# Math Trails in Ottawa

Housed in a spectacular building redolent of crystals and light, the National Gallery of Canada in Ottawa was recently the setting for a highly unusual school event–a mathematics field trip!

For several years, math teacher Ron Lancaster of Hamilton, Ontario, has been creating "math trails" for both students and teachers as a way to demonstrate that mathematics can be found anywhere and everywhere, whether in puzzles and games, magic tricks, scientific investigations, newspapers, architectural structures, cityscapes, or even art galleries.

For the May conference of the Ontario Association for Mathematics Education, he worked with Mary Bourassa of Lisgar Collegiate Institute in Ottawa and her students to produce a detailed itinerary with a variety of math-related activities for a visit to the National Gallery.

The first stop was a set of four, brightly painted pillars standing in a plaza in front of the main entrance to the gallery. The task was to develop and carry out a strategy to show that the pillars are located at the corners of a square. The math trail kit happened to include string and a tape measure, so participants quickly got busy determining the distances between the pillars–not only the sides but also the diagonals.

Even something as mundane as a sign at the entrance showing when the building is open to the public suggested a mathematical question. Participants had to estimate the percentage of time the building is open over the course of a week. They could later use a calculator to check how close their estimate was to the actual answer.

One breath-taking vista within the gallery was a long corridor with a sloping walkway and a high ceiling. Nine banners with the same pattern but different color arrangements hung down from the colonnade's ceiling. Each banner had five regions, so one of the four colors available (red, yellow, blue, and green) was always used twice.

Questions: How many versions of these banners are possible? Did the gallery use every one of the variations?

Lancaster notes that math trails are really about curiosity and being observant. Good questions get students and teachers to look at and think about what they might otherwise miss. In the case of the banners, for example, a close examination revealed striking symmetries involving the particular colors used for different parts of the banners. Taking into account those symmetries and other apparent restrictions on the designs reduced the number of possible combinations.

Another question concerned large cloth triangles that served as window shades high up in the gallery's crystal towers. How would you go about estimating the amount of material required to make one of these triangular sails? One group of students attempted to answer the question by measuring the shadow cast by a shade on the floor far below.

Many questions and activities concerned artworks on display in the gallery (see http://national.gallery.ca/collections/index_e.html).

Stacked oil cans in an installation by George Segal called "The Gas Station" led to a question about triangular numbers (see Next in Line, November 16, 1996). Andy Warhol's "Brillo" prompted an inquiry into tilings and perfect squares. Frank Stella's painting from his "Protractor" series elicited a query about connections between the depiction, its mysterious title, "Firuzabad," and the mathematical instrument for which the series is named. Donald Judd's "Untitled" sculpture led to a question about what measurements you would need to calculate the volume of a trapezoidal prism.

A math trail can take unexpected turns. A Jackson Pollock artwork ("No. 29") sandwiched between two pieces of glass so that it could be seen from both sides led to a discussion of mirror symmetries. And that, in turn, brought up a remarkable magic square that Lancaster had discovered many years earlier while drowsing through a college topology course.

8888 | 1118 | 1181 | 8811 |

1811 | 8181 | 8118 | 1888 |

8111 | 1881 | 1818 | 8188 |

1188 | 8818 | 8881 | 1111 |

The sum of the four numbers in any row, column, or diagonal is 19,998. (For more information about magic squares, see Magic Tesseracts, October 16, 1999.) You get the same result if you look at this magic square in a mirror or, if it is inscribed on glass, from behind.

The National Gallery math trail developed by Lancaster and Bourassa featured 28 stops. Math field trips don't have to be this elaborate, however. "It's best to start small," Lancaster says. "You can do it on school property or even in the classroom."

"I have been encouraging teachers to develop math trails for their own area," Lancaster adds. Eric Muller of Brock University in St. Catharines, Ontario, and his colleagues, for example, have developed math trails for Niagara Falls and the Welland Canal (see http://www.cms.math.ca/Forum/events/bu-63.html).

I emerged from the National Gallery with a sharper eye for intriguing details. On the gallery grounds, I came upon a massive rock inscribed with meandering trails. I couldn't help thinking about worms and knots as I studied this cryptic sentinel.

Across the way, I could see the gothic revival spire of the magnificant Library of Parliament–a structure with precisely 16 equal sides.

Once you start looking, you quickly find that there's mathematics everywhere around you.