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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.
References
- Coxeter, H. S. M. (1981)
- Angels and Devils, In: Klarner, David A. ed.,
The Mathematical Gardner ,
Boston: Prindle, Weber and Schmidt.
- Dunham, D. (1986a)
- Hyperbolic symmetry,
Computers & Mathematics with Applications ,
(1/2), 12B, 139-153.
- Dunham, D. (1986b)
- Creating hyperbolic Escher patterns, In:
M. C. Escher: Art and Science ,
H. S. M. Coxeter et al., eds., Amsterdam: North-Holland, 241-248.
- Locher, J. L. (1982) editor
- M. C. Escher, His Life and Complete Graphic Work ,
New York: Harry N. Abrams, Inc.
- Schattschneider, D. (1990)
- Visions of Symmetry: Notebooks, Periodic
Drawings, and Related Work of M. C. Escher
,
New York: W. H. Freeman.
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}.
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