Shape

294 II Seeing How It Works
But then there are a couple more things to heed. Triangles come in three kinds
with three parts or four, and squares come in three kinds
with four parts or eight. Moreover, the different decompositions of the shape may be
regarded as subalgebras of the Boolean algebra, including all five of the decompositions
obtained in the first scenario. Only there’s no guarantee that things will always work
out conveniently. A Boolean algebra needn’t be defined for every shape when identities
apply wherever they can. Other structures are also possible.
Classifying the parts of a shape in a decomposition is a useful practice that
explains how the shape works and what it means. But topologies aren’t the only way
to approach this, and it’s good to see something else to make the point. Let’s try another
easy example with identities that emphasizes the transformations instead of
embedding and the part relation. After all, transformations are also needed to apply
rules—they play the key role in deciding when shapes are alike. I won’t show much
detail, just enough to suggest that other ways of understanding are worthwhile, too.
There are topologies to get back to. Still, there’s no reason to stick with one way of
describing things. If calculating with shapes shows anything, it’s that nothing works
all the time. It always makes sense to look again.
Suppose I have the identity
to pick out squares in the shape
In terms of this identity, there are fourteen squares that appear to be the same. But I
can use the transformations under which the identity applies to define an equivalence
relation. If I first pay attention to scale, then there are three classes of squares—nine
small ones