Shape

Notes
Introduction
1. S. Pinker, The Blank Slate (New York: Viking, 2002), 79.
2. Plato, The Republic, trans. B. Jowett (Cleveland: World Publishing Co., 1946), 211. Later on,
Plato touts mathematics in secondary education as a prelude to reasoning. Five parts divide the
curriculum, only to ‘‘reach the point of inter-communion and connection with one another.’’
First, arithmetic, including counting and calculating—‘‘The little matter of distinguishing one,
two, and three—in a word, number and calculation:—do not all arts and sciences necessarily partake
of them?’’ But this isn’t all when arithmetic is ‘‘pursued in the spirit of a philosopher, and not
of a shopkeeper! . . . [Then] arithmetic has a very great and elevating effect, compelling the soul to
reason about abstract number, and rebelling against the introduction of visible or tangible objects
into the argument.’’ Plane geometry is next, followed by solid geometry, astronomy, and harmony.
Plato’s mathematics ties to visual calculating in some neat ways, for example, in the anthyphairetic
ratio of the diagonal and the side of a square ð 2 :1Þ. Once, in conversation, Lionel
p ffiffiffi
March told me the diagram
p ffiffiffi
as a geometrical figure and as a proof that 2 :1 repeats without end reminded him of the shape
because squares recur in the same way. In fact, the diagonal of the large square in the diagram
shows the string aba when the side of the little square is a and the diagonal b. As Plato would
p ffiffiffi
have it, there’s more to this than meets the eye. The ratio 2 :1 and its successors in reciprocal
subtraction fix segments in palindromes for sides of squares in my shape, as these segments
are refined in differences. For an account of the diagram, see D. H. Fowler, The Mathematics of
Plato’s Academy: A New Reconstruction (New York: Oxford University Press, 1990), 33–35, and
D. H. Fowler, ‘‘The Story of the Discovery of Incommensurability, Revisited,’’ in Trends in the
Historiography of Science, ed. K. Gavroglu, J. Christianidis, and E. Nicolaidis (Boston: Kluwer,