Today, math and masonry wed, and they give birth to
the trumpet squinch. 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.

The Roman arch seems pretty
straightforward -- just a two-dimensional stack of
stone wedges. But each stone has to be pre-cut in
that wedge shape. And those stones have to
accommodate a thin, uniform layer of mortar.
Well-cut blocks typically provide for exactly a
sixteenth of an inch of mortar between them.

Medieval masons took stonecutting into a whole new
dimension. From the 10th to the 13th centuries,
Arabic and European architects bent masonry into
remarkable forms -- complex rib-work on the roofs
of vaults, helical stairways, arches intersecting
at strange angles. They did it all without any
formal geometry. Geometry was central to medieval
scholasticism, but it was the study of logic -- not
a means for making things. Yet masonry cried out
for geometric technique. Shaping stones to these
complex forms is called stereotomy, and that means
solid-cutting.

By the 13th century, masons began inventing their
own approximate geometry to describe stonecutting.
The glorious age of Gothic cathedrals was ending
just as masons began writing down formal means for
making them. By the time a new architecture emerged
in the 16th century, printing presses had put
Euclid's geometry into the hands of a broader
public. Now exact geometrical methods helped to
drive architecture.

16th-century architects started using precise
intellectual apparatus to design magically spatial
forms: barrel vaults, biased arches, helicoids, and
something called a trumpet squinch. A trumpet
squinch is a conical arch that flows smoothly out
of two walls that meet in a corner. It's a support
to hold the floor above, and it's a beautiful
complicated shape. Just imagine trying to cut the
stone blocks that can be piled into such a form.
Actually, the late 16th century was giving birth to
two new institutions. One was modern mathematical
analysis, and the other was the wonderfully fluid
shape of baroque architecture.

But the new science of architectural stereotomy
drove design for only a century. It gave too much
latitude to designers. When mid-17th-century
architects went back to the restrictive purity of
classical lines, the people who did mathematical
stereotomy became hired help. By the middle 1600s,
the brief, intimate bond between mathematics and
architectural design had ended; and so too had its
hold on our imagination.

Mathematics and architecture went their separate
ways until they were rejoined in the 19th century.
Then they gave us such wonders as the Crystal
Palace, the Brooklyn Bridge, and the Eiffel Tower.
Then they reminded us how invention is always born
from a wedding of different ideas.

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

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