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3/13/2020 How math makes movies like Doctor Strange so otherworldly | Science News for Students MATH How math makes movies like Doctor Strange so otherworldly Patterns called fractals are inspiring filmmakers with ideas for mind-bending worlds Kaecilius, on the right, is a villain and sorcerer in Doctor Strange. He can twist and manipulate the fabric of reality. The film’s visual-effects artists used mathematical patterns, called fractals, illustrate Kaecilius’s abilities on the big screen. MARVEL STUDIOS By Stephen Ornes January 9, 2020 at 6:45 am For wild chase scenes, it’s hard to beat Doctor Strange. In this 2016 film, the fictional doctor-turned-sorcerer has to stop villains who want to destroy reality. To further complicate matters, the evildoers have unusual powers of their own. “The bad guys in the film have the power to reshape the world around them,” explains Alexis Wajsbrot. He’s a film director who lives in Paris, France. But for Doctor Strange, Wajsbrot instead served as the film’s visual-effects artist. Those bad guys make ordinary objects move and change forms. Bringing this to the big screen makes for chases that are spectacular to watch. City blocks and streets appear and disappear around the fighting foes. Adversaries clash in what’s called the “mirror dimension” — a place where the laws of nature don’t apply. Forget gravity: Skyscrapers twist and then split. Waves ripple across walls, knocking people sideways and up. At times, multiple copies of the entire city seem to appear at once, but at different sizes. And sometimes they’re upside down or overlapping. Bringing the twisty other world of Doctor Strange to the big screen required time, effort and computers. Wajsbrot also needed a geometric pattern called the Mandelbrot (MAN- del-broat) Set. This is a type of shape known as a fractal. It’s made of curves and patterns, but those curves and patterns have curves and patterns of their own. There are patterns within patterns. And similar ones show up as you zoom in on an object. This happens in nature, too. Zoom in on a jagged mountain top and you find smaller jagged peaks within the peaks. https://www.sciencenewsforstudents.org/article/math-movies-doctor-strange-otherworldly 1/6 3/13/2020 How math makes movies like Doctor Strange so otherworldly | Science News for Students The Mandelbrot Set is a pattern called a fractal. It looks a little like a bug. Look around the edges, and you can see smaller Mandelbrot “bugs.” If you could zoom in on those bugs, you’d find still smaller copies. WOLFGANG BEYER/WIKIMEDIA COMMONS (CC BY-SA 3.0) The people who worked on special effects for Doctor Strange wanted to use a lot of fractals, says Wajsbrot, who works with a company called Framestore. As characters try to navigate bizarre changes to their reality, scenes zoom in or out on a building, wall or floor. And this reveals more buildings, walls and floors within. The filmmakers’ goal was to use math to create sights that people had never seen in a movie before. To get that type of novelty, Wajsbrot says, they needed fractals. And of all the fractals they worked with, they found special inspiration in one type — the Mandelbrot Set. “The Mandelbrot Set,” says Wajsbrot, “was the cherry on the cake.” Monsters, infinities and snowflakes The Mandelbrot Set is named for Benoit B. Mandelbrot. He was a Polish-born mathematician who studied math in Paris, France. He would go on to spend most of his life in the United States working for IBM, the computer company. He died in 2010. Mandelbrot is most famous for his studies of fractals. (In 1975, he even coined the term fractal to describe these shapes.) Mandelbrot didn’t invent or discover these shapes. Earlier mathematicians had explored them. In 1904, for example, a Swedish mathematician named Niels Fabian Helge von Koch (Fon KOKH) devised one of the most famous fractals in history. Von Koch’s fractal is a little easier to grasp than the Mandelbrot Set. Here’s his recipe: Start with an equilateral triangle (that’s one where each side is the same length). Then remove the middle third of each side. Now, build an equilateral triangle in each of those places where you removed the line. Keep going: Everywhere you find a line segment, remove the middle third and build an equilateral triangle there. The figure is known as von Koch’s snowflake. Mathematicians called shapes like this “pathological curves.” (“Pathological” things cause, or are caused by, physical or mental disease.) They sometimes called them mathematical “monsters” because the shapes don’t follow easy rules. For example: If you keep going with von Koch’s process forever, you’ll end up with an infinitely long line. Von Koch’s snowflake is a fractal. If you zoom in on it, anywhere, you’ll find the same pattern of triangles on triangles. One of Mandelbrot’s early demonstrations of a fractal was This image shows the original triangle and first similar to von Koch’s snowflake. It arose from a question: six steps of a shape known as von Koch’s How long is the coastline of Great Britain? The question snowflake. seems simple. The answer isn’t. ANTÓNIO MIGUEL DE CAMPOS/WIKIMEDIA COMMONS https://www.sciencenewsforstudents.org/article/math-movies-doctor-strange-otherworldly 2/6 3/13/2020 How math makes movies like Doctor Strange so otherworldly | Science News for Students Measure a coastline on a globe or from satellite images, and you can use a ruler to find the solution. But if you hop in a boat and follow the rocky coastline all the way around, you’ll get a larger number. (That’s because you can measure more twists and turns, which add distance.) If you walk the whole length, you’ll get a still bigger number. If you could enlist a crab to do the measurement for you, its report would be even bigger. That’s because it would have to scramble over or around every rock it encountered. Mandelbrot showed that the measured length depends on the size of your ruler. The smaller your ruler, the larger your answer. By that process, he said, the coastline is infinitely long. Nature is truly rough Geometry — the math of curves and other shapes — Explainer: The basics of involves straight lines and neat circles. Mandelbrot argued geometry that those concepts don’t describe the roughness of the natural world. Many objects in nature, including mountains, clouds and coastlines, look the same from far away as they do up close. In order to study these irregular shapes better, Mandelbrot turned to the idea of dimension. A line has one dimension. (The lines making up the letters of this article, for example, are one-dimensional.) A plane, like a sheet of paper, has two dimensions. A box has three. But Mandelbrot’s idea was that rough, natural shapes, such as coastlines or clouds, have a dimension somewhere between two whole numbers. He said they have a fractional dimension, which inspired him to make up the term “fractal.” Mandelbrot’s work opened a new area of math exploration, starting in the 1970s and 1980s. For artists, it led to new ways of creating landscapes. Mandelbrot showed that math could be used to create a realistic scene of mountains, water, clouds or other things in nature. The equations that make fractals soon became tools for artists. Many digital artists now look to fractals like the Mandelbrot Set for inspiration. This fractal-like landscape was created by Hal Tenny, an artist in New Jersey. He contributed drawings to help inspire the filmmakers of Guardians of the Galaxy Vol. 2. HAL TENNY “A lot of people may not even realize they are looking at a fractal design that was created with math,” says Hal Tenny. This New Jersey artist creates his art using fractals. “With the different computer programs we have now, we can create almost photorealistic fractal images that are so different than what we are used to seeing with ordinary images.” The Mandelbrot set grows up — and out The Mandelbrot Set might be the most famous fractal of all. Like the von Koch snowflake, the Mandelbrot Set follows a mathematical recipe that tells you to repeat the same steps over and over and over. Mathematicians call this an iterative process. The basic recipe for a Mandelbrot Set includes only multiplication and addition. These are done over and over, again and again. “It is this amazing thing that comes from such a simple rule,” says Sarah Koch. A mathematician, she works at the University of Michigan in Ann Arbor. Koch is an expert in a field called complex dynamics. https://www.sciencenewsforstudents.org/article/math-movies-doctor-strange-otherworldly 3/6 3/13/2020 How math makes movies like Doctor Strange so otherworldly | Science News for Students Her work often leads her back to the Mandelbrot Set. It looks like a bug with lots of smaller bugs around its edges. Zoom in on those exterior bugs, and still smaller bugs, identical in shape, appear. (Other patterns, with names such as Seahorse Valley, also appear.) Zoom in on the Mandelbrot bug, between the head and body, and you’ll end up in “Seahorse Valley,” which gets its name from curves that look like the snout and body of seahorses. WOLFGANG BEYER/WIKIMEDIA COMMONS (CC BY-SA 3.0) Mathematicians still don’t know everything about the ultimate outermost edge of the Mandelbrot Set. It’s not a neat line or curve.

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