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)