Fractals in nature

R- Hey, smart people, Joe here. Ever notice how if you
look at part of a tree, it looks a lot like
an entire tree? And why does this
underground part of a tree look so much like
the rest of the tree? That's pretty weird. This isn't a tree, but it
sort of looks like one. And so does this, hmm. And these branches,
sure look an awful lot like these branches, except
those are blood vessels and so are these, which also
kind of look like a tree, although this part
reminds me of a river or maybe every river? Lightning, lungs,
cracks in the ceiling, what's going on here? Why do all these
things look so similar? Once you start seeing it,
you see it everywhere. It haunts your dreams! It's like there's
some spooky connection between rivers and lightning
bolts and broccoli and trees and all sorts of living
and non-living things. Well, all these objects
have one thing in common, zoom in or out, and we see
the same branching pattern repeat itself over and
over at different scales. These are fractals, a special
kind of self-similar shape that mathematicians, and the
rest of us, go extra crazy for. And this video is about
why we see them everywhere. I don't know if you've
ever looked at a tree as deeply as I have,
but that weird thing where part of the tree
also looks like a tree, that's called self similarity. It's like one of those triangles with an infinite number of
smaller triangles inside it or whatever this thing is. And unlike the self-similar
shapes we see in nature, these perfectly self-similar
shapes are infinite. We could zoom in or out and continue to see those
patterns repeat forever! Mathematician Benoit Mandelbrot named these self repeating
shapes, fractals, because they exist sort
of in between dimensions or in fractured dimensions. What the heck does that mean? Let's take a quick sidebar to talk about how the way that
mathematicians use a word, it isn't always the same as how you and I use a word. (upbeat music) You and I think of dimensions
as the three that we live in or the two that exist on paper or even the one
dimension of a line, because that's what we
learned in geometry class. What Mandelbrot meant
by, "Dimension," has to do with how
different shapes fill space as they get bigger or smaller and this is kind of
the key thing for us as we explore
fractals in nature. You can 2X the
length of this line and you get twice as much line. Another way of saying
that is you scale it up by two to the power of one. If we do the same to a square, 2X its length and width, you
get four times as much square, or you scale it up
by two to the two. Do it to a cube, 2X
length, width, and height and we get eight times as
much cube or two to the three. This power right here is the dimension Mandelbrot
was talking about and for simple shapes, it matches with our
usual idea of dimension. But what's interesting about
a fractal like this one is when you scale it up by 2X, you get three times
as much fractal. (fractal reverberating) That exponent isn't one or
two, you get 1.585 dimensions. Even though the fractal sits
in a two dimensional plane, just like a regular
triangle does, when you scale it up,
it doesn't fill space quite the same as a
two dimensional object. The same thing is true for
fractals with volume, like this. To a mathematicianologist
or whatever, it's more than two dimensional, but not quite three dimensional. Fractals exist in this
weird in-between space and that's part of what Mandelbrot found
so fascinating about 'em. By the way, you know what Benoit B.
Mandelbrot's middle name is? Benoit B. Mandelbrot. Nerdiest joke I
know right there. Anyway, Mandelbrot pointed out that fractals are not just a toy for mathematicians to
make psychedelic art for your dorm room wall. They can help us
understand nature better, because they're everywhere. To start off, why do trees
even look like trees? Well, the thing is,
biologically speaking, there's no such thing as a tree. Sure, there are things
you and I call, "Trees," because of the way they look. (buoyant music) But if you look at a
tree like this one, many of the plants
we call, "Trees," are more closely related
to things that aren't trees and more distantly
related to other things that do look like trees. So, "Tree," is just a
way of describing plants that look kind of tree-like. It's almost as if growing
fractal-like branches that look similar
at different scales was the solution to some problem that all these
different plants faced and that problem is soaking
up a bunch of sun and CO2. Growing tall is one
solution to that problem or maybe growing just
a few gigantic leaves on top of a trunk or even a
canopy the size of a city block with all the leaves
on the very tip. But all of these options require
spending a bunch of energy to grow for not that much gain, basically, you gotta
make a whole lot of wood for not that much sun. Luckily there's a
better way to do it and that's where being a
fractal is really useful. A perfect fractal lets you
put infinite surface area in a finite amount of space. This snowflake isn't
getting any bigger, but you can keep zooming in then you'll keep finding
another smaller layer just like the first. And you can keep
doing this forever, meaning its outer edge, the line you need
to draw this shape, is infinitely long. Trees do something similar, by growing out each level
as a smaller version of the previous level a tree can pack a bunch of
surface area in its volume, not an infinite amount, like a mathematically
perfect fractal, but it's a pretty cool
way of soaking up more sun without wasting energy
by getting all bulky. And it's no coincidence that trees roots grow
in a similar way, they need lots of surface area to soak up water and nutrients and fractal branching is the
best bang for their buck, maximizing the volume that
the tree can draw from without wasting unneeded energy building plumbing
that's too big. Meanwhile, inside our bodies,
we have our own little trees. A lung's job is
to take in oxygen and an adult body needs around
15 liters of O2 every hour. If our lungs were just two
balloons, they'd never keep up. Fractal branching means our
lungs can hold half the area of a tennis court
while staying packed up nicely inside our chest. (graphics whirring) (crowd clapping) And our lungs aren't the
only trees we have inside us. Our entire circulatory system looks kind of like a bunch
of fractal branches too. We have almost a 100,000
kilometers of blood vessels in our bodies delivering
oxygen and nutrients and removing wastes. Fractal branching lets
our circulatory system pack in as many blood
vessels as we need to protect every point
A with every point B, while also spending the
least possible energy building our body's plumbing and manufacturing all the
blood that runs through it. In a way, it's like each of
these living systems has a goal. A tree wants to soak up
a bunch of light and CO2, a lung wants to take
in a bunch of air, a blood vessel wants
to exchange nutrients with every cell in the body. In all these cases, fractal branches are
the most efficient way to scale up while staying
basically the same size. This secret pattern shows
up in non-living things too. All around the world, from
their sources to their ends, rivers arrange themselves
into branching shapes. And by now you can
probably guess why, at their source,
fractal branching is
the most efficient way to drain water from
a given area of land. And at their mouths we
see fractal branching as sediment piles up
and splits a river into smaller and
smaller strands. Cracks and lightning
bolts are both ways of dissipating energy and
it shouldn't surprise you that fractal branches are the
most efficient way to do that inside of a given space. And when scientists model
all these ways of growing, it turns out that, like
perfect mathematical fractals, these branching shapes
are best described as in between dimensions. At this point, it might
be tempting to think there's one universal rule that underlies every
branching fractal pattern that we see around us, but as usual, nature
isn't so predictable. We also see fractal
branches in crystals, the shapes of snowflakes,
even strange mineral deposits people sometimes mistake
for ancient plant fossils. Similar fractals, but
a different reason. Here, things like
temperature, humidity, and the concentration
of different chemicals act as a set of rules
for building the thing. And as these structures grow, those rules repeat
themselves at multiple scales giving us self-similar
fractal shapes. What's amazing is that as much as these fractal shapes
pop up in nature, there isn't a single gene
or law of physics or brain making all these things
grow fractal branches. But one by one, as each
of these systems evolved to be as efficient as possible, they all landed on
the same solution to their individual problems, letting us look at things in
an interestingly new dimension and making them
infinite