Fun with Fractals


Today let’s have some
fun with fractals. To start with,
what are fractals? You’ve probably seen tons
of fractals in nature, but you just did not know
that they were fractals. One example is a
fern in this image. Here’s what a whole
fern looks like. Now if we zoom into
one leaf, look, it looks like the whole fern. Now let’s zoom in even more. And look at this. It looks like the fern again. Some flowers are also fractal. At first sight if we see
this bunch of flowers we can see that the bunch is
made up of many small flowers. But actually, if
you look closer, each of the small
flowers is made up of even smaller portions. Just the same way that
each portion makes up the small flowers,
each small flower also makes up the whole bunch. So each of these
smaller portions is in fact similar to
each of the flowers, and even to the bunch. This is also an example
of a fractal pattern. Another fractal is a broccoli. And don’t we all
really love broccoli? As you can see, each of
the heads of the broccoli resembles the
vegetable as a whole. Now that you’ve seen these
examples of fractals, you can probably spot your next
fractal in nature by yourself. So what exactly are fractals? The word fractals comes from
the Latin verb to break. Fractals are geometric
patterns in which every smaller part
of the structure is similar to the whole. How do we actually
create a fractal? Let’s first look at the
Sierpinski Triangle. It does look complicated,
but we can create this in a few simple steps. Let’s start with
the black triangle. We can mark the center of
each side of the triangle, and connect the dots to make
a new upside down triangle. We then remove this triangle
and are left with three smaller, black triangles instead of one. We take the centers of each
of the three new triangles and again connect the
dots to make three new upside down triangles. Let’s remove the upside
down triangle at the top. Look, we get three new triangles
instead of the one at the top. We can now remove the
one at the bottom left, and, finally, the one
at the bottom right. You’ll notice that each time
we remove an upside down triangle we get three
new black triangles. So now we have a total of nine
even smaller black triangles. Let’s see the triangles that we
will remove in the next step. Now we can again take the
top part of the big triangle and remove the upside
down triangles. We can continue on to
the bottom left, and then the bottom right. We already have 27
black triangles. If we keep going we can make
more and more black triangles at each step. And if we keep
doing this forever we’ll have an infinite
number of triangles, and all inside the one big
triangle we started with. Let’s have another example. This one is called
the Von Koch Curve, named after the Swedish
mathematician who came up with it. This time we start with a
line and divide it into three. We mark these points,
then take out the middle. We take a line of
equal length to the one we just removed and put it
at an angle of 60 degrees. We can now connect
the two lines, and this line happens
to be equal in length to the line we just removed. So instead of one
middle line, we have two new ones
at the same length, making the curve longer. And notice that instead
of one line segment, we now have four line segments. Now take each of these four new
lines, divide them into three, and take out the middle
from each of them. Starting from the left let’s
put in two new lines instead of the middles. You’ll see that
each of these lines is equal to the length of
the ones we just removed, which means that the length
of the curve increased again. We already have 16 line
segments to work with. Let’s take each of these
and repeat the same steps. It’s starting to look pretty. You can see that at every
step each time we add segments we make the curve longer. If we keep repeating
the steps we make the curve
longer and longer. Since we can keep
repeating this forever, the length of the
curve becomes infinite. This is what makes
fractals special. Now we can move on to a more
complicated and more beautiful structure, the Mandelbrot Set. As we can see if we
zoom in, we recognize the original structure again. If we zoom in again we again
recognize the same structure as the original image. This keeps happening as we
zoom in again and again, and at each step the image
retains its beauty as well. In this video we saw
the beauty of fractals. We showed you how to
recognize them in nature, and even how to create
some simple fractals. We hope you had some
fun with fractals.

About the author

Comments

  1. check out the paisley patterns made in cloth ,its in the brain, hot wired long before computer imagine-it would take a life time to work out on paper this is old maths before technology-blows me,its all thrue nature.

  2. What is "linth"? She says it quite a bit. Btw dont bother responding because I know the answer, I'm just being an ass.

  3. How could someone down vote this? what were they looking for or let down with Fun with Fractals?

    Good job, took a complicated mathmatical concept and explained it while still being fun and SHORT. The expediency of the explanation is almost as interesting as the video.

  4. The word "fractal" comes from contracting "fractional" – because it refers to a geometric object with fractional dimension.
    The triangular Swiss cheese, and the van Koch fractal, e.g., each have dimension between 1 and 2, because each has infinite length, but 0 area, all within a finite 2-ball (circular disk).
    BTW, your justification for infinite length of the van Koch fractal is non sequitur. Just because it gets longer at each of infinitely many steps, doesn't guarantee that its length is infinite.
    But the fact that at each step its length gets multiplied by 4/3 > 1, does, because, as n→∞, so also (4/3)ⁿ→∞.

  5. Dumb as shit comments section omfg… broccoli, triforce, "fractals typically not self similar", try to be goddam original for crissake.

  6. 1:51 the Triforce can teach you about fractals kids, so use this as a reason for getting a legend of Zelda game.

    Actually, please don’t, because this obviously won’t work.

  7. its not broccoli… its a cauliflower. Cauliflower heads resemble those in broccoli, which differs in having flower buds as the edible portion.

  8. 1:56 what happens to the triforce when ganon touches it. It shatters into billions of pieces. Great job now link has to fish up about a bazillion more shards of the triforce.

  9. thank you for the explanation but the vid is poorly titled and the person narrating really sounds like someone reading a paragraph or two. You didn't tell me how to have fun with them either —-I was waiting the whole time. It is the only reason I watched. Why didn't you call it "a quick explanation on what a fractal is"?

Leave a Reply

Your email address will not be published. Required fields are marked *