This article is an online extra from the December 2012/January 2013 issue of MAA FOCUS.
In the summer of 2009, my wife and I led a group of Emporia State University students from Athens to Istanbul on an eighteen-day journey across two continents and 2,600 years. No other area of the world provides a better backdrop than Greece and Turkey to discuss the interplay between the mathematical, historical, artistic, and theological viewpoints of infinity. It was my goal to give students of any mathematical background a glimpse into these intriguing aspects of infinity, while also experiencing different cultures and visiting sites of significant historical interest.
Our tour began with a five-day stay in Athens. Here we enjoyed many sights of classical Greece, including a day trip to Delphi. Our next stop was the Greek island of Samos, birthplace of Pythagoras. We had three days to enjoy its peaceful villages and natural beauty. A two-hour ferry ride delivered us to Kusadasi, Turkey, which is the gateway to visiting ancient sites such as Ephesus and the Temple of Artemis, before arriving in Izmir. After a visit to the Izmir University of Economics, we traveled by bus, train, and ferry to Istanbul, where we spent the remaining week of our tour, visiting remnants of history from the Roman, Byzantine, and Ottoman empires.
One appealing aspect of studying infinity is the wide variety of topics available for discussion, even for students with a minimal background in mathematics. For this reason I chose Eli Maor’s To Infinity and Beyond as the text for the course. For the artistically inclined, we discussed symmetry operations and the seven one-dimensional symmetry groups for infinite strips. Students searched for these groups on designs of 2,600-year-old vases found in the National Archaeological Museum in Athens.
Our time in Turkey provided an opportunity to discuss tiling the plane and the seventeen two-dimensional symmetry groups. Turkish rugs and Islamic artwork found in mosques supplied numerous examples for students to analyze and identify. Maor’s book deals extensively with tessellations, particularly as seen in M. C. Escher’s work. Although Escher is Dutch, we viewed some of his artwork firsthand at the Herakleidon Museum in Athens. Of course, no discussion of art, mathematics, and Greece would be complete without a look at the debate regarding the design of the Parthenon and the golden ratio.
Topics like the golden ratio also allowed us to talk about the more mathematical side of infinity. Students recognized the beauty found in the infinite continued fraction that represents the golden ratio. In fact, we discussed several topics that might be familiar to math majors, but were fresh ideas for these students. For instance:
Limits and infinite series, particularly geometric and harmonic series,
Estimating p as discovered by Archimedes,
The irrationality of the square root of 2 as discovered by the Pythagoreans,
Euclid’s proof of the infinitude of the primes, and
Construction of the cube root of 2 as commanded by the Delian Oracle.
A theme that we returned to throughout our trip was how infinity was feared or embraced throughout history. Greek philosophers, religious beliefs, and scientific findings shaped these views. Here were some of the questions we considered:
Is the universe a "One" or a "Many"?
Does the infinite exist in actuality or only potentially?
Does time have a beginning?
Is the universe infinite, unbounded, or both?
Is matter made of atoms, or is it infinitely divisible?
As we visited Athens and Samos, we debated these questions while walking in the footsteps of Zeno, Socrates, Plato, Aristotle, Pythagoras, and Aristarchus. Visiting ancient Ephesus, St. John’s Basilica, and other sites in Turkey led us into discussions of how Christian and Islamic theology changed the prevailing view to some of these questions.
The Historical and Cultural
One of the advantages of our trip was that we followed the chronological progression of major empires that heavily influenced Western civilization. Fortunately, many sites of historical interest survive. Highlights included
Athens: the Acropolis, Plato’s Academy, and the Ancient Agora where Socrates taught.
Eupalinus tunnel: Built 2,600 years ago from both sides of a mountain on Samos Island.
Western Turkey: The Roman city of Ancient Ephesus and the remains of the Temple of Artemis, one of the seven wonders of the ancient world.
Haghia Sophia in Istanbul: A sixth-century Byzantine church designed by the mathematicians Isidore of Miletus and Anthemius of Tralles.
Grand Bazaar: Opened in 1461, the largest indoor bazaar in the world.
A primary reason for the growing popularity of study-abroad trips is that they enlarge a student’s perspective of our world. New experiences for us included bargaining at the bazaar, a boat tour of the Bosphorus, and viewing a handmade Turkish rug be made from silkworm to finish. Less enjoyable was finding that streets were busier and dirtier than at home, and that hotel amenities and public toilets were not always as comfortable as we wished.
Despite the differences in our cultures, we are alike in some ways. Turkish students we met in Izmir and Istanbul had similar interests in movies and liked eating fast food. Education was important to them, and they were involved in student groups that had meetings and conferences just as an ESU group would. It is interesting that a trip overseas can simultaneously help you to more fully appreciate your own culture as well as that of others.
Enough for Everyone
Would I do it again? Definitely!
Would I change anything? It is tempting to have a calculus I prerequisite next time so students might understand and appreciate the mathematics at a higher level. On the other hand, it was enjoyable having students with a wide range of backgrounds. For instance, an English major appreciated visiting sites of mythological importance such as Delphi and the Temple of Zeus. Another student with an interest in art particularly enjoyed the Museum of Turkish and Islamic Art in Istanbul. Ultimately, this was one of the most attractive things about a trip to Greece and Turkey to study infinity. Something appealed to each participant. —Brian Hollenbeck
Brian Hollenbeck received his Ph.D. from the University of Missouri and is an associate professor of mathematics at Emporia State University in Kansas. His interests include real and complex analysis, probability and statistics, and mathematical modeling.