Does any hyperbolic tiling pattern exists which can be scale smoothly

Question

I know that some (maybe all) tilings patterns on Euclidean plane can be scale up or down at any ratio. For example, the edge length of a square grid can be scale from 2 to 0.5 smoothly, and meanwhile the square tiling pattern still can be hold.

Based on this geometric intuition and the fact that hyperbolic plane is also a homogeneous space. I had thought that hyperbolic tiling patterns can be scale smoothly. But now I am developing a drawing tool to draw hyperbolic tiling patterns on Poincare disk, and I meet some difficulties on this assumption, I was failed to draw a pattern with the length of edge 1.

Thanks for comments from @IvanNeretin , we know that the above geometric intuition on Euclidean plane may not generally hold on hyperbolic plane: for a given negative constant curvature $c$ and a specific tiling pattern, at least we can solve the proper edge length $l$ numerically. ref Edge length of hyperbolic tesselations

So my question is

• Does any hyperbolic tiling pattern exists which can be scale smoothly? or a kind of global rigidity hold for all hyperbolic tiling patterns?

Images are from related Wikipedia articles.

Answer 1

In the question, the scale operation in hyperbolic space is not defined very clearly. So we may clarify it first. As any Euclidean scale operation, a scale operation in hyperbolic are also:

• An one-parameter transformation group with a radius parameter $r$ ($r$ may not related with length)
• these transformations have a special originate point as a shared fix point when radius parameter changes

Under this definition, I constructed several patterns which can be scaled up or down.

The first one is with a construction factor 2

The second one is with a construction factor 3

We may construct a pattern like this with any integer factors.