November 1, 2013
As a mathematical concept, the fractal can be intimidating.
Benoit Mandelbrot, the Polish-born mathematician who coined the term, defined a fractal as “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole.” Fractus, in Latin, means “broken.”
But, the whole idea, I think, becomes a lot more digestible when you look to nature.
The natural world is chock full of fractals. Consider a tree, one of the simplest examples. Whether you look at the entire tree, a branch or a single twig, the shape is generally the same. The same can be said for rivers and their tributaries. This “self-similarity” is a defining trait of a fractal. A fiddlehead—a young fern that is tightly coiled—has little leaflets that form even tinier coils. Similarly, the interior sections of a nautilus shell, all the same crescent shape, get progressively larger from the center of the spiral outwards. Fractal geeks also point to their favorite vegetable: Romanesco broccoli. Each bud of the edible plant is composed of more miniature buds of the same geometric form.
“There’s this moment of awakening where you understand that the natural patterns that you’ve been seeing your entire life are actually based on simple mathematical formulas. And once you’re aware of those patterns—be it the spiral shape of a galaxy or the whirl of a hurricane or the swirls of cream in your morning coffee—you’re able to recognize them anywhere,” says Ben Weiss.
An expert in computer graphics, Weiss has taken it upon himself to make these universal mathematical principles even more accessible. His new iOS app, Frax, which he developed with colleagues Kai Krause and Tom Beddard, puts fractals, as he says, “in the palm of your hand.”
Frax users begin with a basic shape from the app’s fractal library. Then, they manipulate the shape to their own liking, adding depth, shading, color, lighting, gloss and texture. The end result is nothing short of art. The fractals are complex, colorful patterns that conjure any number of things—sea weed, snowflakes, sand dunes and oil spills.
While most will just doodle on their iPhones and iPads, “Some will use it to create more complex works of art, using it as a starting point for fabrics or paintings or digital art installations,” says Weiss. “We’re also hoping that the interaction with these beautiful images will inspire users to want to learn more about the underlying math and geometry, in the same way that looking through a telescope can inspire interest in astronomy and science.”
Weiss’ fascination with fractals took root at an early age. As a 10-year-old, he was writing bits of code and patiently waiting hours for the images to load on the screen of his Apple IIc. For three decades, fractal programs have required users to plug in lots of equations to generate visuals, Weiss explains. He was excited to harness the power of today’s touchscreen devices for this purpose. Frax is built on the famous Mandelbrot and Julia set equations, but, as Weiss told Co.Design, he and his team hid all the mathematical inputs, amounting to almost 100,000 lines of custom code, “under the hood.”
“Not everyone wants to be introduced to something in terms of math,” says Weiss. “There is plenty of complexity hidden away behind the scenes, but the audience is immersed more easily if they don’t see the mechanics behind it all.” (It is a little like slipping fruits and vegetables into desserts.)
Kai Krause, a German software and interface designer involved in the project, has watched kids use Frax. “They clearly have no clue about ‘Mandelbrot’ or the math of it,” he says, and yet they have an appetite for the app, as an entertaining, creative experience. The design team sees Frax as something with broader appeal than other fractal programs on the market, used mainly by math geeks. Krause says they have amplified the play value, without making Frax a game in the traditional sense. “The belief is that you can have serious fun without the need for shooting pigs or people or high scores,” he says.
The experience is immersive, and, as the user zooms in on fractals and makes aesthetic decisions about colors and other effects, he or she picking up skills and developing a more innate understanding of this mathematical art form.
“You’re playing directly with mathematics, but it doesn’t feel dry,” says Weiss. “It feels like an artistic adventure.”
May 31, 2013
For most people, the study of astrophysics means poring over calculations, charts, texts and graphics. But Wanda Diaz-Merced, a graduate student at the University of Glasgow, and fellow researcher Gerhard Sonnert have pioneered a different approach. Its underlying motif is simple: Space produces music.
She grew up with an enthusiasm for science and space, but in her early 20s, as a physics student at the University of Puerto Rico, her vision swiftly deteriorated due to diabetes. When she spent time in an astrophysical observatory, though, and inadvertently heard the hiss and pops of the signals collected by a radio telescope, she realized that there might be a way she could rely solely on her hearing to interpret data.
Since, she’s teamed with computer scientists to use NASA-developed software called xSonify—which converts scientific data of all kinds into synthesized musical sounds, a process called sonification (PDF)—to analyze solar flares on the sun, as well as X-rays coming from the EX Hydrae star system. This software allows users to customize how the data are represented, using pitch, volume, rhythm and even different types of instruments to distinguish between different values and intensities in the electromagnetic spectrum detected by spacecraft over time.
Diaz-Merced listens to these data streams to pick out irregularities and changes in the sounds, and has even convinced some colleagues to adopt the software, because listening while watching data in chart form can help them become more attuned to subtle patterns in the data. “I can listen for harmonics, melodies, relative high- and low-frequency ranges,” she told Physics Today last year. In one case, she said, “I was able to hear [previously overlooked] very low frequencies from gamma-ray bursts. I had been listening to the time series and said to the physicists in charge, ‘Let’s listen to the power spectra.’”
In its raw form, the sounds she listens to seem more like noise than music:
In the spring of 2011, Diaz-Merced was interning at the Harvard-Smithsonian Center for Astrophysics, in Cambridge, when her use of sonification inspired Gerhard Sonnert, a researcher, to do something new with the sounds. He spotted sheet music that represented X-ray emissions from EX Hydrae, collected by the Chandra X-ray Observatory satellite, and noticed a rhythm, common in Afro-Cuban music, called a clave.
A bass player, Sonnert got the idea to convert the sounds from EX Hydrae, some 200 light-years away, into blues, jazz and classical music. As part of the Star Songs project, he teamed with his cousin Volkmar Studtrucker, a composer, to manually convert the data into nine different songs, which the duo then performed with drummer Hans-Peter Albrecht and released as an album.
Listen to the raw sound data that produced the blues track, along with the completed song:
Studtrucker started off by picking select portions of the signal that were suitable for use in composition. As a whole, the sounds are largely irregular, because they result from X-rays emitted in a variable fashion due to the nature of EX Hydrae. The system is actually made up of two stars, with one continually pulling matter away from the other at varying rates, which causes the level of X-ray emissions to fluctuate as well.
But particular portions of the sounds representing the X-ray emissions seemed to have melodies and a beat, and by repeating these short segments—and adding harmonic elements, as well as altering the underlying clave rhythm—Studrucker was able to compose songs based off the data in a variety of styles. In addition to blues, he produced several others:
Jazz Waltz (data, then song):
Of course, there’s an element of abstraction in all these tracks, and with even the raw sounds produced by xSonify that Diaz-Merced uses to conduct her research. But that doesn’t mean that her research—or Studtrucker’s music—is any less representitive of phenomena in space than the work of conventional astronomers.
As Ari Epstein put it in a terrific Studio 360 segment on Diaz-Merced’s research, “Stars and planets don’t give off sounds as they move through the sky. But they don’t draw lines on graphs either. All of these things—graphs, numbers, music—they’re all just tools we can use to understand a complicated universe.”
May 30, 2013
Those who have a neurological condition called chromesthesia associate certain colors with certain sounds. It’s these people that I think of when I see Mark Fischer’s Aguasonic Acoustics project. Fischer systematically transforms the songs of whales, dolphins and birds into brightly colored, psychedelic art.
The software developer from San Jose, California, gathers the sounds of marine mammals in nearby Monterey Bay using a hydrophone and the chirps of birds in his neighborhood with a digital recorder; he also collects audio of other hard-to-reach species from scientists. Fischer scans the clips for calls that demonstrate a high degree of symmetry. Once he identifies a sound that interests him, he transforms it into a mathematical construct called a wavelet where the frequency of the sound is plotted over time. Fischer adds color to the wavelet—a graph with an x and y axis—using a hue saturation value map—a standard way for computer graphic designers to translate numbers into colors. Then, he uses software he personally wrote to spin the graph into a vibrant mandala.
“The data is still there, but it’s been made into something more compelling to look at,” wrote Wired.
The first animal sound that Fischer turned into visual art was that of a blue whale. “I was spending some time down in Baja California. Someone had posted a note on MARMAM [the Marine Mammal Research and Conservation email list] looking for volunteers for a blue whale population survey out of the University of La Paz, and I volunteered. We spent the next three days in the Sea of Cortez looking for blue whales,” says Fischer. “We never did find a blue whale, but I was able to make recordings. I just got fascinated with the sounds of whales and dolphins.”
Fischer concentrates on whales, dolphins and birds mostly, having found that their calls have the most structure. Humpback whales, in particular, are known to have incredible range. “They make very well defined sounds that have extraordinary shapes in wavelet space,” says the artist. The chirps of insects and frogs, however, make for less engaging visuals. When it comes to a cricket versus a humpback, Fischer adds, it is like comparing “someone who has never played a guitar in their life and a violin virtuoso.”
Animal sounds have long been studied using spectrograms—sheets of data on the frequency of noises—but the software designer finds it curious that researchers only look at sounds this one way. Fischer finds wavelets much more compelling. He prints his images in large-scale format, measuring four feet by eight feet, to call attention to this other means of analyzing sound data.
Some researchers argue that little progress has been made in understanding humpback whale songs. But, Fischer says, ”I am concluding that we are looking the wrong way.” The artist hopes that his mandalas will inspire scientists to look at bioacoustics anew. “Maybe something beneficial will happen as a result,” he says.
The Peabody Essex Museum in Salem, Massachusetts, will include a selection of Fischer’s images in “Beyond Human,” an exhibition on artist-animal collaborations on view from October 19, 2013 to June 29, 2014.
March 15, 2013
To say that Henry Segerman is schooled in mathematics is an understatement. The 33-year-old research fellow at the University of Melbourne, in Australia, earned a master’s degree in math at Oxford and then a doctorate in the subject at Stanford. But the mathematician moonlights as an artist. A mathematical artist. Segerman has found a way to illustrate the complexities of three-dimensional geometry and topology—his areas of expertise—in sculptural form.
First things first…three-dimensional geometry and topology?
“It is about three-dimensional stuff, but not necessarily easy to visualize three-dimensional stuff,” says Segerman, when we talk by phone. “Topology is sort of split along low-dimensional stuff, which usually means two, three and four dimensions, and then high-dimensional stuff, which is anything higher. There are fewer pictures in the high-dimensional stuff.”
Since 2009, Segerman has made nearly 100 sculptures that capture, as faithfully as is physically possible, some of these hard-to-grasp lower-dimensional mathematical concepts.He uses a 3D modeling software called Rhinoceros, typically used to design buildings, ships, cars and jewelry, to construct shapes, such as Möbius strips, Klein bottles, fractal curves and helices. Then, Segerman uploads his designs to Shapeways.com, one of a few 3D printing services online. “It is really easy,” he says. “You upload the design to their Web site. You hit the ‘add to cart’ button and a few weeks later it arrives.”
Before 3D printing, Segerman built knots and other shapes in the virtual world, Second Life, by writing little bits of programming. “What cool things can I make in 3D?” he recalls asking himself. “I had never played around with a 3D program before.” But, after a few years, he reached the limit of what he could do within that system. If he wanted to show someone a complicated geometric shape, that person needed to download it to his or her computer, which seemed to take ages.
“That is the big advantage of 3D printing. There is an awful lot of data in there, but the real world has excellent bandwidth,” says Segerman. “Give someone a thing, and they see it immediately, with all its complexity. There is no wait time.”
There is also something to holding the shape in your hand. Generally speaking, Segerman designs his sculptures to fit in someone’s palm. Shapeways then prints them in nylon plastic or a costlier steel bronze composite. The artist describes the 3D printing process, for his white plastic pieces:
“The 3D printer lays down a thin layer of plastic dust. Then, it’s heated up so that it is just under the melting point of plastic. A laser comes along and melts the plastic. The machine lays down another layer of dust and zaps it with a laser. Do that again and again and again. At the end, you get this vat filled with dust, and inside the dust is your solid object.”
While his primary interest is in the mathematical idea driving each sculpture, and in conveying that idea in as simple and clean a way as possible (“I tend towards a minimalist aesthetic,” he says), Segerman admits that the shape has to look good. A Hilbert curve, the 3-sphere—these are esoteric mathematical concepts. But, Segerman says, “You don’t need to understand all of the complicated stuff in order to appreciate the object.”
If viewers find a sculpture visually appealing, then Segerman has something to work with. “You’ve got them,” he says, “and you can start telling them about the mathematics behind it.”
Here are a few selections from Segerman’s large body of work:
Segerman made up the word “autologlyph” to describe sculptures, such as “Bunny” Bunny, pictured at the very top, and this sphere, above. By the artist’s definition, an autologlyph “a word, which is written in a way that is described by the word itself.” With “Bunny” Bunny, Segerman used the word “bunny,” repeated many times over, to form a sculpture of the Stanford Bunny, a standard test model for 3D computer graphics. Then, in the case of this sphere autologlyph, block letters spelling the word “sphere” create the sphere. Minus the bunny, many of Segerman’s autologlyphs have a mathematical slant, in that he tends to use words that describe a shape or some sort of geometric feature.
This cube, shown above, is Segerman’s take on a Hilbert curve, a space-filling curve named for David Hilbert, the German mathematician who first wrote about the shape in 1891. “You start with a curve, really a straight line that turns right angle corners,” says the artist. “Then, you change the curve, and you make it squigglier.” Remember: Segerman does these manipulations in a modeling software program. “You do this infinitely many times and what you get at the end is still some sense a one dimensional object. You can trace along it [the line] from one end to the other,” he says. “But, in another sense, it looks like a three-dimensional object, because it hits every point in a cube. What does dimension mean anymore?” Hilbert and other mathematicians became interested in curves like these in the late 19th century, since the geometries called into question their assumptions about dimensions.
“I had been looking at this thing on a computer screen for a year, and when I first got it from Shapeways, and picked it up, it was only then that I realized it was flexible. It is really springy,” says Segerman. “Sometimes the physical object surprises you. It has properties that you didn’t imagine.”
Round Klein Bottle is a sculpture, much larger than Segerman’s typical pieces, that hangs in the Department of Mathematics and Statistics at the University of Melbourne. (The artist applied a red spray dye to the nylon plastic material for effect.) The object itself was designed in something called the 3-sphere. Segerman explains:
“The usual sphere that you think of, the surface of the earth, is what I would call the 2-sphere. There are two directions you can move. You can move north-south or east-west. The 2-sphere is the unit sphere in three-dimensional space. The 3-sphere is the unit sphere in four-dimensional space.”
In the 3-sphere, all the squares in the grid patterning of this Klein bottle are equal in size. Yet, when Segerman translates this data from the 3-sphere to our ordinary three-dimensional space (Euclidean space) things get distorted. “The standard Mercator map has Greenland being huge. Greenland is the same size as Africa [in the map], whereas in reality, Greenland is much smaller than Africa. You are taking a sphere and trying to lay it flat. You have to stretch things. That is why you can’t have a map of the world which is accurate, unless you have a globe,” says Segerman. “It is exactly the same thing here.”
Segerman is now toying with the idea of moving sculptures. Triple Gear, shown here, consists of three rings, each with gear teeth. The way it is set up, no single ring can turn on its own; all three have to be moving simultaneously. As far as Segerman knows, no one has done this before.
“It is a physical mechanism that would have been very difficult to make before 3D printing,” says the artist. “Even if someone had the idea that this was possible, it would have been a nightmare to try to build such a thing.”
September 21, 2012
The other day I wrote about five horrendously inaccurate scenarios in science fiction movies, all selected by David Kirby, a trained geneticist and author of Lab Coats in Hollywood: Science, Scientists, and Cinema. If you missed it, Kirby’s list touched on asteroid predictions, natural disasters and a cloning incident—all bogus, when dissected by a scientist.
I had heard Kirby talk about the history of science advising in the TV and film industries at “Hollywood & Science,” a recent webinar hosted by the American Association for the Advancement of Science (AAAS). Directors hiring scientists to review the science they portray on screen goes back to the 1920s and 1930s. Kirby is actually quite forgiving when it comes to science fiction movies heralding from those early decades. The “bad science” those movies sometimes portray is not always the fault of filmmakers, Kirby says; in many cases, it is due to the limitations of technology or simply a reflection of the state of scientific knowledge at the time. For instance, Destination Moon, a sci-fi flick from 1950, was one of the first to show space travel in a somewhat realistic way. However, the astronauts could not wear clear, goldfish bowl-type helmets, as they did in real life, because they created too much glare for the camera.
Today, filmmakers have little excuse for error.
The Science & Entertainment Exchange, a program of the National Academy of Sciences, actually matches TV and film professionals, even video game makers, with science consultants for free. “We have Nobel Prize winners on speed dial,” said Ann Merchant, deputy director for communications at NAS and a fellow panelist. “We were told, if we built it, they [directors, screenwriters, etc.] would come—and they did.” Since the program was launched in November 2008, it has received three to five new calls a week and arranged a grand total of 525 consults. The movies Iron Man, Tron, Spiderman, Prometheus and The Avengers and TV shows Fringe, The Good Wife and Covert Affairs have all benefited from the service.
Here are Kirby’s top five “science done right” moments in film:
1. 2001: A Space Odyssey (1968)
“For its time, 2001 is one of the most, if not the most, scientifically accurate film ever made,” says Kirby. Stanley Kubrick, the film’s director, hired former NASA space scientist Frederick Ordway to serve as his science adviser. One of the greatest lengths that Kubrick went to is in acknowledging that gravity doesn’t exist on a spaceship. “Kubrick actually decided to acknowledge this fact by building an artificial gravity wheel for the spaceship,” says Kirby. “On a long-distance space flight, you need to spin it to get the centrifugal force to simulate the idea that there is actually gravity, something pulling you down. That is what this thing did.” The prop cost $750,000 (equal to $5 million today) and took six months for Vickers Engineering Group to build. “That shows incredible commitment to scientific veracity,” says Kirby.
2. Finding Nemo (2003)
As I mentioned in my previous post, animators painstakingly removed all bits of kelp from the coral reef scenes in Finding Nemo after marine biologist Mike Graham of the Moss Landing Marine Laboratories in Moss Landing, California, explained that kelp only grows in cold waters. But, as Kirby points out, this is just one of many measures the filmmakers took to ensure scientific accuracy.
According to an article in the journal Nature, Adam Summers, then a postdoc in fish biomechanics at the University of California, Berkeley, and other experts he recruited gave lessons during the movie’s production on a wide range of topics, including fish locomotion, how fish scales reflect light and the mechanics of waves. Director Andrew Stanton attended the lessons along with animators, producers, writers and character developers involved with the project. Robin Cooper, head shader for the film, gets extra credit though. She actually reached her arm into the blowhole and mouth of a beached, dead gray whale to take some photographs. This way, when Nemo’s dad, Marlin, gets sucked into a whale’s mouth and blasted out through its blowhole, she could accurately portray the inside of the whale. “I’m just amazed at how rigorous these people were,” Summers told Nature.
3. Contact (1997)
Warner Brothers filmed some of the scenes of this movie, adapted from Carl Sagan’s book Contact, at the Very Large Array, a New Mexico branch of the National Radio Astronomy Observatory. (Remember the huge white dishes facing the skies?) Bryan Butler, then a postdoc researcher at the site, served as a science advisor.
In the film, scientist Ellie Arroway, played by Jodie Foster, tries to make contact with extraterrestrial life. According to Kirby, her actions are largely in line with SETI, or search for extraterrestrial intelligence, protocol. “The setting, the dialogue, the way that they are trying to confirm what they are seeing, is real,” says Kirby. “They have to call someone in Australia and say, ‘hey, can you see this too?’ They have to wait for it to be confirmed by somebody on the exact other side of the world before they can actually confirm that it is real. All that type of stuff was accurate.”
4. The Andromeda Strain (1971)
In this sci-fi thriller, based on Michael Crichton’s 1969 novel of the same title, a team of scientists studies an alien virus that infects and kills humans. “There is a scene where they are trying to figure out how big the microbe is that they are dealing with. From modern eyes, it ends up being a very slow, boring scene, but that is because it is realistic,” says Kirby. “It is this idea of, ‘Let’s try two microns. Oh, that’s too big. Let’s try 0.5. Oh, that’s too small. Let’s try one.’ The science in it is accurate. They are experimenting, but it doesn’t make for very gripping cinema.”
5. A Beautiful Mind (2001)
Russell Crowe played the brilliant, schizophrenic mathematician John Nash in A Beautiful Mind. However, the actor had a hand double. Dave Bayer, of Barnard College’s math department, wrote all the mathematical equations so that they had “a natural flow,” according to Kirby.