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)

I feel at least our brain is a fractals creating machine, through our imaginations. Which branches and re branches billions of time in our conscious and subconscious mind, but still keeps the integrity of our human body and capacity. If this holds the natural cure for mental and emotion illnesses then it's a blessings we are born with.

I learned alot from this Ted Talk, which is exactly what they are for. I understand so much more when scientists speak human. Showing us how much more wonderful our world is than even our imagination can be. Amazing how something so simple is also so complex. I LOVE FRCTILES!

how is it possible that this video has only 1.4k views and 3 comments…? this blew my

mind

Wow. My three favourite things.. Fractals, psychedelics and neural networks/machine learning all in one TED video? Unreal!! This talk just got better and better every minute. I hope I can join you guys at google some day. Thank you mr. Weiss!!!

I love this.

the only thing that annoys me is the lip smacking its like

LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACK LIP SMACKdude’s hot🧚♀️

water

Someone get this man some water and s chill pill. jUsT relax dude! Fractals aren’t going anywhere !

The definition of dimension used by Mandelbrot in his first full book is a logarithm. Logarithms are defined on the ENTIRE REAL NUMBER SYSTEM; yes, they include the negative numbers and all the irrationals and transcendentals, ALL "real numbers".

The primary purpose for studying algebraic theory is the principle if you can show a given set fulfills all requirements to be logically equivalent to another type of set, say positive whole numbers, you do not need to prove 1 unit plus 1 unit of your set equals 2 units; proving the equivalency of your set to whole positive numbers is all you need.

All arithmetic that can be done with real numbers can be done with 2 different Mandelbrot sets. We don't have to come up with examples to know this, algebraic theory affirms it.

The "infinitely long surface" on a meter of rough-cut wood becomes smoother, a lower fractal, by the 'subtraction' of sanding.

The sanding action—the making the wood's surface smoother—can be thought of as subtracting the fractal value of the sandpaper from the fractal value of the rough wood. The more sanding, the more subtracting, until the fractal for the

rough wood surface approaches closer and closer to 1, a completely smooth surface. (It never reaches D = 1, molecules are 'bumpy')

Over sand, you will make a hole in the surface (Zero fractal, zero dimension).

Keep sanding like a mindless robot, you go beneath the original surface. The "negative surface" of the sandpaper.

The world can be seen as interacting fractals. You can see the wind acting on sandstone, waves interacting with shore, roots working through soil, a caterpillar eating a leaf, you can see the whole universe as interacting fractals.

That is only the beginning.

Complex Numbers, the "imaginaries", were discovered via algebra. Euler never feared using them exponentially, hence his famous identity. Logarithms are just exponents rearranged.

Dare to create a fractal defined on complex numbers: You will reinvent all the "weird" aspects of quantum mechanics, including Bell's Interconnectedness theorem and quantum entanglement.

The next line of Blake's poem: "To hold infinity in an hour" maybe he was walking along that infinite real-world curve defined by the grain of sand, an infinite trip.

word to the mushrooms .

Watching fractals puts me into a meditative state quickly.

Gotta try this app on lsd

Is there such thing as Reverse Fractal?

As above, so below.

my brain tickles while watching the fractals! AMAZING!

my favorite strategy in trading is fractals, becoz people are fractals

"Jooish" "space" propaganda.

Einstein destroyed physics

Hey this is awesome

Has anyone tried the math in 3d or tried to calculate it with time added in.

This dude is satisfying

Cant find the app, what's name of it?

Those comments are so fake hahaha

The mind is absolutely a fractal. One thought does not become another thought: it becomes ten, a hundred, a thousand other thoughts, ad infinitum.

His voice is so soothing

Did have Frax lite. Now I bought the app.

Life goes so deep. Yet we worry about the smallest things

Was quite interested until i realised this "Frax HD" app created by this guy is Apple store only. Meh whatever

Well. I'm tripping balls right now

Is there a way that I can come up with my own fractal, using numbers that I picked!

I’m downloading your App as we speak!

Autistic people have sensory overload — Mandelbrot was definetly on the autism so was Einstein, and Issac Newton.

10:54

12:07

So NOBODY noticed that the shape his computer made was the same as 6:30? my mind was blownnn

Time is a fractal

I want to create Korean subtitle on this. How can I do this?

I absolutely love it! Thank Ben for all that you have done and continue to do for all of us!

Can't wait to view this on 3D goggles/cardboard!

GOD, GOD, GOD, GOD, GOD, ad infinitum…..