[MUSIC PLAYING]
PROFESSOR: Hi.
What do snowflakes and cell phones have in common?
Well, let me start by drawing a snowflake first.
I draw an equilateral triangle, divide each side
into three parts, and draw another equilateral triangle
on top of each one.
Then take out the middle and repeat the process,
this time with one, two, three, four,
times three, which is 12 sides.
Eventually, this shape will start
looking something like this.
In mathematics, this is called a Koch snowflake.
If I repeated my process again and again,
I would see this same pattern anywhere I looked.
This is a Koch snowflake, this time drawn on a computer.
Such never ending patterns, that on any scale
on any level of zoom look roughly the same,
are called fractals.
Computer scientists can program these patterns
by repeating an often simple mathematical process
over and over.
In the 1990s, a radio astronomer by the name of Nathan Cohen
used fractals to revolutionize wireless communications.
At the time, Cohen was having troubles with his landlord--
the man wouldn't let him put an antenna on his roof.
So Cohen designed a more compact fractal radio antenna instead.
The landlord didn't notice it, and it worked better
than the ones before.
Working further, Cohen designed a new antenna,
this time using a fractal called the Menger sponge.
The Menger sponge is not really the sponge
you'd be scrubbing your back with,
but you can still think of it like that.
Imagine both water and soap getting
through your sponges holes, except the water is Wi-Fi
and the soap is, say, Bluetooth.
The fractal's infinite sponginess
allows it to receive many different frequencies
simultaneously.
Before Cohen's invention, antennas
had to be cut for one frequency, and that was the only frequency
they could operate at.
Without Cohen's sponge, your cell phone
would have to look something like a hedgehog
to receive multiple signals, including
the one your friends use when they call.
Cohen later proved that only fractal shapes
could work with such a wide range of frequencies.
Today, millions of wireless devices,
such as laptops and bar code scanners,
use Cohen's fractal antenna.
Cohen's genius invention, however,
was not the first application of fractals in the world.
Nature has been doing it the whole time, and not
just with snowflakes.
Natural selection favors the most efficient systems
in organisms, often of a fractal form.
The spiral fractal, for example, is
present in sea shells, broccoli, and hurricanes.
The fractal tree is relatively easy to program,
and can be used to study river systems, blood
vessels, and lightning bolts.
So many natural systems previously
thought off-limits to mathematicians
can now be explained in terms of fractals.
Mathematics allows us to learn nature's best practices
and then apply them to solve real world problems.
Much like Cohen's antenna revolutionized the field
of telecommunications, other fractal research
is changing medicine, weather prediction,
and building design here at MIT and everywhere in the world.
Look around you.
What beautiful patterns do you see?