In M.C. Escher's Snakes, you have three snakes going through some loops. I'm more interested in the loops though.
In this image, a ring model of the hyperbolic plane is given. It is given by $w=e^{za}$, where $w$ is a point in the ring, and $z$ is a point in the band model of the hyperbolic plane, for appropriate $a$. It only works if the image in the hyperbolic plane has the appropriate symmetry, since some different points in the band model get mapped to the same point in the ring model.
So my question is, if we interpret the loops in M.C. Escher's Snakes as representing the ring model, do the loops form a (piece of a) regular tilling of the hyperbolic plane? (Note, I'm interpreting the places where three rings intersect as vertices which then connects to three closest vertices.)
(Its interesting to note that M.C. Escher was familiar with hyperbolic geometry.)
This does not correspond to a regular hyperbolic tiling because there are two types of polygonal tiles, octagons and hexagons. I thought this might correspond to a uniform hyperbolic tiling, but that is not the case either. If it were, the tiling would be vertex-transitive (i.e. only one vertex type), but there are multiple types of vertices here. For example, some vertices are surrounded by 3 hexagons and others by two octagons and a hexagon. (There may be further vertex types as well.)
This looks like a tiling with varying geometry. Towards the center, there are only hexagons, 3 meeting at each vertex. This part is a (conformal) inversion of a portion of a Euclidean tiling, with the tiles approaching a parabolic point at the center. Away from the origin, octagons are introduced to give the tiling some negative curvature. That allows the subsequent shrinkage of tiles near the disk boundary. If the inversion of the hexagonal tiling continued, the hexagons would have grown until they covered the entire plane.
I'm thinking of the topology of the tiling as a once punctured disk, vs. a ring.