Remembering The Father Of Fractals
Measuring the scale of Benoit Mandelbrot's achievements
by
Carrie Arnold l ISNS Contributor
WASHINGTON (ISNS) - Benoit Mandelbrot, the
mathematics professor at Yale who coined the word "fractal," passed
away on October 14 at the age of 85. His death recalls the complicated
history of his life's work -- the details of which, like fractals
themselves, depend on how closely one looks.
Some varieties of cauliflower are fractals
Credit: ElbtheProf
After fractals were first described by Mandelbrot in 1967, they
languished unappreciated for decades, much to his consternation. Today,
though, researchers today use this mathematical concept to better
understand everything from the stock market and the human heartbeat to
earthquakes and mobile phone antennae.
"He was a great mathematician," said Harvard University cardiologist Ary
Goldberger, "but he was a polymath who was interested in all sorts of
things."
Fractals are geometric shapes that can be broken down into smaller
parts, each of which resembles the whole -- like broccoli florets or
branches on a bolt of lightning. Mandelbrot first developed the
mathematics behind fractals in order to answer a simple question: how
long is the coastline of Britain?
He imagined measuring the coastline by laying yardsticks end-to-end
around the perimeter of the island and counting how many would be needed
to encircle it. He then imagined repeating the same process but
measuring with a stick only 2 inches long. This second measurement would
be longer because a shorter stick can measure smaller indentations in
the coastline.
Mandelbrot's revelation, published in a 1967 paper, was this: You can't
accurately measure the coastline. Its length depends on how closely you
look. Out of this paradox, he created a new way of looking at
mathematically-difficult phenomena that researchers have continued to
explore and develop.
"You get this mysterious combination of variability and organization that is mathematically describable," said Goldberger.
The
late mathematician Benoit Mandelbrot Credit: Rama
Goldberger uses fractals to help define a healthy human heartbeat.
Physicians once thought that a healthy heartbeat should be as steady as a
metronome, but heart traces, or EKGs, have revealed that healthy hearts
are actually much more irregular. Instead of a metronome, Goldberger
said, they're more like a symphony. Irregularity, he said, "is where
physiology meets fractality." He believes that in order to adapt to the
environment, our bodies can't be locked into one mode of functioning.
Variability in heartbeats, he said, is essential to life and repeats itself across different scales.
The peaks and valleys of an EKG look the same over 10 minutes as they do over 10 milliseconds.
This similarity of fractals over different timescales also exists in
patterns found in landscapes and geography. That's why photographs of
rocky terrain often contain a reference object, such as a Swiss Army
knife, to provide a sense of scale. Without the knife, it would be
impossible to tell the size of the rock.
"That's what 'scale invariance' is," said geologist Donald Turcotte of
the University of California, Davis. "Everything looks the same," and
you can't tell whether you are looking at a one square centimeter of
rock or a one kilometer landscape, he said.
Besides describing the appearance of the earth's surface, the
mathematics of fractals also help scientists to better predict the
frequency of earthquakes, floods, and other natural disasters.
Mathematical models that don't use fractals tend to forecast far fewer
major natural disasters than actually occur, said Turcotte. By using a
fractal models, however, geologists have been able to obtain more
accurate predictions of the frequency and severity of such natural
disasters.
Yahya Rahmat-Samii, an electrical engineer at the University of
California, Los Angeles, uses fractals to improve cell phones' ability
to pick up signals. Mobile phone antenna once picked up only one radio
wave frequency. In order to pick up rather faint signals, the antenna
itself had to be rather large.
Fractals repeat the same pattern over and over again on ever-smaller scales
Credit: Nevit Dilmen
In the mid-1990s, though, engineers discovered that bending an antenna
into a fractal-like shape enabled a miniature antenna to pick up an
array of signals.
Each portion of the fractal could be designed to pick
up a different frequency, which has allowed cell phone companies to
provide Bluetooth and Wi-Fi capabilities (all of which operate at
different radio frequencies) on the same phone.
The next step in fractal
antenna design, said Rahmat-Samii, is continued miniaturization.
"There's a lot of room for miniaturization," Rahmat-Samii said, "because
fractals come with so many different features, and there are likely
some that have not been exploited effectively."
All three of these researchers have met Benoit Mandelbrot, and all three
describe the late mathematician as having had a defining influence on
their life's work. They also acknowledge that he was a character who at
times could be as complicated as his mathematics.
"A lot of people didn't like him," Turcotte said. "He was extremely
arrogant and a bit prickly. But he introduced this concept [of fractals]
in the 1960s, and he didn't receive recognition for another 20 years.
So if you've been fighting a battle for 20 years, I think you have the
right to resent things a bit."
