200 years ago, William Blake
wrote this verse: “To see a World in a Grain of Sand And a Heaven in a Wild Flower, Hold Infinity in the palm of your hand And Eternity in an hour.” Now, poetry is often
subject to interpretation. But I think we can all agree that this
poem is about fractals. And that’s pretty impressive because fractals weren’t discovered
until 170 years later. But Blake’s poem
captures something essential about the natural world around us. It draws a connection between
the very large and the very small, and it observes that similar patterns
can occur at vastly different scales, across time and space. And we see this in the universe around us:
similar patterns, from the shape of immense spiral galaxies to the weather patterns of a hurricane to little eddies in a stream. And whenever you have a single object
that contains these patterns repeating over and over again
at many different scales, and where every small part
resembles the whole, that’s a fractal. And we see them around us everywhere,
from the jagged shapes of lightning bolts to the rough, ragged edges of coastlines. But fractals are not just outside of us. Our lungs are fractals. That’s how they manage to pack
the surface area of a tennis court folded down into your rib cage. And our circulatory system
is a fractal as well, which is how 60,000 miles
of branching blood vessels and capillaries fit inside every human being. So I’m able to stand here
and talk to you today, because I’m a living, breathing fractal... and so are you. But for hundreds of years, these complicated geometric shapes
were swept under the rug. (Laughter) They were considered too complicated
to mathematically analyze, because they don’t behave like circles
or squares or triangles. And the assumption was
that any explanation for these shapes would have to be as complicated
as the shapes themselves. Because ordinarily, simple things
have simple explanations and complex things
have complex explanations. But every once in a while something comes
along that breaks the mold. Take the universe: 100 years ago,
there was a crisis in physics. Measurements of the natural world
weren’t working out the way that scientists expected. And physicists
were tying themselves in knots coming up with more and more
convoluted theories, trying to explain away
these discrepancies. But then this guy showed up
with a very simple equation. And this led to the discovery
of the theory of relativity that opened up an entire new era
in physics. 70 years later, geometry was in a similar crisis, because mathematicians
still couldn’t explain the properties of these complex shapes. But then this mathematician,
Benoit Mandelbrot, found himself studying
an equally simple equation. And from this simple equation, this unlocked the mysteries
of these complex shapes. And it led to the development
of the entire new fields of chaos theory and fractal geometry. So let's take a look at this equation. z goes to z squared plus c. And the way that this works
is that you take a small number, you square it,
you add the original number, then you feed that
back into the equation, square it, add the original number, and you process it over and over, each time feeding back in
the original number to keep the process going. And for some numbers,
not much happens, it just is small. For other numbers,
it just bounces back and forth. But for other values,
it grows very quickly, and it quickly escapes to infinity. And it’s possible
to compute this by hand for 10 or 100,
or 1,000 different numbers. But to get a true idea
of what’s really going on, you need to do this with a computer for millions and millions
of numbers in two dimensions. And when you do that,
and plot the results, you get this fantastic shape: the Mandelbrot set, a shape of infinite complexity generated from such
an incredibly simple formula. This shape changed my life. Now, my parents’ first inkling
that I was mathematically inclined was when I was quite small. And I announced to them
that I was going to count to infinity. And so they humored me and they bought me
a 70 page notebook. And I got started. And it was going along pretty well. I think I was actually getting
most of the way there. (Laughter) And so as you can imagine, when I discovered that I could make
a computer do this sort of thing for me, and also produce images as complicated and as amazing
as the Mandelbrot set, I was completely hooked. I would write a computer program,
go off to summer camp for a week, and run back to find
that it had almost finished drawing one picture of a fractal. This image was from a paper I wrote
in the ninth grade, and it literally took the computer
a week to draw. But in that paper,
I had this crazy idea. I thought that if you could make
enough of these pictures and put them back to back,
you could make a movie of flying through the Mandelbrot set. Of course, in the 1980s, that was a crazy pipe dream
on a home computer. But fortunately, computers have gotten
a little faster since then. For perspective, in the Apollo era,
in the 1960s, the sum total of all
the computing power in the world was about 10 gigaflops,
about 10 billion instructions per second. By the late 90s, IBM was able to pack
all of that power into a single supercomputer
the size of a refrigerator. By 2011, it could fit into your pocket. And extrapolating ahead
just a few years, all that power will be able to fit
into a single grain of sand. So let’s do that zoom
into the Mandelbrot set. And now, it can be done
in real time on a cell phone. And so I’ll zoom in just a little bit, and by a little bit,
I mean a factor of 100 million, because mathematically,
there’s no reason to stop there, this shape literally goes on forever. You really can hold infinity
in the palm of your hand. And in addition to all
of the geometric complexity, you can see the self-similarity, that burried deep down
within the Mandelbrot set, are little copies of the original shape. In 2010, Mandelbrot himself
gave a talk at the TED Conference in Long Beach and I was lucky enough to be there
and to get to meet him. And as you can imagine,
that was a real TED moment for me. (Laughter) And I think he would have appreciated
that very deep down within my fractal talk is a little copy of his fractal talk. (Laughter) (Applause) And so I showed him what I was working on, this program on the cell phone
to fly into the Mandelbrot set. And he was amazed. He said it was superb that such a thing
could now be done on such a small device. And it felt like the passing of a torch. And I felt like I should run with that. And I wanted to run with that
as far as I could. Now, the Mandelbrot set
is very beautiful all by itself. But when you look at
natural fractal shapes that occur in the real world, they have color and shading,
lighting and texture. And I wanted to bring that
into the images that I was creating. And I found that if I took the space
surrounding the Mandelbrot set, and distorted it, using techniques similar
to that used for paper marbling, not only it would give it much
more visual richness and complexity, but it also reveals more of
what’s happening mathematically in the area outside the shape. Just as when you put smoke
into a wind tunnel, it reveals patterns in the air flow that would otherwise be invisible. And so using this technique, here are just a few
of the textures and patterns that we’ve uncovered
hidden in the math. Occasionally, someone will tell me
that these images belong in a museum. But I think that's a terrible idea. Because at the museum,
the first thing they tell you is, don’t touch the art. But we’re all walking around now
with touchscreens, supercomputers in our pockets. Fractals should be all about
touching the art. It should be an interactive,
immersive experience. And so that's what I set out to create. Working with the legendary
graphical user interface designer, Kai Krause, and another fractal expert,
Tom Beddard, this is the app that we came up with. And you might notice
that something is missing. There are no numbers,
or equations, or scary math. And that's very intentional. Because we do want to get people
excited and interested about math. And we think that
the best way to do that is not to shove a bunch of math
in their faces. But to show them something inspiring
that can be done with math and let them discover the inspiration
all for themselves. And so this is the app in action. And when you’re actually
interacting with these fractals, the interface completely goes away. And you can just interact
completely immersively. And even when you
zoom into the fractal, you don’t even need to touch
to tell it where to go. You can just tilt your phone to steer, and it feels just like you’re flying. And you can change the colors as well, pinching and stretching
the color gradients around the fractal, swiping to animate them, rotating to change the colors
and the hues. Or randomizing your way through any number
of different color palettes until you find one that you like. And you can change the lighting as well
in a similar way. Move the lights around,
swipe to animate them. Pinch and stretch to make the fractal
more or less depthy. And my favorite is the texture, which you can interact with
directly as well, changing the surface
characteristics and properties, stretching and twisting it, and also randomizing through
many different variations on a theme. And a whole community has built up
around these tools of artists from all over the world. And we’re consistently amazed
by their artistry and creativity. But fractals also tend to draw a crowd. A few years ago,
I was invited out to Burning Man to show these fractals on a big video wall
in the middle of the desert. And people in the audience could come up and control and steer the fractals
themselves from a tablet. And of course, being Burning Man,
I had to make a costume. (Laughter) So this is
an electroluminescent Mandelbrot stitched on to the back of a cape. Because fractals and Burning Man
go together perfectly, because Burning Man is psychedelic
and fractals are psychedelic. But why is there this automatic
association between fractal images and mind altering substances? I have a theory about this. Don’t worry, Mom, it’s just a theory. (Laughter) And that is, ordinarily when you’re
looking at something or someone, or a face, for instance, you don’t consciously perceive
every last detail all at once, every pore, every eyelash. Your brain is constantly simplifying
and filtering what you’re looking at to make it more comprehensible. But people who’ve had
psychedelic experiences sometimes say that it feels as if
those filters are removed, and all of the information
gets through at once. And it's sort of sensory overload. And I suspect that when you’re looking
at fractals like this, that the reason it has a similar effect
is that they’re so complicated, your brain doesn’t know
how to simplify them. And so the filters don’t work and all the information
gets straight through. And it creates an overwhelming
sensory experience. And there’s some interesting
implications of this. Because recently, there have been some
preliminary studies that seem to show that controlled doses
of chemicals like psilocybin can actually have a very measurable
positive effect for some patients suffering
from PTSD or depression. And if these fractal images are activating some of
the same areas of the brain, it’s possible they might have
some of the same beneficial effects. And although it’s anecdotal, some of the feedback
that we’ve received seems to bear this out. [“I am very much impressed with your app. As a disabled person and
100% disabled Vietnam vet, anything that helps me is important.”] [“I’m a Speech-Language Pathologist that works a lot with kids
on the autism spectrum and other sensory needs. We’ve been using your app like crazy. It’s really hard to get them
to stop, actually.”] [“Due to a failing Autoimmune System, I constantly suffer from chronic pain... I use this kind of program to escape the grip and boundaries
my inflicted pain keeps me in. I just wanted to let you know
how much your app (Frax HD) has seemed to help me
in my personal struggle.”] And so if fractals can affect the brain
to this extent, that leads to another
interesting question. Because we know that
the lungs are a fractal. And the circulatory system is a fractal. Could the brain be a fractal as well? One way to think about this question is through the lens of artificial
intelligence and machine learning, because these are some of the best tools
that we have to model and simulate
what’s going on inside the brain. And some of the most successful
of these deep neural networks operate on a principle
where they’re processing information over and over again,
while periodically feeding back in the original information
to keep the process going. And that might sound
a little bit familiar, because it actually is quite similar to the way that Mandelbrot’s
fractal equation works. And what that suggests is that, whether or not the brain is a fractal, it’s something much more powerful. It’s an engine for generating
fractal levels of complexity. So it could be that the brain
is not a fractal, but the mind is. And perhaps that's why these fractal
images resonate so deeply with us. It’s because when we’re looking at them, in a way, we’re seeing
a reflection of ourselves. And it suggests that these images
might have another practical use, which is to shed further insight
onto the mysteries of human perception and consciousness. And so these fractals have
so much to teach us. They're a constant source of inspiration
about science and math. They remind us that when we’re looking at
something extremely complicated, the key to understanding it
might turn out to be something surprisingly simple. And when I’m looking at these images, just as William Blake envisioned, I do see eternity in an hour
and the world in a grain of sand. And I hope you do too. Thank you. (Applause)
