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## Some Circular Designs

I receive occasional e-mail messages about curve-stitch designs, in part, I think, because there are few Web sites dealing with such designs. This seems to make me, by default, a curve-stitch expert. (I am embarrassed to admit that I have never touched cardboard and thread to make a design, however.)

I have received several messages from Belgian Bob Beckers. I’m afraid I have not been of much help in providing Bob with an easy way to draw designs on his computer. (Writing PostScript programs is the best technique I have discovered, but doing so is a pretty arcane skill.) Undiscouraged, Bob eventually sent me a new design and asked if I could write a program to generate it. Responding to that request led to what follows.

The design that Bob sent—a design made, he tells me, for Elke—used curve-stitch parabolas to produce a circular pattern that could be extended indefinitely by adding layers—rather like layers of an onion—to an existing construction. When I indicated that the pattern underlying Bob’s design was not completely obvious, he kindly provided a page of helpful diagrams that were crystal clear. Bob’s basic design is illustrated here:

This design may be best thought of as a “pie” constructed by drawing a “wedge” (or “slice”), which is repeatedly replicated and rotated. In the figure above, there are six wedges making up the pie. Here, there are four “layers” in each wedge, and therefore, in the pie. The first layer in a wedge is a single curve-stitch parabola whose axes, in this case, form a 60° angle. Each added layer places identical parabolas at the end-points of the parabolas of the previous layer and adds parabolas on the “backs” of adjacent parabolas in the new layer. The pattern of this construction can be seen in the figure below, which shows a single wedge of one, two, and three layers. To make the construction clearer, I have used more lines in each parabola than in the figure above, and the parabolas on the “backs” of other parabolas have been rendered in a different color.

Once this pattern is understood, it is obvious that various parameters can be manipulated to generate a family of designs. Most obviously, the number of wedges in the pie can be 3, 4, 5, etc. One can also vary the number of segments into which the axes of the parabolas are divided. In the figures immediately above, the axes are divided into 15 segments, whereas, in the top figure, only eight segments are used. Although it is not immediately obvious, the number of segments can be as small as 1, an explanation of which will require a brief digression later in our discussion. Several variations on Ed’s design can be seen on the next page. Additional variations could be constructed by manipulating the overall size of the figure, line width, line color, and background color. The entire figure can also be rotated, which sometimes changes the psychological impact of the figure, even though this may be of little mathematical interest.