Today, we ask medieval architects, "When is
mathematics mathematics?" The University of
Houston's College of Engineering presents this
series about the machines that make our
civilization run, and the people whose ingenuity
created them.

Suppose you're a contractor,
and you're asked to erect a quarter-mile-long
structure, as tall as a typical downtown office
building. Then you're told you must design the
structure as well -- that it is to be made of
stone, no steel beams, no concrete -- that the only
available power is at ground level, limited to
horses, oxen, and maybe a three-horsepower
water-driven saw. One more thing: you've got to do
the job with no stress calculations and no set of
working drawings.

Sound impossible? Well, that's what scores of
medieval cathedral builders did. We read a lot
about cathedrals, but we seldom weigh them as
engineering construction projects. In fact, they
took remarkable coordination of people and
materiel.

The master mason was in charge. He was architect
and builder rolled into one. He often directed a
work force numbering into hundreds. But he also
worked among his people. He cut stone and installed
plumbing. That puzzles us, wed as we are to the
notion that academic and manual knowledge don't
mix. That's a mistake!

The subtle grace of the Gothic cathedral touches us
powerfully on so many levels. But that grace is
vested in engineering design. Those barrel vaults,
flying buttresses, Gothic arches, and spiral stone
staircases had to be born of mathematics.

Mathematics means handling numerical quantities
symbolically, not a subject medieval masons
studied. In fact, some couldn't even read. As we
comb the rich medieval record, we find not only no
mathematical basis for these glorious buildings, we
don't find architectural drawings. We find only the
crudest sketches. Yet the medieval cathedral is
geometry and proportion -- from labyrinths in
mosaic floor tiles to the criss-crossing ribs that
hold the ceiling. It just doesn't make sense. Then
we realize:

The building is the geometry text. The master
mason, with his fingers touching stone, used stone
to express geometry. If mathematics is the symbolic
expression of magnitude, that's what the cathedral
itself is. The balance of mass and space goes by
square roots of 2 and 3, and the so-called Golden
Section.

Medieval iconography regularly shows one
mathematical instrument in the hands of the mason
-- a pair of dividers. When medieval artists show
us God, He often appears as the Master Craftsman,
holding a great pair of dividers. With dividers and
a carpenter's square alone, you can prove the
Pythagorean theorem, and you can create any of
those seemingly sacred proportions.

The cathedrals were not so much designed by
mathematics as they are mathematics. They are
mathematics that flowed straight from the mind's
eye to the fingers of masons who built them.

I'm John Lienhard, at the University of Houston,
where we're interested in the way inventive minds
work.

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