A Mathematical History Tour: Reflections on a Study Abroad Program - Travelogue, Florentine Delights

The students and I arrived in Italy in late May to study the history of mathematics as it relates to Renaissance art and architecture. On our first full day, we climbed to the top of the Cathedral of Santa Maria del Fiore to see how Fillipo Brunelleschi used geometry to construct the impressive dome. We were among the first people to climb the dome that Tuscan morning, and our group discussion at the top was a memorable way to begin our tour.

Figure 2: Enthusiastic about the history of mathematics at Santa Maria del Fiore in Florence. Photo by the author.

In fact, we attracted a crowd. This happened throughout our travels; at the top of the Duomo in Florence, while we discussed the place of geometry in architecture, several strangers stopped to listen. I stifled a laugh as a woman near me elbowed her husband and muttered, “Pay attention, Henry.” Everywhere, people are thirsty for knowledge.

The nearby Duomo Museum contains the original “Gates of Paradise” by Lorenzo Ghiberti, which is a masterpiece of three-dimensional representation in bronze. The museum also contains works by Donatello and Michelangelo, as well as a recently discovered “practice dome” by Brunelleschi. In the same piazza is the Baptistery of St. John, a celebration of symmetry that Brunelleschi used to demonstrate the power of linear perspective in painting.

Another highlight of our time in Florence was the Accademia museum, home of Michelangelo’s "David", as well as his "Prisoners".

Figure 3: One of Michelangelo's "Prisoners”. Wikipedia.

Michelangelo claimed that his sculptures existed within the marble itself, and that his job was to reveal the hidden creatures. For my students, these "Prisoners" sparked a lively conversation about Platonism and mathematics. The excitement was enhanced by the fact that I had neglected to obtain appropriate permissions to “lecture” inside the Accademia[1]

Serendipity

One afternoon, I led the students on a “history of mathematics walking tour” of old-town Florence. Rain clouds gathered as we began, and by the time we reached the Orsanmichele Church, it was raining hard. One of the striking features of that church is the set of fourteen architectural niches on its external walls. One niche is occupied by Donatello’s statue of St. Mark, believed to be the first explicit use of trigonometry in a free-standing statue since ancient times. Donatello computed the exact proportions necessary to make the statue appear realistic when viewed from below (e.g., at street level). According to the story, when the statue was first presented to the wool guild that paid for it, the guild members were underwhelmed by its “inappropriate” proportions. But when the statue was finally lifted into its niche, the importance of trigonometry in sculpture was revealed.

Figure 4: “St. Mark” by Donatello (1413). Wikipedia.

I tried to explain all of this wonderful history in a downpour, as the students made a “roof” of umbrellas over our heads. Eventually the rain forced us inside the church. I had not planned to go inside, but took advantage of the cool, dimly-lit setting to tell the story of Savonarola and the end of the Florentine Renaissance. The ambiance was perfect for a horror story, and the students also had a chance to see a fine example of a Gothic church. In a final twist, we were observed by a local tour guide who had her own group in the church at that same time. She gave us a private tour of the Uffizi Gallery the following day—a neat connection enhanced by a rainstorm.

A note on guided tours

I was hesitant about taking the students to the Uffizi because of the crowds, but the opportunity to trace the development of linear perspective in Renaissance art was not to be missed. Highlights (depending on one's perspective) included “St. Francis Receiving the Stigmata” by della Croce, “The Battle of San Romano” by Uccello, and “Triumph” by della Francesca. Our experience at the Uffizi supports the argument that paying to hire a local guide is money well spent. Our guide had the perfect mix of knowledge, passion, and humor. She really brought mathematics to life through art, and engaged the students with plenty of questions. We paid for guides throughout our tour of Europe (many arranged ahead of time), and each one greatly enhanced our experience. In terms of cost, our tour of the Uffizi was the most expensive, about $17 per student—a good value for a 2.5-hour tour with an expert.

Day Trips from Florence

Florence is within an easy train ride of several interesting cities. We spent a day in Pisa, and another in Bologna.

Pisa is a city that both Leonardo of Pisa (Fibonacci) and Galileo called home at different times in their lives. Our discussions in Pisa centered on the importance of trading ports in bringing knowledge from the Islamic Empire into Western Europe. From the top of the Leaning Tower (remarkably uncrowded in May), we talked about the history of numerals. From the Camposanto Monumentale, in front of the monument to Fibonacci, we discussed phi, phyllotaxis, and the Fibonacci sequence. Ever since taking a history of mathematics course as an undergraduate, I wanted to teach those ideas to students in Pisa, at Fibonacc's monumenti. It was a powerful moment. We also visited the cathedral where (perhaps apocryphally) Galileo was inspired to study pendulums by the swinging of a chandelier (which still swings to this day).

We rambled through the medieval streets of Pisa, thinking about Hindu-Arabic numerals and algebra. We sat at the feet of the statue of Fibonacci in the Camposanto Monumentale and worked problems from Liber Abaci on a portable whiteboard we’d carried with us from home. We discussed the meaning of such problems to a 13th-century merchant. We explored the fascinating properties of the Fibonacci sequence and spirals. Other tourists watched us with curiosity. A woman took out her phone and snapped a picture of the Americans working math problems in a cemetery.

Figure 5. Paying homage at the monument to Fibonacci in Pisa. Photo by the author.

Two days later, the students and I fell in love with Bologna. It is a charming university town, with a laid-back atmosphere that is completely different from Florence. The university is connected with the solution of the cubic equation, and with well-known characters such as Cardano, Tartaglia, and del Ferro. Bologna was also the home of Bonaventura Cavalieri who did foundational work with the geometry of indivisibles. The account of his work, and the persecution he suffered at the hands of the Catholic Church, comprise several thought-provoking chapters in the book Infinitesimal by Amir Alexander.

Serendipity Lost

While in Bologna, we visited Santa Maria della Mascarella, the church connected with Cavalieri. As the students and I discussed infinitesimals on the sidewalk, a priest from the church came outside and asked (in Italian) if we were Catholics, or tourists. I speak very little Italian, and tried to tell him we were mathematicians.  He became very excited, and pointed to a plaque honoring Cavalieri on the wall of his church. He spoke at length, and with passion. We listened in awe, not understanding a single word. In the end, we all thanked him. He smiled, gave a little wave, got into his tiny Italian car and drove away. But what did he say? What knowledge of Cavalieri and the geometry of indivisibles was he trying to impart? This is the curse of being a monolingual American.

Figure 6: Plaque honoring Bonaventura Cavalieri in Bologna. Photo by the author.