Curve-stitch Designs

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More Generalizations

As was clear from the last section (Some Circular Designs), one idea for a curve-stitch design often leads to other, unexpected patterns. In this section, I will describe a line of thought that led me to devise more designs and to build more tools for producing them.

Many of my designs (particularly those in A Family of New Curve-stitch Designs) have begun with a square. What would happen, I asked myself, if squares were nested, as shown here:

Ten nested squares

The triangles at the corners of each square could be inscribed with parabolas, and this design could be repeated indefinitely, in the progressively smaller squares. (Such patterns are more suited to computer-generation than to construction with cardboard and thread, of course!) To facilitate drawing such a design, I wrote PostScript to draw parabolas on adjacent sides of arbitrary polygons. Given my immediate objective, I began using the code to produce triangles, such as would be needed for the proposed construction:

Triangle with inscribed parabolas

I quickly realized that I could do more with this code than I needed for my design. More on this later.

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