The Symmetries of the Baptistery
and the Leaning Tower of Pisa
David Speiser
Scuola Normale Superiore, Pisa and
Institut de Physique Théorique, Louvain-la-Neuve
The round Leaning Tower of Pisa consists of a massive ground
floor with a blind arcade, six tiers of open arches, and a
smallish drum (tamburo). While the symmetry of a circle is an
infinite one, the number of architectural elements of the tower
reduce it to a finite one. The 15-fold and 30-fold symmetry of
the Leaning Tower is extremely rare, if not unique. The Pisans
of those times delighted in mathematical relationships in
building design: they knew what they were doing when they
constructed a pentadecagonal
campanile.
The Baptistery is not as harmonious as the Leaning Tower. It
consists of four levels, and counting a garland of arches, there
are five elements to be analyzed. The Romanesque ground floor
consists of a blind arcade of 20 arches resting on 20 pilasters.
The second level displays a striking "symmetry clash": a 12-fold symmetry. That this clash is not more manifest is due to
the garland, which consists of 60 columns supporting 60 arches.
Since
3 x 20 = 60 = 5 x 12
it stands in a consonant relation to the levels below and above
as well as to the one it conceals. The third level returns to
a 20-fold symmetry. The 12-fold symmetry of the cupola is again
sharply at odds with the first and the third level.
Entering the Baptistery, one is captivated by the upward surge
of the pillars and columns of the innermost space, crowned by the cupola. Everything in the interior is subordinated to the
dominant 12-fold symmetry. Yet another "clash" is strikingly
apparent: not counting the cupola, only two interior levels
correspond to three exterior levels!
The symmetry clashes help reconstruct the history of the
Baptistery construction. In 1185 the original plan for the
exterior was replaced by that of Guidolotto, who, if not the
designer of the Leaning Tower, was strongly influenced by it.
His design involved the regular pentagon and pentadecagon, which
had appeared for the first time in the construction of the
Leaning Tower. But Guidolotto's capolavoro was not to
everybody's liking. Circa 1221, the quarrels ended with the 12-fold symmetry triumphant. A third reversal, circa 1278, revived
the icosagonal symmetry.
The consequence of the symmetry clashes was a continuous striving
for reconciliation. Despite this, the Baptistery is
overwhelming, especially in the light of a full moon.
