Mystic Rose and Other Designs
One of the oldest straight-line designs is the mystic rose. It is the
result of connecting every vertex of a regular polygon by a straight
line to every other vertex. Here, for example, is a mystic rose based on
an octagon:
It is easy to see that whether the number of sides of
the polygon used to create a mystic rose is even or odd has an effect on the
appearance of the figure. If the polygon has an even number of sides (s),
such as in the image above, s/2 lines will pass through the center of the
figure. If s is odd, no lines pass through the center. In that case,
slightly less obviously, the center of the figure is surrounded by an s-sided
polygon, as can be seen in the illustration below.
Because the mystic rose is well-known and, I thought,
not particularly interesting, I ignored it for a long time. Instead, I focused
on polygons whose sides are subdivided as in the designs
shown earlier. I then connected points—not just vertices—to other points a fixed
number of points distant. This strategy is shown in the design below. The sides
of the octagon are divided into three segments, and each point is connected to a
point 6 points distant.
Whereas the above figure is not very inspiring, it
suggests some interesting possibilities. In the figure below, a 12-sided polygon
has sides divided into 7 segments. Points various distances from one another are
connected by lines of different colors.
Alternatively, rather than using a 12-sided polygon
with divided sides, we can simply make all the points vertices. That idea yields
the figure below based on an 84-sided polygon.
Here’s a minor variation:
Given a willingness to use various polygons, various
strategies for connecting vertices, and various line colors, the options for new
designs are legion. The next figure is a stand-in for the many possibilities.
Of course, there are no actual circles in the figures
above. But designs based on polygons with sufficient numbers of sides can yield
pleasing pseudo-circular images. I was frustrated that the traditional
curve-stitch parabolas, in some circumstances, would be more pleasing were they
semicircles.
Consider this figure, a square with four curve-stitch
parabolas drawn within it:
The center of the figure above looks as though it
should be circular, but, of course, it isn’t. We can change our curve-stitch
parabolas to curve-stitch semicircles, however, by subdividing the axes
differently. Rather then dividing them into equal-length segments, we can vary
the width of the segments so that the lines connecting intermediate points on
the two axes are tangents to an imaginary circle. (The details of how to do this
involve some trigonometry, which I have described
elsewhere.) Using
appropriately subdivided axes, we can generate the figure below.
Being able to create curve-stitch semicircles opens up a whole new world of
possible designs. This final design is suggestive of the possibilities.
— LED, 3/23/2024 |
