From satellites to cupolas, Mario Martelli P'06, professor of mathematics and department chair, has a unique perspective on math's mysteries and universal appeal, from roles with NATO, the University of Florence, and the Royal Society of England, to authorship of nine books and more than 80 papers. Governor of the Southern California-Nevada section of the Mathematical Association of America since 2000, Martelli is the Association's second vice presidential nominee and has been invited to organize a Session on Dynamical Systems at the August 2005 Math Fest in Albuquerque. Under his leadership the Undergraduate Student Poster Session has become one of the major events of the Joint Annual Meeting of the American Mathematical Society, the Mathematical Association of America, and the Association for Women in Mathematics.

We invited mathematics major and Newton Fellowship recipient Carrie Staples '04, studying at New York University's Steinhardt School of Education and Courant Institute of Mathematical Sciences, to talk with Martelli about their joint research on the chaotic motion of a pendulum, the puzzles presented by Brunelleschi's cupola at the Cathedral of Santa Maria del Fiore, and why math—seemingly everyone's self-professed "worst subject"—is an essential part of a liberal arts education.

CS: I wanted to take your class because I heard great things about you; your students love you. What do you do to excite students about math?

MM: I like teaching and I like my students. I learn from them as they learn from me. I constantly renew my courses and adjust the material to the students' background and needs. Most people believe that you are born to be a mathematician. Either you have it or you don't. If you say to them, "I am a mathematician," they respond, "Wow. That was my worst subject." Don't they?

CS: Yes, all the time.

MM: Why is it the worst subject? It shouldn't be that way. Mathematics is an art, and it gives us a necessary key for unlocking the secrets of our universe. It combines reasoning, imagination, memory, and common sense. I chose to study mathematics because in math you cannot defend as true a statement you are not able to prove.

CS: How do you challenge your students?

MM: I firmly believe that students like to be challenged, and are capable of doing much more than a teacher normally expects. I talk about my own research, discuss what other students have done or are doing with me, show how math can be used in other fields and in everyday life, and give extra credit problems. I want my students to gain from my courses the maturity and confidence they will need when they leave CMC.

CS: Can you describe a problem on which you've worked with students?

MM: Let me first talk about the problem we solved together with the collaboration of Mike O'Neill [associate professor of mathematics] and Massimo Furi [Universita` degli Studi di Firenze]. As you know, from Earth we always see the same side of the moon. However, if we could go to Saturn and observe its satellite, Hyperion, we would see its face changing continuously in an unpredictable manner. Our goal was to find the reason for this unpredictable behavior, which many astronomers believe was the norm when the solar system began. We proposed a model based on the motion of a pendulum with a vertically oscillating pivot. We proved the existence of infinitely many chaotic orbits of the pendulum [Electronic Journal of Differential Equations, Vol. 2004, No. 36, pp. 1-14; http://www.emis.de/journals/EJDE/ index.html] and, although our result could not be applied to satellites, it constituted a very important first step.

I came back to the problem one year later when Adam Landsberg [associate professor of physics], Massimo Furi, and I proved the existence of chaotic changes for certain satellites on elliptical orbits and with non-spherical bodies, by modifying the original model of the pendulum proposed in our joint paper by adding a periodically forcing term [Journal of Difference Equations and Applications, April 2005]. I worked on another problem with Adam Cox '06, Christopher Jones '06, and Allison Westfahl '07. Our goal was to find simple formulas for the area and perimeter of a plane region bounded by two parallel curves. Adam presented the research at the August 2004 Math Fest of the Mathematical Association of America in Providence, R.I. During the same conference I gave an invited talk on the cupola of Santa Maria del Fiore, Florence's cathedral. I impersonated Brunelleschi, and Adam acted as Paolo Toscanelli, a mathematical consultant for the architect.

CS: Your lecture on Brunelleschi's cupola is the most requested talk among parents and alumni. Why is the cupola such an interesting puzzle?

MM: We don't know how Brunelleschi built it! In 1420 the cupola's platform was an imposing octagon, located 55 meters from the ground! The use of scaffolding to support the cupola until its completion would have posed significant stability problems and would have been very expensive. Brunelleschi's design was revolutionary: a very slender cupola with no scaffolding. However, he categorically refused to explain how his model worked because he was afraid that somebody would steal his idea. Today we are left with a mystery: How did he do it?

CS: What were some of the techniques Brunelleschi used?

MM: We can easily recognize three strategies used by the famous architect: the spinapesce, the incorporation of stone chains in the masonry, and the use of the corda blanda. The Romans, who built the dome of the Pantheon around 67 C.E., knew the spinapesce, or herringbone strategy, which requires the spacing of horizontal bricks with spirals of vertical bricks. A semispherical cupola like the Pantheon can be constructed without scaffolding with the appropriate use of three spinapescespirals.

Neri di Fioravanti suggested the second strategy in 1366, requiring the incorporation of stone chains in the masonry of the cupola to balance the lateral push of its walls. Brunelleschi used four stone chains.

The corda blanda was Brunelleschi's idea, observed by Jesuit priest Padre Leonardo Ximenes for the first time in 1657. The beds of bricks are not flat and the profile of a bed on the eight sails is a curve (the corda blanda) that has a tangent line at each point, including where the curve intersects the ribs of the cupola. We still do not know how the corda blandawas designed.

In my opinion, the combination of the corda blanda and the spinapesce strategies allowed Brunelleschi to build the cupola without scaffolding.

CS: How did he adapt his design to the base of the cupola's preexisting octagonal shape?

MM: Some experts have suggested that Brunelleschi built a semi-spherical wall inside the octagonal cupola. This idea is incorrect. It would require a wall very thin at the base and very thick at the top. Moreover, it would enormously complicate the use of the corda blanda. Brunelleschi stayed with the octagon, keeping in mind both proportion and appearance. For example, the height of the rib of the interior cupola is to its radius as the height of the rib of the exterior cupola is to its radius. Moreover, the longest diagonal of the octagonal eye of the cupola is six meters, just like the diameter of the eye of the Pantheon.

CS: In addition to holding leadership roles within the American Mathematical Society and the Mathematical Association of America, you chair CMC's mathematics department. What are your goals, and how are you going to meet them?

MM: We have five long-term goals: to provide support for the other sciences, including economics, biology, physics, and chemistry; to provide all CMC students—particularly freshmen—with modern, challenging, and well-designed courses to satisfy the general education requirement in mathematics; to provide students interested in a career in mathematics with a full range of courses; to offer a comprehensive program in statistics; and to significantly expand our offerings in computer science.

Many assume that mathematics is a department that is more of a service to other departments—needed for the sciences and economics but without a role of its own. We can change that assumption by improving interdepartmental collaboration. Our role will be appreciated when experts from other areas experience how better mathematical training can dramatically improve their students' performance. For example, we have reached a tentative agreement with the economics department to offer Game Theory in alternating years, and are working with the biologists of Joint Science on the design of a yearlong mathematics course tuned to the needs of their students and potentially interesting for all students interested in sciences or in economics/accounting. Jorge Aarao [assistant professor of mathematics] developed a course in Financial Economics, and I have developed a course in Stochastic and Deterministic Modeling that has been incorporated in the Financial Economics sequence. I believe the government program, also, could be enhanced with an appropriate course blending together statistics, probability, and computer science.

The work ahead of us is not minor, and cannot be accomplished in a short period of time, but it will be beneficial to all students.

CS: What role should mathematics play as part of the College's curriculum?

MM: Mathematics is an essential component of the educational goals of CMC. Mathematics is the bridge between the humanities and the sciences. On the one hand, mathematics has the philosophical underpinnings characteristic of the humanities, and on the other hand, mathematics has the rigor and reasoning characteristic of the sciences.

Many people feel that mathematics is esoteric, far removed from their life. There is nothing esoteric about it, like there is nothing esoteric about a symphony or a painting. Mathematics is a very sophisticated product of the human intellect and, not infrequently, mathematical ideas developed hundreds of years ago continue to find application today.

Mathematics never ends. Our work will never be complete. The truth is complex, and our minds will never grasp it in its entirety. There will always be room for exploration and expansion. We have barely scratched the surface of our universe. This is the case in math, and because math is the bridge between the humanities and sciences, it means that the window of every other field also is open to infinite possibilities. Mathematics plays an essential role in the program of a prestigious institution like CMC. Our faculty knows that. It is our duty as teachers to convince our students that a good understanding of math will greatly enhance their private life and their contribution to society.

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Allison Westfahl '07, pictured with Professor Mario Martelli, presented a poster at the Undergraduate Student Poster Session in Atlanta. Attended by 120 teams of undergraduates and evaluated by 150 professional mathematicians, the session took place during the Joint Annual Meeting of the American Mathematical Society, the Mathematical Association of America, and the Association for Women in Mathematics. Westfahl's poster, based on a research project conducted with Christopher Jones '06 and Adam Cox '06 (currently attending the Budapest Semester in Mathematics) under Martelli's supervision, was recognized with a prize.

Martelli and Christopher Jones '06