Shape

272 II Seeing How It Works
descriptions that keep them numerically distinct. Nonetheless, there are no units for
higher-dimensional things. The shapes in rules don’t have parts that I can count before
I calculate. These depend on how the rules are used. The description that makes sense
for a rule now isn’t binding later. What the rule is doing may change freely as it’s
applied. A square is four lines the first time the rule
is tried and eight lines the second time around. Descriptions don’t count. They alter
too erratically for numbers to make sense.
The difference between rules defined for shapes and generative grammars is
evident in other kinds of devices for calculating. They’re described in various ways—
spatially and not—but the method is always combinatorial. And, in fact, they’re all
easy to define in zero-dimensional algebras in which labels or weights are associated
with points.
Turing machines are typical. They’re defined for symbols on tapes—labeled
points evenly spaced on a line—that are modified according to given transitions that
work like my previous rule AqR fi AAqAL. But this is too obscure, and it’s easy to be
clear. The rule
in the algebra V 02 corresponds to a machine transition in which the symbol A is read
in the state q and replaced by the symbol AA. The state q is changed to the state qA, and
the tape is moved one unit to the left. I’ve assumed that the rule applies just under
translations. Three points are needed to mimic the transition if all of the transformations
are allowed. This rule
does the trick nicely. It’s also a cinch to do Turing machines when i is more than zero.
Try it with lines using triangles, and with planes using tetrahedrons, so that the parts
that rules pick out behave like units. This series of scalene triangles