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First of all, apologies if the question in the title is too vague. I would have liked to have stated it in a more precise manner, but I don't unfortunately know enough about the subject to do so.

As I was reading the Wikipedia article on the Cantor set, this picture caught my attention:

The name of the file was "Cantor binary tree", and it seemingly represents the process of removing middle thirds from the interval [0,1]. According to the article, this is an example of a finite division rule. What I found interesting is that it bears a certain resemblance with some well-known images of regular hyperbolic tilings I had seen, this one for instance (source):

enter image description here

Looking at the article on finite division rules for more information, I read that "subdivision rules have been used [...] in the study of hyperbolic manifolds", but it doesn't give any reference for this claim, and the Google searchs weren't neither much helpful.

I'm aware that the self-similarity in the hyperbolic tiling is only apparent, as one should think of each of the tiles as having the same size, but I was curious to know if one could "construct" an hyperbolic plane following an interative process, in the likes of the one that underlies the Cantor set construction... for instance, "gluing" tiles together following some simple rule. I also wondered if there could be a way of imbuiding a flat surface with intrinsic negative curvature by iteratively "cutting" and "reconnecting" the right fractions of it, in a similar fashion as one iteratively removes fractions of the [0,1] segment in order to obtain the Cantor set. However, those were just vague attempts to phantom some kind of relationship between the two concepts.

To make the question a bit more concrete: appart from self-similarity, are there some other interesting mathematical properties shared between those two objects? Any thoughts or references coming from a more knowledgeale person would be greatly appreciated.

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I don't see a direct relationship. However, there is the following, more advanced relationship- we can "hyperbolize" the ternary Cantor set into a metric space with non-positive curvature in the sense of Gromov. In other words, one can prove that the ternary Cantor set is isometric to the hyperbolic boundary of some Gromov hyperbolic space.

Reference: Uniform Cantor Sets as Hyperbolic Boundaries.

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  • $\begingroup$ Thank you for your answer. Do you know of any similar results for more general hyperbolic spaces? $\endgroup$ – David Herrero Martí Nov 16 '16 at 22:02

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