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Today's featured picture on Wikipedia shows a deep zoom into a Mandelbrot set: http://upload.wikimedia.org/wikipedia/commons/a/a4/Mandelbrot_sequence_new.gif

If one could pick any coordinate to zoom into, is there one which eventually takes one back to the original image, preferably without rotation, so that the animation can loop satisfyingly?

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Windows of Periodicity Scaling gives a formula $\beta \Lambda_p^2$ for the size (and orientation, taking it as a complex number) of a minibrot island with nucleus $c$ of period $p$ relative to the top level continent (the $\frac{9}{4}$ on that page is the length of its intersection with the real axis, and not relevant for scaling). The relevant formulae from that page:

$$z_1 = 0 ; z_{n+1} = z_n^2 + c \\ \lambda_n = 2 z_n \\ \Lambda_n =\lambda_2 \lambda_3 \cdots \lambda_n \\ \beta = 1 + \Lambda_2^{-1} + \Lambda_3^{-1} + \dots + \Lambda_p^{-1}$$

This can be used to render a looping video, however the smaller copies are distorted to varying degrees, and the surrounding filaments/decorations will differ (as hinted at in lhf's answer) which will spoil the smoothness to some extent. For example:

looping zoom animation

No small copy is geometrically similar to the whole set as there is only one perfectly circular component. To make the looping GIF I zoomed about the point $c / (1 - r)$ where $c$ is the nucleus (corresponding to $0$) and $r$ is the complex-valued size estimate $r = 1 / \left(\beta \Lambda_p^2\right)$.

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The Mandelbrot set is only approximately self-similar: it does not contain an exact copy of itself.

See this question for a discussion.

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  • $\begingroup$ depends what you mean by exact copy. there are topological copies of the Mandelbrot set that are dense in the Mandelbrot set. some of those copies are quasiconformal, other are quasiconformal except at a cusp. a priori none is isometric or even biholomorphic to the original mandelbrot set. $\endgroup$
    – Albert
    Jun 16 '17 at 14:41
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    $\begingroup$ @Glougloubarbaki If the OP wants to zoom in "so that the animation can loop", then it's clear that "exact copy" means geometrically similar. $\endgroup$ Jun 17 '17 at 10:54

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