The Mandelbrot set contains (countably) infinitely many baby Mandelbrot set copies. Each hyperbolic component (cardioid-like or disk-like shape) has a positive integer period $p$: the center of the component $c$ has the property that iteration of $z \to z^2 + c$ starting from $0$ returns to $0$ after $p$ steps. Each component is surrounded by a larger atom domain $A_c \subset \mathbb{C}$, where iteration of $z \to z^2 + a_c$ for each $a_c \in A_c$ reaches a new minimum $|z|$ at the $p$th iteration. The size of the atom domain can be estimated as $$R_c = \left|\frac{ F^q(0, c) }{ \frac{\partial F^p}{\partial c}(0, c) }\right|$$ where $F(z, c) = z^2 + c, F^0(z, c) = z, F^{n+1}(z, c) = F^n(F^1(z, c), c)$ is the iterated quadratic polynomial, and $q$ is the iteration number of the last minimum $|z|$ strictly before the $p$th iteration.
Each hyperbolic component can be identified by an angled internal address, which describes a path through a sequence of hyperbolic components increasing in period. An artistic technique colloquially known as Julia morphing (repeating a series of zooms towards mini Mandelbrot set islands, going off-center in a similar way at each level) can result in angled internal addresses with a repeating structure, often ending with a sequence like $$\cdots p \overset{1/3}\longrightarrow 2p+k \overset{1/3}\longrightarrow 2(2p+k)+k \cdots$$
I noticed empirically that the zoom level when re-centering from period $p$ to period $2p+k$ (which is approximately the last time two-fold rotational symmetry occurs when zooming towards $c_p$; all deeper zooms have four-fold symmetry or higher, at least until a close vicinity of the mini-Mandelbrot set where symmetry is broken) is related to the atom domain size: $$|c_p - c_{2p+k}| \approx R_{c_p}^\frac{9}{8}$$
Why does $\frac{9}{8}$ occur here?
Example:
$$1 \overset{1/2}\longrightarrow 2 \overset{1/2}\longrightarrow 3 \overset{1/12}\longrightarrow 36 \overset{1/2}\longrightarrow 40 \overset{1/2}\longrightarrow 76 \overset{1/3}\longrightarrow 188 \overset{1/3}\longrightarrow 412 \overset{1/3}\longrightarrow 860 \overset{1/3}\longrightarrow 1756$$
| $p$ | $R_{c_p}$ | $R_{c_p}^\frac{9}{8}$ | $|c_p - c_{2p+k}|$ |
|---|---|---|---|
| 188 | 6.02e-12 | 2.38e-13 | 3.35e-13 |
| 412 | 5.99e-17 | 5.61e-19 | 7.99e-19 |
| 860 | 1.90e-24 | 2.02e-27 | 2.94e-27 |
| 1756 | 1.08e-35 | 4.59e-40 | - |
Coordinates of period 1756 minibrot island:
Re: -1.7486089471559587445618364830459565964411309584255777888057575
Im: 5.2725711650258050977059680687255394855123946220224103429391918e-4
Zoom: 6.75417723706178e50
UPDATE I noticed other magic numbers: $$|c_p - c_{2p+k}| \approx R_{c_{2p+k}}^\frac{3}{4}$$ $$R_{c_{2p+k}} \approx R_{c_p}^\frac{3}{2}$$ and $\frac{9}{8} = \frac{3}{4}\cdot\frac{3}{2}$. So it's partially explained in terms of other relations, but all of these three relations are still mysterious.