Shape

16 Introduction: Tell Me All About It
they’re normally defined and used to calculate. This takes me back to what happened
as I grew up.
When I tried Birkhoff’s measure, I knew how to calculate with numbers. That’s
what you learn in grade school and later. I was good at it. Numbers made Birkhoff’s formula
easy, yet it wasn’t design. There wasn’t anything to see, only to count. I had to
find something else. I learned what I needed as an undergraduate at MIT. It still wasn’t
design, but it got me started in the right way. Marvin Minsky taught symbol manipulation—he
explained Turing machines, Post’s production systems and their nifty way
of defining rules, and sundry equivalent or otherwise related devices. It was amazing
how many different ways there were to calculate. The details varied, but a single idea
tied it all together: one way or another, calculating is combining symbols according to
given rules. An easy example is enough to show how this works.
Suppose I have two symbols, left and right angle brackets h and i, and I want to
combine them in strings, so that left and right brackets are paired in the way left and
right parentheses normally are. I’m after strings that have the following form
h i
hh ii
h i h i
hh i hh iii
I’m using angle brackets instead of parentheses because the brackets look like something
I’m going to do later on. You’re probably getting used to this by now. One way
to define strings of brackets is to calculate. I can use the two rules
(1) h i fi hh ii
(2) h i fi h i h i
Both rules let me replace the string h i whenever it’s contained in another one, that is,
if h i is part of it. When I use rule 1, hiis replaced with the new string hh ii. The idea
is to put a pair of brackets around h i, or equivalently to move the brackets in h i
apart to insert a pair. And when I use rule 2, h i is replaced with hihi. A pair of
brackets is added either to the left or to the right of hi. (When these rules add brackets
to a string, everything else in the string moves to make room for them. Practical—
concrete—details are ignored. This won’t be the case for shapes. They aren’t abstract
in this way.) I always start to calculate with the same string h i. It’s the shortest string
in which angle brackets are paired correctly. I can use my rules three times like this
h i Þ hh ii Þ hh i h ii Þ hh i hh iii
step 1 step 2 step 3
to generate the string hh i hh iii with four pairs of brackets. The double arrow Þ
indicates the process of applying a rule. The rule is used to replace the string h i that’s
bold. In step 1, I use rule 1, in step 2, rule 2, and in step 3, it’s rule 1 once more. Every