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.
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