Every summer, scores of tourists take to the bustling streets of Barcelona, Spain, a city known for its breathtaking architecture. Nicolás Atanes Santos, a young Spanish mathematician, sees this as an opportunity to engage more people in his favourite subject.
In partnership with the regional government of Catalonia, Santos created what he calls “math trails,” self-guided walking tours for visitors to explore landmarks of Spanish cities, one maths problem at a time.
Mathematics is used in architecture for both function and design. Trigonometry helps calculate angles and heights. Symmetry creates proportion and balance. Geometry brings shape and form to life.
For Santos, maths can also inspire ideas about structures that have already been built — a new way to see and appreciate the world. Inspired by this idea, The New York Times created a tour of some of the most striking architecture in Barcelona, a city where maths and exploration meet.
Santos is an undergraduate student at the National University of Distance Education and has studied mathematics in Burgos, Pamplona and Barcelona. His walking tours are part of a personal mission to bring maths to the streets, a response to the broad distaste he perceives for his field of study.
“People are often afraid of mathematics,” Santos said. “They remember nothing but the stress of learning things they couldn’t see practical applications for.” Or they find it boring and mechanical, he explained, consisting only of formulas to memorise.
But for Santos, mathematics means more than that. It is something to be discovered, an international language that connects people with place.
Santos has helped create maths trails for tourists who stroll the streets of Barcelona, Girona, Lleida, Tarragona and Tortosa. Everywhere, he said, there are “many puzzles, many problems, that are very beautiful”.
Mathematics, it seems, is anywhere you choose to find it — in architecture, but also in music, the flow of traffic and bowling.“It’s all around us,” Santos said. “It transcends borders, countries, cultures and even eras.”
At the heart of the city lies perhaps the most iconic structure in Barcelona. La Sagrada Família, a Catholic church that has been under construction since 1882, is the magnum opus of Spanish architect Antoni Gaudí. Visiting the unfinished church immerses you in shapes: circles, spirals, hyperboloids. “The straight line belongs to men,” some believe Gaudí once said. “The curved line to God.”
The Passion facade is one of three fronts of the church. Its columns, 77 feet long and 48 feet apart, form an arch that can be approximated by two right triangles. Use the Pythagorean theorem, a² + b² = c², to estimate the height of the arch. Here, a = 24 feet, half the distance between the columns; c is roughly 77 feet. With the Pythagorean theorem, that makes b, the height of the arch, about 73 feet. The shape of the arch mirrors the broader design of the church: the tallest tower, still under construction, will be in the centre, flanked by smaller towers.
Crowds once walked through the arches of La Plaza de Toros Monumental de Barcelona to watch matadors whip capes at angry bulls. The circular bullring, known to many simply as La Monumental, hosted one of the last bullfights in the region. La Monumental now functions as a venue for concerts, food festivals and other events. It also contains a bullfighting museum.
The bullring measures about 168 feet across. What is the length of its perimeter? To compute a circle’s perimeter, known as its circumference, multiply its diameter by pi. Written as a formula, C = πd. The circumference of the bullring is about 528 feet. La Monumental’s round shape allows optimal viewing for spectators in the audience.
Torre Glòries is one of the tallest buildings in Barcelona. Designed by Jean Nouvel, a French architect, the building is reminiscent of a geyser shooting water into the sky. Its exterior is made of thousands of aluminium plates and glass panes, which shimmer in the sunlight. Both materials help with thermal insulation, making the skyscraper an example of bioclimatic architecture.
Each glass pane has an area of about four square feet. Approximate Torre Glòries as a cylinder to estimate how many glass panes cover its exterior. The cylinder has a height h = 472 feet and a radius r = 64 feet. Its surface area can be calculated using the formula A = 2πrh + 2πr². This yields a surface area of 2,15,538 square feet. Dividing by four square feet, the area of one glass pane, gives 53,884 — an estimate of the number of glass panes covering the building’s exterior. (The exact amount — 52,744 glass panes — is smaller, because Torre Glòries is not a true cylinder.)
The fantastical look of Casa Batlló is a trademark of Gaudí, its designer. Its wavy facade, adorned with blue-green mosaics and a scaly roof, alludes to features of the sea. Balcony railings shaped like carnival masks add to the avant-garde look.
Thirteen of the windows on Casa Batlló have shutters. Assume the shutters can be either both open or both closed. How many different arrangements of shutters exist? One window has two possible arrangements: open or closed. Two windows have 2 x 2 = 4 arrangements. Three windows have 2 x 2 x 2 = 2³ = 8 arrangements. Following this pattern, 13 windows have 2¹³ = 8,192 arrangements. The colour of the shutters and the stained glass on some of the windows further play to the motif of water.
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