More Generalizations (Page 14)
I have not considered irregular figures, as they are likely merely to
seem bizarre if their shape has no particular meaning for the viewer.
Familiar shapes can be decorated with parabolas, however, to good
effect. Our last figure is a keystone, which has 8 vertices. As before,
sides are divided into 20 segments, and an offset of 20 is used:
This figure raises a question I will leave to the
reader to ponder. As with all the figures that have been considered here, all
sides on which parabolas are drawn are divided into the same number of segments.
Aesthetically, this may not always be the ideal construction. In the keystone, for
instance, we might want to keep the lengths of all the segments equal, or at
least divide the shorter sides into fewer segments. This idea also occurred to
me when I was working on the nested squares. Should the inner squares have sides
divided into fewer segments to prevent the figures from looking too busy near
the center? No simple, universal, and elegant answers to such questions seem to
be available. In the keystone, it is easy to feel that fewer segments should be
used near the top of the figure, but using the same length segments as on the
long sides does not look like the right solution either. When I first began working on
the nested squares, I found myself drawing parabolas on one side of a line using
twice as many points as the parabola on the other side of the line. It seemed an
attractive idea to use the same number of points on both sides, but this would
have reduced the number of segments per side by 2 with the addition of each
inscribed square. When scores of squares were nested, this was clearly reducing
the number of segments too fast.