7. Future Work

Directions of future research include:
      (1) finding different motif transformations,
      (2) allowing the fundamental region to be a non-isosceles triangle or quadrilateral, and
      (3) transforming between any of the three classical geometries.
The motif transformation described above is not conformal. Theoretically, the Riemann Mapping Theorem says there is a holomorphic (and hence conformal) isomorphism between the Poincaré isosceles triangles of any two tessellations {p,q} and {p',q'}. A natural fundamental region for Escher's Circle Limit III is a quadrilateral divided into two triangles whose sides are two hyperbolic line segments and a segment of an equidistant curve - the above methods may extend to such triangles. Finally, transforming from spherical to Euclidean (and hence hyperbolic) patterns would only involve finding a mapping from isosceles spherical triangles to isosceles Euclidean triangles.


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PostScript Files for the Figures

You can access the PostScript versions of figures from the paper by clicking on the following links.
Figure 1 (1.5Mb) A rendition of Escher's Circle Limit IV pattern.
Figure 2 (1.5Mb) Circle Limit IV with the {6,4} tessellation.
Figure 3 (550Kb) Escher's Circle Limit I fish pattern with isosceles triangles.
Figure 4 (490Kb) A transformed Circle Limit I based on {4,6} with one isosceles triangle.
Figure 5 (2.3Mb) Another transformed Circle Limit I based on {6,6}.
Figure 6 (3.1Mb) The pattern of Figure 5 colored with three colors.
Figure 7 (1.5Mb) A rendition of Escher's Circle Limit II pattern.
Figure 8 (1.3Mb) A pattern of 5-armed crosses based on Circle Limit II and {10,3}.
Figure 9 (1.7Mb) A rendition of Escher's Circle Limit III pattern.
Figure 10 (5.4Mb) A pattern of fish based on Circle Limit III and {10,3}.
Figure 11 (2.8Mb) A pattern of angels and devils based on Circle Limit IV and {4,5}.