3-D Fractals Offer Clues to Complex Systems · Quanta Magazine January 3, 2017 3-D Fractals Offer Clues to Complex Systems

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https://www.quantamagazine.org/3-d-fractals-offer-clues-to-complex-systems-20170103/ January 3, 2017

3-D Fractals Offer Clues to Complex SystemsBy folding fractals into 3-D objects, a mathematical duo hopes to gain new insight into simpleequations.

By Kevin Hartnett

Olena Shmahalo/Quanta Magazine; original figure by Laurent Bartholdi and Laura DeMarco

If you came across an animal in the wild and wanted to learn more about it, there are a few thingsyou might do: You might watch what it eats, poke it to see how it reacts, and even dissect it if yougot the chance.

Mathematicians are not so different from naturalists. Rather than studying organisms, they studyequations 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 newways to look at familiar things, the possibilities for insight multiply.

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That’s the promise of a new idea from two mathematicians: Laura DeMarco, a professor atNorthwestern 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 schoolmath student: f(x) = x2 – 1. Instead of graphing it or finding its roots, they take the unprecedentedstep of transforming it into a 3-D object.

Read the related Abstractions post:How Curvature Makes a Shape a Shape

With polynomials, “everything is defined in the two-dimensional plane,” Lindsey said. “There isn’t anatural place a third dimension would come into it until you start thinking about these shapes Lauraand I are building.”

The 3-D shapes that they build look strange, with broad plains, subtle bends and a zigzag seam thathints at how the objects were formed. DeMarco and Lindsey introduce the shapes in a forthcomingpaper in the Arnold Mathematical Journal, a new publication from the Institute for MathematicalSciences at Stony Brook University. The paper presents what little is known about the objects, suchas how they’re constructed and the measurements of their curvature. DeMarco and Lindsey alsoexplain what they believe is a promising new method of inquiry: Using the shapes built frompolynomial equations, they hope to come to understand more about the underlying equations —which is what mathematicians really care about.

Breaking Out of Two DimensionsIn mathematics, several motivating factors can spur new research. One is the quest to solve an openproblem, such as the Riemann hypothesis. Another is the desire to build mathematical tools that canbe used to do something else. A third — the one behind DeMarco and Lindsey’s work — is theequivalent 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 beunderstood!” DeMarco said by email, referring to the shapes.

http://www.math.northwestern.edu/~demarco/

http://math.uchicago.edu/~klindsey/

https://www.quantamagazine.org/20170109-how-curvature-makes-a-shape-a-shape/

https://arxiv.org/abs/1602.02327

https://arxiv.org/abs/1602.02327

http://amj.math.stonybrook.edu/

http://www.claymath.org/millennium-problems/riemann-hypothesis

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Laura DeMarco, aprofessor at Northwestern University.

“It’s sort of been in the air for a couple of decades, but they’re the first people to try to do somethingwith 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 theearly 2000s, while she was doing graduate work with him at Harvard. DeMarco then went off to dopioneering work applying techniques from dynamical systems to questions in number theory, forwhich she will receive the Satter Prize — awarded to a leading female researcher — from theAmerican Mathematical Society on January 5.

Meanwhile, in 2010 William Thurston, the late Cornell University mathematician and Fields Medalwinner, heard about the shapes from McMullen. Thurston suspected that it might be possible to takeflat shapes computed from polynomials and bend them to create 3-D objects. To explore this idea, heand Lindsey, who was then a graduate student at Cornell, constructed the 3-D objects fromconstruction paper, tape and a precision cutting device that Thurston had on hand from an earlierproject. The result wouldn’t have been out of place at an elementary school arts and crafts fair, andLindsey 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 mindthat made him think this was really important,” said Lindsey. “Then unfortunately when he died, Icouldn’t ask him anymore. There was this brilliant guy who suggested something and said hethought it was an important, neat thing, so it’s natural to wonder ‘What is it? What’s going on

http://www.math.harvard.edu/~ctm/

http://www.ams.org/news?news_id=3222

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here?’”

In 2014 DeMarco and Lindsey decided to see if they could unwind the mathematical significance ofthe shapes.

A Fractal Link to EntropyTo get a 3-D shape from an ordinary polynomial takes a little doing. The first step is to run thepolynomial dynamically — that is, to iterate it by feeding each output back into the polynomial as thenext input. One of two things will happen: either the values will grow infinitely in size, or they’llsettle into a stable, bounded pattern. To keep track of which starting values lead to which of thosetwo outcomes, mathematicians construct the Julia set of a polynomial. The Julia set is the boundarybetween 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.

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If you

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shade the region bounded by the Julia set, you get the filled Julia set. If you use scissors and cut outthe 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 originalpolynomial, 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,” DeMarcosaid. “If you glue them together you get a shape that’s polyhedral.”

The algorithm was Thurston’s idea. When he suggested it to Lindsey in 2010, she wrote a roughversion of the program. She and DeMarco improved on the algorithm in their work together and“proved it does what we think it does,” Lindsey said. That is, for every filled Julia set, the algorithmgenerates the correct complementary piece.

The filled Julia set and the planar cap are the raw material for constructing a 3-D shape, but bythemselves they don’t give a sense of what the completed shape will look like. This creates achallenge. When presented with the six faces of a cube laid flat, one could intuitively know how tofold them to make the correct 3-D shape. But, with a less familiar two-dimensional surface, you’d behard-pressed to anticipate the shape of the resulting 3-D object.

“There’s no general mathematical theory that tells you what the shape will be if you start withdifferent types of polygons,” Lindsey said.

Mathematicians have precise ways of defining what makes a shape a shape. One is to know itscurvature. Any 3-D object without holes has a total curvature of exactly 4π; it’s a fixed value in thesame way any circular object has exactly 360 degrees of angle. The shape — or geometry — of a 3-Dobject is completely determined by the way that fixed amount of curvature is distributed, combinedwith information about distances between points. In a sphere, the curvature is distributed evenlyover the entire surface; in a cube, it’s concentrated in equal amounts at the eight evenly spacedvertices.

A unique attribute of Julia sets allows DeMarco and Lindsey to know the curvature of the shapesthey’re building. All Julia sets have what’s known as a “measure of maximal entropy,” or MME. TheMME is a complicated concept, but there is an intuitive (if slightly incomplete) way to think about it.First, picture a two-dimensional filled Julia set on the plane. Then picture a point on the same planebut very far outside the Julia set’s boundary (infinitely far, in fact). From that distant location thepoint is going to take a random walk across two-dimensional space, meandering until it strikes theJulia set. Wherever it first strikes the Julia set is where it comes to rest.

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Test Your Mathematical SculptingSkillsCan you turn a two-dimensional fractal into a 3-D object? Break out your scissors and tape for achance to win a 3-D printed sculpture.

The MME is a way of quantifying the fact that the meandering point is more likely to strike certainparts of the Julia set than others. For example, the meandering point is more likely to strike a spikein the Julia set that juts out into the plane than it is to intersect with a crevice tucked into a region ofthe set. The more likely the meandering point is to hit a point on the Julia set, the higher the MME isat that point.

In their paper, DeMarco and Lindsey demonstrated that the 3-D objects they build from Julia setshave a curvature distribution that’s exactly proportional to the MME. That is, if there’s a 25 percentchance the meandering point will hit a particular place on the Julia set first, then 25 percent of thecurvature should also be concentrated at that point when the Julia set is joined with the planar capand folded into a 3-D shape.

“If it was really easy for the meandering point to hit some area on our Julia set we’d want to have alot of curvature at the corresponding point on the 3-D object,” Lindsey said. “And if it was harder tohit some area on our Julia set, we’d want the corresponding area in the 3-D object to be kind of flat.”

This is useful information, but it doesn’t get you as far as you’d think. If given a two-dimensionalpolygon, and told exactly how its curvature should be distributed, there’s still no mathematical wayto identify exactly where you need to fold the polygon to end up with the right 3-D shape. Because ofthis, there’s no way to completely anticipate what that 3-D shape will look like.

“We know how sharp and pointy the shape has to be, in an abstract, theoretical sense, and we knowhow far apart the crinkly regions are, again in an abstract, theoretical sense, but we have no ideahow to visualize it in three dimensions,” DeMarco explained in an email.

She and Lindsey have evidence of the existence of a 3-D shape, and evidence of some of that shape’sproperties, but no ability yet to see the shape. They are in a position similar to that of astronomers

https://www.quantamagazine.org/20170105-test-your-mathematical-sculpting-skills/



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who detect an unexplained stellar wobble that hints at the existence of an exoplanet: Theastronomers know there has to be something else out there and they can estimate its mass. Yet theobject itself remains just out of view.

A Folding StrategyThus far, DeMarco and Lindsey have established basic details of the 3-D shape: They know that one3-D object exists for every polynomial (by way of its Julia set), and they know the object has acurvature exactly given by the measure of maximal entropy. Everything else has yet to be figuredout.

In particular, they’d like to develop a mathematical understanding of the “bending laminations,” orlines along which a flat surface can be folded to create a 3-D object. The question occurred early onto Thurston, too, who wrote to McMullen in 2010, “I wonder how hard it is to compute orcharacterize the pair of bending laminations, for the inside and the outside, and what they might tellus about the geometry of the Julia set.”

Kathryn Lindsey,a mathematician at the University of Chicago.

In this, DeMarco and Lindsey’s work is heavily influenced by the mid 20th-century mathematicianAleksandr Aleksandrov. Aleksandrov established that there is only one unique way of folding a givenpolygon to get a 3-D object. He lamented that it seemed impossible to mathematically calculate thecorrect folding lines. Today, the best strategy is often to make a best guess about where to fold thepolygon — and then to get out scissors and tape to see if the estimate is right.

“Kathryn and I spent hours cutting out examples and gluing them ourselves,” DeMarco said.

DeMarco and Lindsey are currently trying to describe the folding lines on their particular class of 3-D objects, and they think they have a promising strategy. “Our working conjecture is that the foldinglines, the bending laminations, can be completely described in terms of certain dynamicalproperties,” DeMarco said. Put another way, they hope that by iterating the underlying polynomial inthe right way, they’ll be able to identify the set of points along which the folding line occurs.


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From there, possibilities for exploration are numerous. If you know the folding lines associated tothe polynomial f(x) = x2 – 1, you might then ask what happens to the folding lines if you change thecoefficients and consider f(x) = x2 – 1.1. Do the folding lines of the two polynomials differ a little, alot or not at all?

“Certain polynomials might have similar bending laminations, and that would tell us all thesepolynomials have something in common, even if on the surface they don’t look like they haveanything in common,” Lindsey said.

It’s a bit early to think about all of this, however. DeMarco and Lindsey have found a systematic wayto think about polynomials in 3-D terms, but whether that perspective will answer importantquestions about those polynomials is unclear.

“I would even characterize it as being sort of playful at this stage,” McMullen said, adding, “In a waythat’s how some of the best mathematical research proceeds — you don’t know what something isgoing to be good for, but it seems to be a feature of the mathematical landscape.”

Try your hand at shaping a two-dimensional fractal into a 3-D polyhedron.

This article was reprinted on Wired.com.


https://www.wired.com/2017/01/build-beautiful-3-d-fractals-simplest-equations/

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3-D Fractals Offer Clues to Complex Systems · Quanta Magazine January 3, 2017 3-D Fractals Offer Clues to Complex Systems