**O**ne of the most typical
elements of Gothic architecture is the tracery found in windows,
on walls, and in many other places in Gothic churches. What is
mathematical about it? Tracery is exclusively constructed from
circular arcs and straight line segments! It is the most mathematical
kind of art known to me. In many of the thousands of Gothic churches
and other buildings of that time surviving in Europe you can
find nice examples, take photos and analyze them geometrically
at home.
Traceries appear some 60 years after the first examples of
Gothic churches in the 1200's in Reims. There their construction
is, like that of the typical pointed Gothic window, based on
the equilateral triangle. In the course of stylistic development,
the constructions became more and more elaborate and less determined
by geometry until we find whole windows covered by wavy ornaments
in the flamboyant late Gothic of about 1500.
Much more than the usual geometric designs of traceries can
be found in the cloisters of the Cistercian monastery of Hauterive
near Fribourg, Switzerland. Here the theme of the windows is
geometry itself. Regular n-gons are shown for n = 3,4,5,6 and
8. Variations of the pentagon show the pentagram and a delicately
constructed rose. The whole cloisters seems to be a commentary
to Euclid's book IV on the regular n-gons carved in stone. |