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

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.

• @Rhaul well, the first one is graphic theoretically the same as this, and the second one seems to be non-periodic (mostly due to "mistakes" due to M.C. Escher). The question is though if the reptiles are all roughly the same shape and size in hyperbolic space. Another interesting question is if the color is periodic in the first one. Commented Jan 18, 2018 at 6:06
• @Rahul oh, it's not uniform Commented Jan 18, 2018 at 20:22
• I do not think that working in the half-plane model is that good for a modern artist, because it is nice to see the end result. Rather than that, I think that they should use a tool where they can draw in any model, immediately see their brushstrokes periodicized according to the tesselation, and change between the various models. (The texture editor in my HyperRogue works like this -- while not a fully featured graphical editor, it can be used to create a sketch at least.) Commented Feb 9, 2018 at 22:49
• Also, it also has features to load a PNG of a tesselation and transform it to another model, though it might be hard to get it work on these pictures (and I think I cannot legally post the results if I try). Commented Feb 9, 2018 at 22:53
• @ZenoRogue I don't think Escher intentionally worked in the half plane model, but I am wondering if his work could be interpreted that way. Commented Feb 9, 2018 at 22:54

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.

• Although it doesn't appear intentional, I think the first image might accidentally correspond to an half-plane model. That's because the size decreases exponentially in both the image and the half-plane model. Commented Dec 17, 2017 at 5:45
• Also, the half-plane model is invertible. Given a half-plane, you can convert it to an image in the hyperbolic plane, and examine it there. Commented Dec 17, 2017 at 7:48

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.

• Escher choose the disc model because he wanted to include infinitely many shapes in finite space. Also, I don't understand what you mean by "in this case the Euclidean plane is generated within the Klein model of the hyperbolic plane". Commented Dec 17, 2017 at 9:18
• @PyrRulez: (1) other models of the hyperbolic geometri like the Klein model would be suitable if one wanted to create infinitely many figures within a finite part of the embedding Euclidean geometry. (2) The Euclidean plane can be modeled within the hyperbolic plane. The hyperbolic plane can be modeled (say by the Klein model) in the Euclidean plane. So, the Euclidean plane can be modeled within itself. My drawing second drawibg is a transformation of the first drawing via modelling the Euclidean plane in the hyerbolic one and then the latter in the (by Klein) in the Eu plane again.
– zoli
Commented Dec 17, 2017 at 9:35
• (1) I'm not sure, but I guess the poincare model is more popular since it preserves angles (2) that would require an embedding of the euclidean plane into the hyperbolic plane. I'm not aware of any such embedding. (I am aware of disk models of the euclidean plane, but not any that for thorough the hyperbolic plane first). Commented Dec 17, 2017 at 9:38
• @PyRulez: (1) You are not sure in what? (2) What is the advantage of seeing proper angle? (Especially from the point of view of art?)
– zoli
Commented Dec 17, 2017 at 9:53
• (1) I'm not sure if that's the reason (2) Aesthetic purposes? Another advantage is you can generally see more shapes in the poincare model. Commented Dec 17, 2017 at 9:55