[size=150]The work [i]Circle Limit III[/i] is directly referenced by Coxeter in one of his papers (see [2]). Undoubtedly the most complex and elaborate.[br][br]With a GeoGebra analysis, we can verify that its adjustment to the conditions of the Poincaré Disk model are deficient for the white color lines drawn by Escher. The resulting angles of these lines at the disk boundary are quite less than 90º, and that the sum of the angles of a triangle does not correspond to that of hyperbolic geometry. [/size]

[size=150]However, this difference isn't an error at all. It is sought to allow coloring the fish of each line with a single color and maintain a single flow from head to tail in each line.[br][br]Actually the work resulted in one (8.3) tessellation of the hyperbolic disk, where 3 polygons of 8 sides meet at each vertex. The central octagon is characterized by keep angles of 120º between its hyperbolic sides, which is achieved by placing its vertices exactly where the 3 heads of the fish and the 3 fins of these meet, alternately. Any displacement of the marked point located on the head of the 3 fish causes a change in the angles of the octagon, which completely undoes the tiling, with exponential effects on the adjustment error of the set, depending on this displacement.[br][br]Between the green arc and the white arc of Escher's drawing (these aren't hyperbolic lines) the hyperbolic line is placed in red. Between the first two, the sides of the polygons of the tessellation run in zig-zag, forming what is called a "Petrie polygon".[/size]