut out comprehending Dinkydau's feats.