4

Impossible Constructions

The Brotherhood

It was a powerful secret society of men bound together by common be-
liefs in religion and mysticism. Then, one day, their entire belief struc-
ture would be shattered by a discovery so profound it would transform the
thinking of the civilized world.
It sounds like a description of Opus Dei, the powerful clandestine Catho-
lic secret society that played a pivotal role in the blockbuster novel The Da
Vinci Code. Alternatively, it might have described the core of the church
during the seventeenth century, when it was confronted by Galileo’s shat-
tering discovery that the moons of Jupiter orbited a celestial body other
than Earth. This secret society, however, existed some two millennia prior
to Galileo. Founded by the philosopher-mathematician Pythagoras, the
motto of the society—“All is number”—ref lected the view that the uni-
verse was constructed of either whole numbers or their ratios. The discov-
ery that was to rock their world was that the square root of 2, the ratio of
the length of the diagonal of a square to its side, was incommensurable—
that is, it could not be expressed as the ratio of two whole numbers.