Urban Figures (Lisbon)

Becket, November 15, 2013

Known as the "Baixa", literally meaning "lower town", the center of Lisbon's historic core sits on a narrow valley running north-south. To the east, the old Arab neighborhood of Alfama climbs up the hills of Sao Jorge and Sao Vicente. To the west, the neighborhoods of Chiado and Barrio Alto rise on top of the hills of Chagas and Santa Catarina (yes, yes, Lisbon is another city of seven hills, like Rome, Constantinople and a few others.)

In response to these steep slopes, Lisbon developed a number of ingenious and rather extraordinary forms of transportation, including several funiculars and elevators. Among them, the Santa Justa Elevator cuts a particularly memorable silhouette in the skyline of the city. Designed by the Luso-French engineer Raoul Mesnier du Ponsard and completed in 1901, the lift is a freestanding iron tower sitting on a set of wide steps a the end of the Rua de Santa Justa in the Baixa. Its slender shaft rises up to the level of Chiado, where it is crowned by a two-story lookout pavilion cantilevering on all sides. From that point, there is a bridge-like structure that connects the tower to the ruins of the Carmo Convent and the Largo do Carmo further west.

Visiting Lisbon in the early 1950s the great French filmmaker Agnès Varda took a number of beautiful photographs, among them one that shows two of Lisbon's legendary "varinas" balancing their characteristic baskets on top of their heads. Standing behind, the Santa Justa elevator seems to replicate the figures of the fishwives at a much larger, monumental scale, almost as a guardian angel for the city and its inhabitants.

Urban Figures (Istanbul)

Providence, November 12, 2013

Often, travelers arriving to Istanbul have depicted a city of hills, domes and minarets. Instead, when the legendary French photographer Henri Cartier-Bresson visited Istanbul in 1964, he chose to point his camera deep into the life of the street. Among his photographs there is one showing the Camondo Stairs in Galata. It looks like a scene in play, with a small cast of characters in their urban roles, walking along the street, smoking against a parapet, coming down the steps, stopping at a landing, looking back towards the residential neighborhood up the hill.

But don't you think that the main character in the photograph is the stair itself? After all, it's an extraordinarily expressive urban piece, with its baroque contortions and heavy Art-Nouveau architecture, clearly a figure (not a backdrop) wedged tightly in the narrow slot sloping between the façades of the adjacent buildings.

The stairs were built in the 1870s by the Camondo family, a prominent Jewish family of financiers often compared to the Rothschilds. This is the part of Istanbul known as Galata, originally the Genoese concessions of Constantinople on the northern shore of the Golden Horn. The Camondo Stairs connect Bankalar Caddesi--the banking street and financial heart of Istanbul--with the residential neighborhood up the hill, where families like the Camondos lived at the time. Actually, you can think of the Camondo Stairs as the daily routines in the life of the city turned into architecture.

(With thanks to my former student Başak Öymen, who introduced me to the history of the Camondo family.)   

The Bridges of Königsberg (Urban Math)

Cambridge, October 8, 2013

Many cities have rivers running along or through them. Sooner or later most of them have built bridges crossing their rivers. And in turn, those bridges have inspired countless artists and writers. From the top of my head I can think of Art Crane's exquisite poem about the Brooklyn Bridge and I'm that sure you'll be able to think of many other great examples. But how many examples of bridges that inspired mathematicians can you think of?

In 1735, the great Swiss mathematician Leonhard Euler solved the riddle known as the "Seven Bridges of Königsberg", demonstrating the first theorem of graph theory and establishing the foundations of topology.

The city of Königsberg in Easter Prussia (now Kaliningrad, Russia) is traversed by the Pregel River and includes two islands. At the time, the city and the islands were connected by seven bridges. Euler managed to demonstrate that it was not possible to devise a route through the city crossing all seven bridges once and only once. In mathematical terms, a trail that visits every edge (the bridges in Euler's example) of a graph (the path) exactly once, is referred to as an "Eulerian path", and one that accomplishes that beginning and ending at the same place as an "Eulerian circuit".

And no, the Königsberg Bridges graph is not Eulerian. But go ahead, pick a city with bridges and give it a try.