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Beauty Formula

In mathematics, we now talk not only about four-dimensional and million-dimensional spaces, but also 1.89-dimensional spaces.

Junpei Sekino’s father was a thoughtful man with a gentle smile who became one of the most respected printmakers of the 20th century, with major museums in New York, Boston, Paris and London vying to purchase his prints. Sekino’s father had hopes that his son would become an artist as well, and the younger sekino didn’t lack for talent, winning first place in the junior division of a national Japanese printmaking contest at age 10. But instead of ink and wood blocks, Junpei Sekino took up mathematics and computers.

Junpei Sekino

In 1985, seven years after Sekino came to Willamette University to teach math, he adopted his father’s trade, in a roundabout way. He began writing programs to create stunning fractal images based on mathematical equations. His computer art is now recognized around the world, and a Google search for “fractal gallery” brings up Sekino’s site as fourth out of 1.8 million entries. Encyclopedia Britannica calls Sekino’s gallery one of the “Web’s Best Sites.” The online exhibit has received more than 75,000 visits from computer hobbyists, fractal artists, and math teachers and students.

Sekino began generating fractal images six years after IBM researcher and Harvard scholar Benoit Mandelbrot sat down at an ancient, oversized computer and plotted the epoch-making fractal Zn+1 = Zn2 + c, now called the Mandelbrot set. “It probably took him several days to plot what now takes a minute,” Sekino says. Mandelbrot’s fractal eventually resulted in a pop culture craze, with its startling patterns appearing on posters, T-shirts, coffee mugs, book covers and a screen saver.

In the early 20th century, long before Mandelbrot’s time, French mathematicians Gaston Julia and Pierre Fatou used pencil and paper to guess at a world they couldn’t yet see—the world of fractal geometry. Astoundingly, they could see fairly accurate fractal images in their minds, even though fractals require millions or billions of computations, something that wasn’t possible until the advent of computers.

Unlike traditional Euclidean geometry, based on solid lines, triangles and circles, fractal geometry points to what now seems obvious: The world is not composed of regular shapes, but features a seemingly infinite number of irregularly shaped objects and non-uniform phenomena. Many of the fractals are infinitely complex—the closer you look, the more detail you see—and they are self-similar. If you zoom in on a small portion of the image, you find a geometric pattern that resembles the whole.

“Fractals tied with chaos provide us with tools to explain natural phenomena, such as how hurricanes like Katrina develop.”

Fractals are everywhere in nature—trees, shorelines, blood vessels—and can be found in complex objects engineered by humans as well, even in stock market graphs. “A small flake of rock, when magnified, is indistinguishable from a boulder, and atoms might be said to represent solar systems,” says Douglas Martin, a fractal enthusiast who has his own website.

Euclidean geometry fueled numerous scientific and technical advances, but fractal geometry was needed to understand natural phenomena such as the spread of bacterial colonies, changes in climate, the distribution of galaxies and the grammar of DNA. Although presidential candidate Al Gore took some ribbing over who invented the Internet, scientists agree that without fractal geometry, the Internet would not exist in its present form. Fractals are key to understanding the frenetic behavior of signals linking the world’s computers, and an understanding of fractals makes it possible for engineers to compress information efficiently enough to transmit images through cyberspace.

Fractal geometry has also opened up new fractional dimensions. “In mathematics, we now talk not only about four-dimensional and million-dimensional spaces, but also 1.89-dimensional spaces,” Sekino says. Fractal geometry has opened up new vistas in music as well; witness the provocative “Bach to Chaos: Chaotic Variations on a Classical Theme,” based on fractals and composed by an electrical engineer.

Spring Reflection

Some mathematicians like to say that fractals are chaos made visible. The theory of chaos was born in 1974 when biologist Robert May used the word to describe wildly unpredictable population dynamics, based on the logistic equation. The American Mathematical Monthly published an article about chaos the following year that created such a sensation among younger mathematicians that it trickled over into popular culture. For example, in “Jurassic Park,” Steven Spielberg introduced a “chaotician” who worried about an unpredictable fluctuation in the dinosaur population. “Through fractals, we see that chaos, which appears to be—well—chaotic in terms of numbers, is actually quite orderly,” Sekino says. “Fractals tied with chaos provide us with tools to explain natural phenomena, such as how hurricanes like Katrina develop.”

When Sekino took up high-tech art 21 years ago, computers worked in slow motion, often taking an entire night to generate an image. Now his computer can process most images in several minutes, although some images are so complex they take 10 days of computing time. The black and white palette Sekino worked with when he began has evolved into a palette of 16 million colors, enough to create three-dimensional images and build luster and brilliance. Many of his artworks are reminiscent of the traditional Asian paintings he knew as a child.

Sekino has programmed fully automated fractal-plotting software with built-in equations that change input colors in a variety of ways. Using the software, he types in an equation, specifies basic colors, pushes the start button and waits for the fractal image to fill his screen. When the math professor plots a 3D image, he instructs his computer to build topography directly from a fractal using its digitized colors or to create contours around the fractal using metric topology.

Millions of people try their hand at fractals, and thousands have posted them to the web, but most artists keep their techniques and equations to themselves. Some 3D landscapes require great sophistication, and individuals and corporations jealously guard their methods. Hollywood’s special effects industry has made them extremely valuable. Sekino’s technique with coloring and contouring sets him in a class by himself, and his 3D images with an Asian flare are uniquely his own.


His sensitivity to color and shape was a gift from his father, Jun-ichiro. “On our walks in the fields, he would point out the delicate shades of blue on a butterfly against the sky or the intricate mix of lines and curves on a shrine,” he says. Most of the houses in Tokyo were burned down during World War II, leaving many open fields. “I didn’t know about war,” says Sekino, who was born in 1942. “I remember I was a happy child because I had open fields to play in with my friends. I ran around with grasshoppers and butterflies all day long.”

Sekino’s father died before he saw his son’s art, but Sekino believes his father would have been pleased. The math professor retired in May and will pour his energies into his own sons, now 10 and 12. “I want to spend a lot of time with them, before they become teenagers and stop intermingling with their parents.” He’ll play baseball with his sons, hunt for mushrooms in the mountains with his wife, and create magical places onscreen, where children of all ages can wander.

A How-To Guide to Fractals


What skills will I need?

Bone up on algebra, the geometry of complex numbers and beginning calculus, along with basic computer programming skills in a graphics mode, topology and measure theory.

How do I select colors?

According to Sekino’s website, “Each pixel color is determined by the number of iterations the corresponding sequence undertakes before escaping from a circle of a prescribed radius or before getting sufficiently near a fixed point in the complex plane.” Got that?

Dragon Rock 1

What’s the secret to a good 3D image?

If the 2D aspect is too busy, it cannot be converted to a 3D image that is aesthetically pleasing. “In 3D plotting,” Sekino says, “chaos can work against you.”

Can I predict how an image will turn out?

An accomplished artist usually has a vague idea what kind of image will emerge, but surprises are common. Simpler equations don’t necessarily produce simpler images.

What kinds of numbers are we talking about?

Many fractals emerge from an area as small as 0.000000000000000000001 times a pixel. In the future, a more powerful generation of computers will be able to unlock smaller images buried deep within today’s images, computing pixel sizes that are a trillionth of a trillionth of a trillionth.

Seahorse 2

Can I generate different images from one equation?

You can magnify different areas of the fractal to unlock an infinite number of images, or use color, contouring and other mathematical tools to discover endless possibilities.