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## Math

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# What is a fractal (and what are they good for)?

## Video transcript

[MUSIC PLAYING] [SINGING] Science out loud. What do snowflakes and
cellphones have in common? The answer is never ending
patterns called fractals. Let me draw a snowflake. I'll start with an
equilateral triangle. Then I'll draw another
equilateral triangle on the middle of each side. Pull out the middle and repeat
the process, this time with 1, 2, 3, 4 times 3,
which is 12 sides. If I do this over
and over, the shape will look something like this. This is called a Koch snowflake,
and it has a special property. No matter where I look
or how much I zoom in, I will see the same
pattern over and over. Never ending patterns like
this that on any scale, on any level of zoom
look roughly the same are called fractals. We can actually draw a Koch
snowflake on the computer by having it repeatedly graph
a mathematical equation. Each time we add a triangle,
one side of the Koch snowflake will turn into four. After the first repetition,
we'll get three times four to the first, or 12 sides. After the second repetition,
we'll get three times four to the second, or 48 sides. After repetition number n,
we'll have three times four to the n sides. If we do this an
infinite number of times, we'll get infinitely many sides. So the perimeter of the Koch
snowflake will be infinite. But the area of the Koch
snowflake wouldn't be infinite. If I draw a circle with a finite
area around the snowflake, it will fit completely inside
no matter how many times we increase the number of sides. So the Koch fractal has
an infinite perimeter, but a finite area. In the 1990s, a radio
astronomer named Nathan Cohen used the
fractal antenna to rethink wireless communications. At the time, Cohen's
landlord wouldn't let him put a radio
antenna on his roof, so Cohen decided to make a more
compact, fractal like radio antenna instead. But it didn't just hide the
antenna from the landlord. It also seemed to work
better than the regular ones. Regular antennas have to be
cut for one type of signal, and they usually work
best when their lengths are certain multiples of
their signals' wavelengths. So FM radio antennas can only
pick up FM radio stations, TV antennas can only pick
up TV channels, and so on. But fractal antennas
are different. As the fractal repeats
itself more and more, the fractal antenna can pick up
more and more signals, not just one. And because the perimeter
of the Koch snowflake grows way faster than its
area, the fractal antenna only takes up a quarter
of the usual space. But Cohen didn't stop there. He designed a new antenna, this
time using a fractal called the Menger sponge. The Menger sponge is kind of
like a 3D version of the Koch snowflake and has infinite
surface area but finite volume. The Menger sponge is sometimes
used in cellphone antennas. It can receive all
kinds of signals while taking up even less
area than a Koch snowflake. Now, these antennas
aren't perfect. They're smaller, but
they're also very intricate, so they're harder and
more expensive to make. And though low
fractal antennas can receive many different
types of signals, they can't always receive
each signal as well as an antenna that was cut for it. Cohen's invention was not the
first application of fractals. Nature has been doing
it forever, and not just with snowflakes. You can see fractals in river
systems, lightning bolts, seashells, and even
whole galaxies. So many natural systems
previously thought off limits to mathematicians can now be
explained in terms of fractals, and by applying
nature's best practices, we can then solve
real world problems. Fractal research is changing
fields such as biology. For example, MIT
scientists discovered that chromatin is
a fractal, and that keeps DNA from getting tangled. Look around you. What beautiful
patterns do you see? Hi. I'm Yulia. Thanks for watching
"Science Out Loud." Check out these other awesome
videos, and visit our website. Good. Now wait for it. OK. That's it.