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Consider the irregular quadrilateral tiling of the Euclidean plane depicted by the log-log coordinate grid:

log-log grid

I'm wondering if in the Hyperbolic plane exist some analog of this kind of tiling where the polygons are of the same number of sides, but the sizes/lengths change according to some prescription?

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  • $\begingroup$ Yes there are... Do you maybe want to be more specific? $\endgroup$ – Dan Rust Oct 25 '14 at 3:02
  • $\begingroup$ some examples would be nice $\endgroup$ – lurscher Oct 25 '14 at 16:37
  • $\begingroup$ Have you looked at en.wikipedia.org/wiki/Uniform_tilings_in_hyperbolic_plane There you can find many tilings with uniform tiles (same number of sides, same configurations around vertices etc.). Not sure what exactly you are looking for. As Daniel said, try to specify your question a bit further. $\endgroup$ – MHS Oct 25 '14 at 20:59
  • $\begingroup$ Those are regular tilings, all the tiles have the same exact size, and cannot be rescaled since regular tilings in hyperbolic space don't satisfy affine symmetry. That's what the 'semi-regular' part of my question means $\endgroup$ – lurscher Oct 25 '14 at 21:36
  • $\begingroup$ You can just use a square tilling and then subdivide the squares differently. $\endgroup$ – PyRulez Jan 18 '18 at 23:24
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In the upper half plane model, take the line $\{y=ai-\frac{1}{2}\mid a\in\mathbb{R}^+\}$ and $\{y=ai+\frac{1}{2}\mid a\in\mathbb{R}^+\}$ and some sequence of circular arcs between these lines whose center is at the origin, and whose radii decrease to zero at whatever rate you like. Now take all of these lines and arcs and translate it by all the integers horizontally. This is a tiling of the hyperbolic plane by hyperbolic quadrilaterals.

There are of course many ways to alter this, and it's hard to know what properties you exactly want this tiling to have. Please be more specific if you want it to have a particular property for which you are not sure if it actually exists or not.

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