This is the simplest construction, but thereby an triangle has got an edge glued to a neighbouring edge (so now the dual_rot around the vertex there fixes an edge label), and since poly=3 as a side-effect this results in a loop for the remaining edge, which is perfectly valid but conflicts a little bit with the way we draw edges as straight lines (we could program something for this circumstances, draw some circle or so). The cartographic groups and Grothendieck's dessins d'enfants (also see the section called “M(24)”) perfectly fit into our framework in principle.

See [Bre00] and [Atlas] for more information about this regular representation using the period triple (2,3,7) and orbit genus 3. The dual graph has got the same group of course. And again a basic map together with some additional identification of boundaries could be more meaningful (Klein's configuration in [TS]). Actually we could avoid self-intersections by allowing triangles with non-straight lines involving Klein's quartic (see [Egan05] and [Christy]).

Figure 7.17. L2(7) Regular

the simple group L2(7) of order 168 (hyperbolic geometry) with some self-intersections