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## A Family of New Designs

The interest that visitors showed in the first curve-stitch design I posted here prompted me to explore other designs. This turned out to be a more time-consuming enterprise than I had planned, but it
has been interesting.

I began with a straightforward design within a square:

##### Segments = 24, Offset = 25 (see explanation below)

(There is no particular significance to the colors in this and later figures.)

There is a simple way to describe this figure. I began with a square, each of whose sides was divided into s equal-length segment. The endpoints of these segments (including, of course, the corners of the square) form a set of 4s equally spaced points on the perimeter of the square. The figure is formed by connecting each point with a line to the point which is 1+s points away from it, traveling in a counterclockwise direction along the edges of the square.

Anyone familiar with curve stitching will describe this as placing four parabolas inside a square. (I learned to make my parabolas by connecting points 1+s points apart, but some people use an offset (call it o) of only s points. The resulting figures are identical except at the ends, and the figures look pretty much the same unless s is very small.)

What would be the result, I wondered, if o exceeds 1+s? Some lines of the “parabola”—it isn’t clear whether this would give lines tangent to an actual parabola— would start on one side of the square and end on the opposite, rather than an adjacent, side. A few hand-drawn figures convinced me that this idea had possibilities, so I proceeded to write a PostScript program to draw figures of this sort. I wrote it so that I could easily change the size of the square, the number of segments, s, and the offset, o. Of course, what was really of interest was how the figure changed as the offset was changed. The first figures I got were about what I expected on the basis of my hand drawings. The figure below, for example, has = 24, as does the figure above, and o = 29:

##### Segments = 24, Offset = 29

Then, things got interesting. (Continued on next page)