More Generalizations (Page 13)
Just as a 5-pointed star is formed from a pentagon by connecting the
vertices in a different order, we can do the same thing with the points
of a square, forming a kind of hourglass or figure-8. In the figure
below, our path goes through the points at the lower left, lower right,
upper left, upper right, and back to the lower left. The point in the
center of the figure is not part of our “polygon.” Twenty segments are
used on each side, and the offset is 20:
If our “polygon” really is a polygon but is not
convex—that is, it is possible to connect two points on the perimeter and have
at least some of the points on the line between them be points outside the
polygon—we get a kind of 3-D effect, resulting from parabolas being drawn on
both the inside and outside of the figure. Here, for example, is a hexagon with
a “waist”:
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