Square Pursuit

The sculpture standing in the plaza near the main entrance to the National Gallery of Canada in Ottawa consists of four rectangular pillars, each one a different color. The steel pillars have a square cross section, and the four pillars mark the corners of a square.

Created by prominent Canadian artist Guido Molinari (1933-2004), the sculpture is titled Homage to Samuel Beckett.

Some years ago, teachers Ron Lancaster and Mary Bourassaa created a math trail to lead high school students on a mathematical adventure through the National Gallery. Each student carried a detailed itinerary, featuring a variety of math-related activities, for the visit. Molinari's columns were the first stop on the tour.

Supplied with string and tape measure, the students had to determine whether the four columns are truly located at the corners of a square. They had to realize that measuring the distances only between adjacent columns (the lengths of the sides) is not enough.

For the second exercise, one person stood at each column. Each person then looked directly at the person to his or her left and started walking toward that person. If the four students all started at the same time and walked at the same speed, what path would each person follow? Where would they end up?

The resulting path is known as a pursuit curve. In this case, the four "pursuers" would each follow a spiral path toward the center, where they would meet.

John Sharp has proposed the following set of steps for drawing pursuit curves when the pursuers start at the vertices of a regular polygon.

1. Draw a regular polygon. One pursuer starts off at each corner of the polygon.
2. Mark the spot on each side of the polygon that each pursuer has reached after a fixed period of time.
3. Join the new points to show the new chasing directions and create a new polygon.
4. On the new polygon, mark off the same length from each corner as you did for the original polygon and join the resulting points to create another, smaller polygon.
5. Continue in the same fashion until you can't comfortably fit a new polygon into the remaining area.

The first two steps in drawing pursuit curves based on a square.

Courtesy of John Sharp.

In the case of a square, it's possible to mimic the resulting pattern by constructing a series of larger and larger right-angled triangles, starting with a square in the middle. Such an arrangement of triangles and squares can serve as the basis for a colorful, eye-catching quilt.

Reference:

Sharp, J. 1999. In pursuit of pursuit curves. In ISAMA 99, N. Friedman and J. Barrallo, eds. University of the Basque Country.

Photos by I. Peterson