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

Fractals are complex, never-ending patterns created by repeating mathematical equations. Yuliya, an undergrad in Math at MIT, delves into their mysterious properties and how they can be found in technology and nature.

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  • duskpin ultimate style avatar for user Jack Smith
    Did anyone notice at , that is the Star of David?
    (5 votes)
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  • blobby green style avatar for user Rafael Pous
    Infinite number of sides does not prove infinite perimeter, as the length of the sides approaches zero. A better proof is needed.
    (2 votes)
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  • starky sapling style avatar for user ForgottenUser
    How could chromatin be a fractal if it is made of a finite amount of matter? After you get down to the level of quarks, surely the pattern it creates cannot continue?
    (2 votes)
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  • blobby green style avatar for user Rafael Pous
    The first work on fractal antennas happened in Barcelona, at Technical University of Catalonia. The first paper, thesis, and patents where published there by Carles Puente and Rafael Pous.
    (1 vote)
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  • leaf green style avatar for user colin
    This does not make any sense, in terms of how the area is finite while the perimeter increases infinitely. Wouldn't the increase of perimeter in a fixed area lessen the surface area? What are the mathematical formulas and models associated with fractals?
    (0 votes)
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    • blobby green style avatar for user Tyson Dunn
      In the video, the circle drawn around the fractal is a primitive demonstration of the finite area of the fractal. As you "add triangles", you would never exceed the area of the circle, and thus we find that the fractal cannot have an infinite surface area. You could effectively (I think) calculate the limit which the fractal's surface area would approach, but due to the infinite addition of triangles, you would never be able to definitively calculate the exact surface area of the fractal. It may be possible to calculate its area with certainty to a computable number of decimal places, because at some point, any additional number of smaller triangles would not contribute any more to the larger decimal places (bringing us back to the fractal's approaching a limit).
      I think your main confusion was (and correct me if I'm wrong) that you are adding triangles infinitely, and thus adding an infinite surface area, but you must keep in mind that the triangles become infinitely small at the same time.
      (1 vote)

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.