Is this Escher artwork a tessellation of the half-plane model of hyperbolic space?

Question

One Escher's prints look like this. A similar one is this.

These look suspiciously like Poincaré half-plane models of the hyperbolic plane (there are pieces of artwork by Escher specifically based on the hyperbolic plane).

Note that these pieces do not contain the whole half-plane, of course. But it is not too hard to imagine extending the artwork left, up, and right into the whole half-plane. The second piece would also need to be extended downwards, but since the lizards are shrinking exponentially, they would converge to a line, it appears.

My question is, as interpreted as half-plane models, would these correspond to tessellations of the hyperbolic plane? (It would also be interesting to see them changed into other models of the hyperbolic, such as the Poincaré disk model.)

(It's interesting, because if so, this presents a much easier way to replicate Escher's work. Drawing in a half-plane model appears to be much simpler than in a circle model. So the artists could draw in the half-plane model, and then use a computer to convert to the circle model.)

EDIT: For example, figure 1 of this paper shows a tessellation by square like shapes of the hyperbolic plane, displayed in the half-plane model. It looks very much like the second print.

Answer 1

As you wrote, there are Escher works that are based on hyperbolic plane, but these came after previous attempts which had no connection to hyperbolic plane and showed he was searching for some kind of aesthetic effect he wanted to produce. For the first print see this link about resizing tessellations. The second print looks more like a hyperbolic tessellation but I would have to study it to make sure. One problem is that hyperbolic plane models are usually conformal, but not all tessellations are designed that way. Another problem is area distortion which is different between models.

Answer 2

This is rather a comment

Yes, the shown pieces of art are related to Poincaré's half plane model of the hyperbolic geometry. There are similar works of Escher shown in the Poincare disc model. Here is one:

I don't know either if this is a tessalation or not. It looks so.

My comment on the question is this: I don't really understand why the Poincaré models are so popular. Similar artworks can be created based on other models of another geometry. Look at the following insignificant piece of art, a tessallation of the Euclidean plane (with squares) with some decorration (circles).

The drawing given above turns more interesting if one depicts it in a model of the Euclidean geometry: in this case the Euclidean plane is generated within the Klein model of the hyperbolic plane.