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 segments. 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 s = 24,
as does the figure above, and o = 29:
Segments = 24, Offset = 29
Then, things got interesting. (Continued on next page)
|
