By folding fractals into 3-D objects, a mathematical duo hopes to gain new insight into simple equations.
Olena Shmahalo/Quanta Magazine; original figure by Laurent Bartholdi and Laura DeMarco
Kevin Hartnett writes in Quanta:
If you came across an animal in the wild and wanted to learn more about it, there are a few things you might do: You might watch what it eats, poke it to see how it reacts, and even dissect it if you got the chance.
Mathematicians are not so different from naturalists. Rather than studying organisms, they study equations and shapes using their own techniques. They twist and stretch mathematical objects, translate them into new mathematical languages, and apply them to new problems. As they find new ways to look at familiar things, the possibilities for insight multiply.
That’s the promise of a new idea from two mathematicians: Laura DeMarco, a professor at Northwestern University, and Kathryn Lindsey, a postdoctoral fellow at the University of Chicago. They begin with a plain old polynomial equation, the kind grudgingly familiar to any high school math student: f(x) = x2 – 1. Instead of graphing it or finding its roots, they take the unprecedented step of transforming it into a 3-D object.
With polynomials, “everything is defined in the two-dimensional plane,” Lindsey said. “There isn’t a natural place a third dimension would come into it until you start thinking about these shapes Laura and I are building.”
The 3-D shapes that they build look strange, with broad plains, subtle bends and a zigzag seam that hints at how the objects were formed. DeMarco and Lindsey introduce the shapes in a forthcoming paper in the Arnold Mathematical Journal, a new publication from the Institute for Mathematical Sciences at Stony Brook University. The paper presents what little is known about the objects, such as how they’re constructed and the measurements of their curvature. DeMarco and Lindsey also explain what they believe is a promising new method of inquiry: Using the shapes built from polynomial equations, they hope to come to understand more about the underlying equations — which is what mathematicians really care about.
Breaking Out of Two Dimensions
In mathematics, several motivating factors can spur new research. One is the quest to solve an open problem, such as the Riemann hypothesis. Another is the desire to build mathematical tools that can be used to do something else. A third — the one behind DeMarco and Lindsey’s work — is the equivalent of finding an unidentified species in the wild: One just wants to understand what it is. “These are fascinating and beautiful things that arise very naturally in our subject and should be understood!” DeMarco said by email, referring to the shapes.
“It’s sort of been in the air for a couple of decades, but they’re the first people to try to do something with it,” said Curtis McMullen, a mathematician at Harvard University who won the Fields Medal, math’s highest honor, in 1988. McMullen and DeMarco started talking about these shapes in the early 2000s, while she was doing graduate work with him at Harvard. DeMarco then went off to do pioneering work applying techniques from dynamical systems to questions in number theory, for which she will receive the Satter Prize — awarded to a leading female researcher — from the American Mathematical Society on January 5.
Meanwhile, in 2010 William Thurston, the late Cornell University mathematician and Fields Medal winner, heard about the shapes from McMullen. Thurston suspected that it might be possible to take flat shapes computed from polynomials and bend them to create 3-D objects. To explore this idea, he and Lindsey, who was then a graduate student at Cornell, constructed the 3-D objects from construction paper, tape and a precision cutting device that Thurston had on hand from an earlier project. The result wouldn’t have been out of place at an elementary school arts and crafts fair, and Lindsey admits she was kind of mystified by the whole thing.
“I never understood why we were doing this, what the point was and what was going on in his mind that made him think this was really important,” said Lindsey. “Then unfortunately when he died, I couldn’t ask him anymore. There was this brilliant guy who suggested something and said he thought it was an important, neat thing, so it’s natural to wonder ‘What is it? What’s going on here?’”
In 2014 DeMarco and Lindsey decided to see if they could unwind the mathematical significance of the shapes.
A Fractal Link to Entropy
To get a 3-D shape from an ordinary polynomial takes a little doing. The first step is to run the polynomial dynamically — that is, to iterate it by feeding each output back into the polynomial as the next input. One of two things will happen: either the values will grow infinitely in size, or they’ll settle into a stable, bounded pattern. To keep track of which starting values lead to which of those two outcomes, mathematicians construct the Julia set of a polynomial. The Julia set is the boundary between starting values that go off to infinity and values that remain bounded below a given value. This boundary line — which differs for every polynomial — can be plotted on the complex plane, where it assumes all manner of highly intricate, swirling, symmetric fractal designs.
If you shade the region bounded by the Julia set, you get the filled Julia set. If you use scissors and cut out the filled Julia set, you get the first piece of the surface of the eventual 3-D shape. To get the second, DeMarco and Lindsey wrote an algorithm. That algorithm analyzes features of the original polynomial, like its degree (the highest number that appears as an exponent) and its coefficients, and outputs another fractal shape that DeMarco and Lindsey call the “planar cap.”
“The Julia set is the base, like the southern hemisphere, and the cap is like the top half,” DeMarco said. “If you glue them together you get a shape that’s polyhedral.” . . .
Continue reading. More at the link, including a good sidebar on the Julia set.