Thomas Delaney asked a question about the Mandelbrot set:

Do all primary bulbs have the following property: their periodicity is less than or equal to the periodicity of any bulb smaller in area than them.


1 Answer 1


The size of the primary (child of period $1$ cardioid) period $q$ bulb at internal angle $\frac{p}{q}$ is approximately:

$$ r \approx \frac{\sin \left(\pi \frac{p}{q}\right)}{q^2} $$

So near $\frac{p}{q} = \frac{1}{2}$, $r \approx \frac{1}{q^2}$, and near $\frac{p}{q} = 0$, $r \approx \frac{\pi p}{q^3}$ by the well-known relation $\sin x \approx x$ for small $x$.

The property can be rephrased as "lower period bulbs are larger than all higher period bulbs".

The property is false, because eventually you can find a period $q_0$ bulb near internal angle $0$ that is smaller than the bulb of period $q_0 + 1$ near internal angle $\frac{1}{2}$.

For example, pick $q_0 = 42$, then the bulb at $\frac{1}{42}$ has size approximately $\frac{\pi}{42^3}$ and the bulb at $\frac{21}{43}$ has size approximately $\frac{1}{43^2}$, which is much bigger despite having a higher period.


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