6. Examples Based on Circle Limits III and IV

M. C. Escher's pattern Circle Limit III is his most beautiful hyperbolic pattern, if not the most well known. Figure 9 below shows a rendition of this pattern. This pattern answers all three of Escher's criticisms of Circle Limit I: the fish all swim the same direction along backbone lines, they are all the same color along each backbone line, and they are rounded, realistic-looking fish. This pattern was the first repeating hyperbolic pattern ever created with perfect 4-color symmetry.

Figure 9: A rendition of Escher's Circle Limit III pattern.

It is tempting to surmise that the white backbones form hyperbolic lines, but this is not the case. A close examination of those lines reveals that they make angles of approximately 80 degrees with the bounding circle. The backbones are so-called equidistant curves, which are represented in the Poincaré model of hyperbolic geometry as circular arcs that are not orthogonal to the bounding circle. All points of an equidistant curve are the same hyperbolic distance from the hyperbolic line (orthogonal circular arc) that has the same endpoints on the bounding circle.

Circle Limit III is also based on the tessellation {8,3} (as is Circle Limit II). However it is more complex since there are two kinds of 3-fold rotation points: at meeting points of fish noses, and meeting points of left fins. Thus a fish motif is contained in two adjacent isosceles triangles instead of only one triangle. The computer program was not designed to handle such a situation, but with some editing of the motif data file it was possible to create a Circle Limit III - like pattern based on the tessellation {10,3}. This pattern is shown below in Figure 10. It certainly requires five colors to achieve perfect color symmetry since five fish meet at right fin tips. In fact, it turns out that six colors are required.

Figure 10: A pattern based on Circle Limit III and the tessellation {10,3}.

For completeness, we show an application of the transformation process to the Circle Limit IV pattern (Figure 1). As with Circle Limit II this pattern could be transformed to a pattern of angels and devils with any number meeting at the feet and at the rotation points at the wing tips. Again, such a pattern can be constructed from the tessellation {p,q} as long as p is even (due to the bilateral symmetry of the angels and devils). Then p/2 angels and devils meet foot-to-foot and q meet at the wing tips. As shown in Table 1, the pattern will be hyperbolic if (p-2)(q-2) > 4. Figure 11 below shows a pattern of angels and devils based on the tessellation {4,5}.

Figure 11: A pattern of angels and devils based on Circle Limit IV and the tessellation {4,5}.

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