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Does the [Cantor set](https://en.wikipedia.org/wiki/Cantor_set) qualify?

This animation loops infinitely, and shows that you get back to where you started every time you zoom in by a factor of 3.

<details>
<summary>Show animated GIF</summary>

![An animation of the Cantor set](https://upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Cantor_Set_Expansion.gif/1200px-Cantor_Set_Expansion.gif)

</details>