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[MUSIC PLAYING]

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PROFESSOR: Hi.

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What do snowflakes and
cell phones have in common?

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Well, let me start by
drawing a snowflake first.

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I draw an equilateral
triangle, divide each side

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into three parts, and draw
another equilateral triangle

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on top of each one.

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Then take out the middle
and repeat the process,

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this time with one,
two, three, four,

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times three, which is 12 sides.

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Eventually, this
shape will start

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looking something like this.

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In mathematics, this is
called a Koch snowflake.

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If I repeated my
process again and again,

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I would see this same
pattern anywhere I looked.

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This is a Koch snowflake,
this time drawn on a computer.

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Such never ending
patterns, that on any scale

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on any level of zoom
look roughly the same,

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are called fractals.

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Computer scientists can
program these patterns

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by repeating an often
simple mathematical process

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over and over.

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In the 1990s, a radio astronomer
by the name of Nathan Cohen

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used fractals to revolutionize
wireless communications.

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At the time, Cohen was having
troubles with his landlord--

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the man wouldn't let him
put an antenna on his roof.

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So Cohen designed a more compact
fractal radio antenna instead.

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The landlord didn't notice
it, and it worked better

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than the ones before.

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Working further, Cohen
designed a new antenna,

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this time using a fractal
called the Menger sponge.

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The Menger sponge is
not really the sponge

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you'd be scrubbing
your back with,

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but you can still
think of it like that.

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Imagine both water
and soap getting

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through your sponges holes,
except the water is Wi-Fi

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and the soap is, say, Bluetooth.

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The fractal's
infinite sponginess

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allows it to receive many
different frequencies

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simultaneously.

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Before Cohen's
invention, antennas

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had to be cut for one frequency,
and that was the only frequency

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they could operate at.

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Without Cohen's
sponge, your cell phone

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would have to look
something like a hedgehog

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to receive multiple
signals, including

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the one your friends
use when they call.

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Cohen later proved that
only fractal shapes

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could work with such a
wide range of frequencies.

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Today, millions of
wireless devices,

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such as laptops and
bar code scanners,

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use Cohen's fractal antenna.

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Cohen's genius
invention, however,

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was not the first application
of fractals in the world.

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Nature has been doing it
the whole time, and not

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just with snowflakes.

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Natural selection favors
the most efficient systems

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in organisms, often
of a fractal form.

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The spiral fractal,
for example, is

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present in sea shells,
broccoli, and hurricanes.

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The fractal tree is
relatively easy to program,

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and can be used to study
river systems, blood

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vessels, and lightning bolts.

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So many natural
systems previously

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thought off-limits
to mathematicians

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can now be explained
in terms of fractals.

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Mathematics allows us to
learn nature's best practices

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and then apply them to
solve real world problems.

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Much like Cohen's antenna
revolutionized the field

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of telecommunications,
other fractal research

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is changing medicine,
weather prediction,

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and building design here at MIT
and everywhere in the world.

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Look around you.

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What beautiful
patterns do you see?