Shape

121 A Second Look at Calculating
For example, if I go on drawing and extend the lines in each of the stars in the
series
so that every two lines intersect—no two are parallel and no three share a point—then
I get the corresponding superstars in this series
Only the five-pointed star is both a star and a superstar—five lines intersect in ten
points. But all of the others stars have greater potential as superstars. For 2n 1 lines,
there are 2n 2 3n þ 1 intersections. And I can count in additional ways, too. For example,
every superstar is made up of n 1 concentric shells—you can think of them as
orbits—and 2n 1 intersections lie in each. Moreover, anywhere I look, 2n 2 intersections
are on a line. It’s all pretty neat. Still, some things about superstars are far easier
to see than to count. Let’s try—it turns out to be even neater with a host of real
surprises.
Novel behavior is common when polygons interact. With stars, it’s two polygons,
and with superstars, it’s two or more. In fact, there can be a cast of thousands
that changes dramatically. It’s a question of what I see as I calculate. Suppose I define
rules according to the schema
x fi tðxÞ
so that x is any chevron and the transformation t keeps shapes similar—or congruent if
you want. Then I have rules like this one
that I can apply recursively to separate chevrons and superstars in the following
manner