Shape

283 Parts Are Evanescent—They Change as Rules Are Tried
according to the twin formulas for applying rules. The scenarios are good for shapes
with points and lines, etc. In fact, for lines, they’re already familiar from part I. But
my scenarios are just examples. There are other ways to talk about rules and what
they do to divide shapes into parts. I’ll show this below in an easy approach with identities
and transformations. It expands a little on my second scenario to define equivalence
relations that classify parts. Decompositions and other kinds of taxonomies are
an ideal way for reasoning to conclude. There’s a sense of accomplishment, and a real
chance to understand what happened. Parts are charged with meaning. My scenarios
provide the means to get them, and to change them whenever I go on.
In the first scenario, only rules of a special kind are used to calculate. Every rule
A fi
erases a shape A—or some transformation of A—by replacing it with the empty shape.
So, when I calculate, it looks like this
C Þ C
tðAÞ Þ Þ r
I start with any shape C. Another part of C—in particular, the shape tðAÞ—is subtracted
in each step, when a rule A fi is applied under a transformation t. The shape
r—the remainder—is left at the end of this process. Ideally, r is the empty shape. But
whether or not this happens, a Boolean algebra is defined for C. The remainder r when
it isn’t empty and the parts tðAÞ, picked out as rules are tried, are the atoms of the algebra.
This is an easy way for me to define a vocabulary, or to see just how well a given
one works in a particular case.
In the next scenario, rules are always identities. Every identity
A fi A
has the same shape A in both its left and right sides. When only identities are applied,
calculating is monotonous
C Þ C Þ Þ C
where, again, C is a given shape. In each step, another part of C is resolved in accordance
with an identity A fi A, and nothing else is done. Everything stays exactly the
same because tðAÞ is part of C. Identities are constructively useless. In fact, it’s a common
practice to discard them. But this misses their value as observational devices. The
parts tðAÞ that are resolved as identities are tried, the empty shape, and C can be used
to define a topology for C when they’re all combined in sums and products. The first
scenario is included in this one if the remainder r is empty, or if r is included with the
parts tðAÞ. It’s easy to do—for each application of an erasing rule A fi , simply use
the identity A fi A. But a Boolean algebra isn’t necessary when identities are applied.
Complements aren’t defined automatically. They may have to be specified explicitly.
In the final scenario, rules aren’t restricted. Each rule
A fi B