Shape

99 A Second Look at Calculating
A Second Look at Calculating
I’m going to show a few additional examples of rules that aren’t identities and the way
they work to calculate with shapes. I want you to see how shapes are redescribed as I
apply rules of a general kind that includes identities and more. There’s no set way to
divide shapes that rules can’t alter. The ability to handle all of the ongoing changes in
what I see—there’s always a chance for more, even with a single rule—is what makes
calculating visual. If you first want to check the formal technicalities that are needed
for rules to work—there aren’t many—you can jump to part II. It’s safe to wait,
though, because they aren’t really necessary. It’s always possible to follow along with
your eyes. You simply have to look to understand what I’m doing. Nothing is hidden.
I can draw it all when I calculate by seeing.
The rule
and its inverse
—it’s the rule defined when the left and right sides of a rule are flipped—move triangles
back and forth along a fixed axis
keeping their orientation constant. Together they’re like the identity