Shape

300 II Seeing How It Works
combine in this way
to define parts (atoms) in the topologies of the second and third shapes in the series,
and ultimately to define the large outside triangle and the small inside one in the topology
of the fourth shape. Moreover, the pieces in the fourth and fifth triangles combine
in like fashion to make the three triangles in the topology of the sixth shape that
are needed for the production of the final shape. Looking ahead, the topologies of the
shapes in the series appear to anticipate what the rule is going to do the next time it’s
applied. But this is because the topologies are given retrospectively. I can always describe
what I’ve done as a continuous process after I stop calculating. This explains
what I did and shows how everything goes together in a coherent fashion. Expecting
more—so that the future is laid out in advance—is preposterous.
Every time the rule
is tried, its right side is divided with respect to a different topology. This shows in yet
another way that the shapes in a rule have diverse descriptions that vary as it’s used
to calculate. The rule doesn’t apply in terms of any one description that measures its
complexity or anything else. How the rule is described is an artifact of what’s been
going on.
Before I finish up, it’s also worth noting that the series
isn’t isolated. There are many others like it. And in fact, they can be defined around
each of the shapes in this array of nested polygons