More Generalizations (Page 2)
I now had the basic tools to draw my proposed design. (I will spare you
the details of getting the PostScript code to produce exactly what I had
in mind.) Here is a figure with 10 nested squares, parabolas drawn
within triangles divided into 10 segments, and an offset of 10 used to
draw the parabolas:
Several observations can be made about this figure.
Perhaps the most obvious is that there is a lot going on here, and the structure
of the design is obscured by it. Next, it is worth noting that this and many
similar designs could not reasonably be reproduced with cardboard and thread
because of the crowding of lines near the center. Having taken note of this
fact, however, it is interesting that the center of the figure is not “filled.”
This last observation is worth some attention. In
this, a physical representation of a mathematical ideal, all the lines have the
same, nonzero width. Therefore, if enough squares are added to the picture,
individual lines will no longer be capable of being distinguished from one
another. (In the above figure, this happens when about 17 squares are used.) In
the abstract, of course—that is, using “mathematical” lines of zero width—we may
add as many squares as we like, making the size of the central square as small
as we like. Although we could not see such an abstract design, we can
certainly visualize it by imagining it to be made of finite-width lines.
As we enlarge the figure (think of “zooming in” toward the center), the
seemingly solid center resolves into distinct lines, and, no matter how great
the enlargement, the figure always looks essentially the same.
That was a bit of a philosophical digression. Look
again at the last figure, and let us consider something a bit more
concrete. One can identify 9 concentric circles in the figure. (Well, they are
polygons that can be made to approximate circles as closely as you like by
adding more segments and lines to our curve-stitch parabolas.) As we move toward
the center of the figure, the diameter of the circles decreases, though this
does not happen linearly. The diameter of the circles must asymptotically
approach 0. What is the diameter
of successive circles?