The Museum of Science in Boston tallies 1.4 million visits a year. Many of those visitors tour the Mathematica exhibit, a beloved if low-tech attraction installed in 1981 that features sculptures, activities and a dizzying array of numbers, symbols and equations to explain mathematical concepts.
While engrossed in the museum’s offerings, Virginia high schooler Joseph Rosenfeld spotted what he believed to be an error in an equation when he visited in early June, a development that grabbed headlines, including in his hometown paper, the Winchester Star, the Boston Globe and as far away as the Times of Israel. The 15-year-old math whiz said he saw a minus sign where he believed there should be a plus sign in an equation related to the golden ratio, a number and a concept that has intrigued mathematicians since the age of Pythagoras (or about 2,500 years ago).
“I was excited. I was very happy that I found the mistake,” said Joseph, who is taking advanced math and will take AP Calculus as a sophomore this coming school year. “It doesn’t happen every day.”
After Joseph alerted the museum, staff there wrote him to thank him and said they would attempt to change it.
“Thank you for taking the time to leave your feedback after your recent visit to the Museum of Science, Boston. You are right that the formula for the golden ratio is incorrect. We will be changing the – sign to a + sign on the three places it appears if we can manage to do it without damaging the original,” wrote Alana Parkes, an exhibit content developer, in an e-mail to Joseph in mid-June.
But did the 15-year-old spot an error that had eluded millions of visitors and the mathematician designers of the exhibit, which had been on public display for more than three decades? Well, not exactly. Upon further examination, it turned out that both Joseph and the museum’s exhibit were correct.
The museum has since retracted the original admission to the error. As it turns out, the exhibit was displaying the equation for the lesser-known sibling of the golden ratio — its reciprocal, also known as the golden ratio conjugate.
Eve Torrence, a math professor at Randolph-Macon College, was shown a photo of the exhibit and said the museum’s math checks out, even though it’s an unusual way to talk about the concept.
“I’d call it uncommon, not the way most people think of the golden ratio nowadays,” Torrence said. “It’s not what most people think of, but it’s not incorrect.”
The golden ratio can be seen in what is known as a “golden rectangle”: When the sides of a rectangle are such that if you cut the rectangle in two using the shorter side to make a square — and then discard the square — you end up with a second, smaller rectangle of equal proportions to the original rectangle. The process can be repeated an infinite number of times.
The golden ratio can be found in the geometry of perfectly drawn pentagrams and pentagons and even in nature: the seeds in the center of sunflowers and the scales of pine cones arrange themselves roughly according to the golden ratio. The ratio is roughly equal to 1.62 and is represented by the lowercase Greek letter phi (φ). Torrence said the constant has taken on a sort of mythology, with some claiming it can be found in the geometry of the perfect human face. (“Silliness,” she quipped.)
The Mathematica exhibit talks about “golden proportions.” In the equation in question, the exhibit designers took the ratio of the short side of a rectangle to the long side of the rectangle. That’s why the equation the placard displayed contained a minus sign where there would normally be a plus sign. The reciprocal is represented by the uppercase phi (Φ), which yes, looks a lot like the lowercase letter.
Still, Torrence said she was impressed Joseph spotted the discrepancy.
“I’m proud of him for noticing that it had a minus sign instead of a plus sign, but it’s not technically wrong,” Torrence said. “He’s to be commended for questioning authority.”
Joseph said he was disappointed when he found out that he had not actually found an error. But he’s been enjoying the media attention. Thursday, he traveled to the Fox network’s studio in Washington to appear on Fox & Friends.
“Even though it wasn’t a mistake, I’m getting a lot of publicity,” Joseph said. “It can’t hurt with colleges.”
It also will give Joseph a chance to return to Boston: Joseph’s father, Scott, said Friday that John Henry — principal owner of the Boston Globe and the Boston Red Sox — has invited the family to a Red Sox game at Fenway Park.
The organization that represents Charles and Ray Eames, the late designers behind the exhibit, has also reached out telling the family that it is “impressed with Joseph’s findings.”
Even after back-tracking on the error, the Museum of Science in Boston released a statement commending the teen for spotting the difference. The museum has invited him back for a special tour of a new exhibit on the science and math behind Pixar films:
The Museum of Science is thrilled at Handley High School sophomore Joseph Rosenfeld’s enthusiasm about math and our Mathematica exhibit. And it’s not at all surprising that this enterprising student noticed the minus signs because the way the Museum presents the golden ratio in its exhibit is in fact the less common — but no less accurate — way to present it. It’s exciting that people around the country are talking about math and science and that, in the process, we learned something too. Let’s hear it for STEM education and for Joseph Rosenfeld!