In Conway, Burgel and Goodman-Strauss' book The symmetries of things, Chapter 17, the following picture by Escher was analysed using orbifold notation. It's a hyperbolic pattern in the Poincare disk.
If we ignore that some of the angels and devils are facing away from the viewer, the group $G$ of symmetries has signature $4*3$. Let $C$ be one of the regular quadrilaterals defined by $4$ adjacent reflection lines. Then the picture can be recovered from a single tile one quarter the size of $C$ by applying all the symmetries in $G$.
According to the authors, if we take into account that some of the shapes are facing away from the viewer, then the group $H$ of symmetries has signature $*3333$, and has index $4$ as a subgroup of $G$.
Here is where I'm getting a bit confused, because the book says that in situations like this, $H$'s orbifold should be $4$ times as big as $G$'s, which makes sense, but then the "fundamental" tile that generates the image should also be $4$ times as big. That is, it should have exactly the same area as $C$.
But if you take for instance $C$ to be one of the central quadrilaterals, with one angel and one devil facing away (as in the book), then no symmetry in $H$ seems to send points in $C$ to other points in $C$. But also, there are different quadrilaterals, where for example $4$ devils face away. So $C$ is not big enough to generate the tiling, and a tile of bigger area is required.
What's going on here? Am I missing some symmetry of $C$?
Here's a picture to clarify what I'm talking about. It's one choice of quadrilateral $C$, consisting of $4$ pieces. Each of them generates the pattern by the action of the group $G$, which includes the order $4$ rotation around the center of the quadrilateral and reflections along the sides of the quadrilateral.
I now think the problem for the subgroup $H$ —that takes into account which devils and angels are facing away— comes from the following fact: not only the rotation disappears, but in fact some reflections disappear too. Otherwise all the quadrilaterals could be obtained from $C$ via reflections, but that's impossible because there are different quadrilaterals.
I believe the authors did not take into account this fact, because they write:
If you look closely at one of those quadrilaterals, you will find that one of its four devils seems to have no eyes (this is because you can’t see through the back of his head!), as does one of its four angels.
Which is not true for all the quadrilaterals. So if I'm right the question is: how to classify the group of symmetries of the picture? Is the fundamental tile finite? I'm not sure what system Escher used to decide which pictures should be facing away.
M.C. Escher says the following in M. C. Escher: The Graphic Work Introduced and explained by the artist:
Rephrased, what he is saying is that he broke this into regions where he imagined "Heaven and Hell" fighting; in regions where you see faces of demons, but not angels, "Hell" is "winning" and vice versa. In regions where you see both it is "earthly".
In the below picture the center angel and demons "determine" what is happening on the outside: If a region contains (significant) portion of angel or demon in the center then the "outside/boundary" will have the "details" of angel/demon filled in, otherwise it won't.
Basically, with how I partitioned the regions, if the center has the face of one, then the outside won't have the "details" of the other. If the region shares wings then both "details" will be filled in (earthly area).
So you are essentially right, they did not seem to keep into account some of the larger structure of the tiling, which becomes pretty obvious when you look closer at the boundary of the hyperbolic plane. In the "North" pretty much all faces of demons are showing but slightly to the left and right of that region almost none of the demons are showing and Escher is clear this is no accident. There is a similar pattern for angels but shifted a bit.
If we take this interpretation, and my terrible break down, then we get the dihedral group $D_3$ because of the 3-fold rotational symmetry plus you can reflect across a geodesic going through the center that cuts in a nice way (say "North-South" geodesic).