“I want to get weird with pi,” MAA visiting scholar James Tanton told the crowd that packed the MAA Carriage House on December 5 for the organization’s third annual Martin Gardner Celebration of Mind event.
Ever since the 2010 death of the popular mathematics and science writer, yearly Gatherings for Gardner have been held more or less around Gardner’s October 21 birthday. At puzzle parties and math lectures worldwide—more than 100 in 2012, according to Celebration of Mind organizing committee member Bill Ritchie—recreational math junkies get together to keep Gardner’s legacy alive through playful mathematizing.
Sometime university professor and high school teacher James Tanton told listeners Wednesday night that delivering his talk, titled “Weird Ways to Work with Pi,” in honor of Martin Gardner was “a treat beyond belief.”
Tanton’s first step toward the promised weirdness came in the form of a classic puzzle. Imagine taking a long rope, Tanton instructed his audience, and snugly wrapping it around the Earth’s middle. Then lengthen the rope by 10 feet. Say that the rope is magic, able to hover at a uniform distance above the equator. The rope’s added length will create a gap between the ground and the rope. But, Tanton asked, how big of a gap? And what if a similar procedure were conducted on Mars or on a “planet the size of a pea”?
It turns out, as Tanton demonstrated, that the height of the gap is independent of the radius of the planet being circled with rope: It is always 5/pi feet, or a bit more than 19 inches.
“I could squirm my way under that [rope] if I had to,” Tanton marveled. “That’s phenomenal.”
The planet problem was Tanton’s way of introducing the idea that there might be analogs of pi associated with plane figures other than circles.
“Who says planets have to be round?” he asked, and he proceeded to work out on the Carriage House whiteboard that, for a cubical planet of any size, the height of the gap between the rope and the ground would be 5/4 feet, or 15 inches.
Tanton pointed to the four in the denominator of his answer. “Clearly that’s pi for a square,” he said.
But what does it mean to be pi for something other than a circle? Here Tanton cited the “two basic features of pi,” namely, its appearance in the formulas for area and circumference (or, more generally speaking, perimeter).
So four is pi for a square with side length x because, at least if you take x/2 as the “radius,” the formulas work out.
“Why stop at squares?” Tanton asked, swiftly deriving an expression for the pi value associated with any regular n-gon and then for any polygon, regular or not, that circumscribes a circle.
To show how his unorthodox pi’s come in handy, Tanton recalled the “classic wire- cutting problem.” Where should you chop a one-meter length of wire such that when you form one of the resulting pieces into a circle and the other into a square you maximize the sum of the areas of the figures formed?
“This is a very natural need,” Tanton deadpanned. “Everyone needs to know these things.”
The wire-cutting problem typically appears in calculus textbooks, but Tanton solved it without appeal to the optimization techniques generally applied. He used “pi for a square” instead.
Tanton concluded his lecture with a series of questions. While he had encouraged listeners throughout the evening to challenge received wisdom—we operate as if pi has the same value for every circle, but “has anyone checked a circle the size of the solar system?”—his wrap-up urged the audience to dive into the wild world of plentiful pi’s his talk had laid bare:
“Could any real number be a value of pi for some shape?” Tanton asked.
Is the usual pi the smallest pi? (Hint: don’t be straitjacketed by convexity.)
If a shape has a pi value of four, do you “know that it has to be a square, or could another shape have a pi value of four”?
Evidently, the opportunities for exploration—and attendant celebrations of mind—are endless. —Katharine Merow