Ivars Peterson's MathTrek

June 5, 2000

# Sliding Pi

The entrance to the Downsview subway station in Toronto presents visitors with a striking vista. A vast mosaic of square tiles sweeps across a curved wall, inviting viewers to trace its lines and ponder intriguing irregularities in its color scheme of blues, greens, roses, and other hues.

Of the hordes of people who hurry through the station, perhaps only a few take a moment or two to contemplate the curious blend of the regular and the seemingly random in the bands of this cryptic, ethereal spectrum writ large. Far fewer people even suspect that the artist responsible for this mosaic based her remarkable design on the decimal digits of the number pi (p), the ratio of a circle's circumference to its diameter.

The artist is Arlene Stamp, now a resident of Calgary, Alberta. In 1993, at the height of her interest in nonperiodic patterns, she was intrigued by the possibility of using such designs in public spaces for enlivening large areas, such as floors and walls, more often covered with simple, repeating motifs.

To win the commission for the Downsview subway station, Stamp had to come up with an interesting design that would work equally well on various surfaces in different parts of a large station yet stay within a very limited budget.

"It became clear that any kind of design based on. . .modules would be very repetitive within such a restrictive budget," Stamp says. "So I came up with a design that was rich in information but inexpensive in terms of its realization."

The basic unit in Stamp's design was a rectangle, 10 units wide, made up of square tiles. Instead of placing these rectangles side by side along a baseline to create a regular, linear pattern, she overlapped adjoining rectangles. The amount of overlap was governed by successive decimal digits of pi.

"I used pi as a source for a string of unpredictable digits because the circle and curved walls were a design feature of this station," Stamp remarks.

The first digit of pi after the decimal is 1, so the first two rectangles overlap by 1 unit, just as if the second rectangle were sliding over a little bit to rest on top of the first. The second digit is 4, so the third rectangle overlaps those already in place by 4 units. As the process continues, some regions of overlap may end up as many as three or four layers thick.

Stamp used four sets of eight colors for the project, deploying them in different ways on surfaces in various parts of the station. "I assigned colors by thinking of the first layer as the lightest, with the colors deepening according to the number of layers of overlap," she says.

"One of the big challenges of this commission was to find a way of encoding the color information for the tile installer in some kind of space-efficient way," Stamp notes. "With the help of an architect and a number-coded color system, we got all the information on a single sheet of the architect's plans."

Stamp's design doesn't repeat itself anywhere in the station. Indeed, it's fun to look for all the different "pieces of pi" scattered throughout the structure.

"It's true that few visitors to the station would have any idea that there is a connection between the patterns there and a math concept," Stamp says. "But surely some must wonder how such a seemingly random, ever-changing pattern of color happened--because it is such a rarity in public design."

Moreover, upon glimpsing the various manifestations of Stamp's pi-based mosaics, a few may even sense an underlying plan of some sort--that there is method to the randomness.