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00:00you can find it in the rainforest on the frontiers of medical research in the movies and it's all
00:11over the world of wireless communications one of nature's biggest design secrets has finally
00:18been revealed my god of course it's obvious it's an odd-looking shape you may never have
00:25heard of but it's everywhere around you the jagged repeating form called a fractal they're all over in
00:33biology their solutions that natural selection has come up with over and over and over again
00:40fractals are in our lungs kidneys and blood vessels flowers plants weather systems the rhythms of the
00:50heart the very essences of life but it took a maverick mathematician to figure out how they work
00:58I don't play with formulas I play with pictures and that is what I've been doing all my life his
01:06was a bold challenge to centuries-old assumptions about the various forms that nature takes the
01:14blinders came off and people could see forms that were always there but formerly were invisible
01:21making the invisible visible finding order in disorder what mysteries can it help us unravel
01:33coming up next on NOVA hunting the hidden dimension
02:00major funding for NOVA is provided by the following
02:03in 1978 at Boeing aircraft in Seattle engineers were designing experimental aircraft
02:33exotic things with two wings or two tails or two fuselages and just weird stuff because who knows
02:39it might work a young computer scientist named Lauren Carpenter was helping them visualize what
02:46the planes might look like in flight I would get the data from them and make pictures from various angles but
02:53I wanted to be able to put a mountain behind them because every Boeing publicity photo in existence has a
02:58mountain behind it but there was no way to do mountains mountains have millions and millions of
03:02little triangles under polygons or whatever you want to call it and we had enough trouble with a hundred
03:06especially in those days when our machines were slower than the ones you have in your watch
03:10Carpenter didn't want to make just any mountains he wanted to create a landscape the planes could fly through but there was no way to do
03:20that with existing animation techniques from the time movies began animators had to draw each frame
03:28by hand thousands of them to make even a short cartoon
03:32that's why they call me thumper
03:36but that was before Lauren Carpenter stumbled across the work of a little known mathematician named Benoit Mandelbrot
03:44in 1978 I ran into this book in a bookstore fractals form chance and dimension by Benoit Mandelbrot and it has to do with the fractal geometry of nature
03:53so I bought the book and took it home and read it cover to cover every last little word including footnotes and references twice
04:00in his book Mandelbrot said that many forms in nature can be described mathematically as fractals a word he invented to define shapes that look jagged and broken
04:12he said that you can create a fractal by taking a smooth looking shape and breaking it into pieces over and over again
04:21Carpenter decided he tried doing that on his computer
04:26within three days I was producing pictures of mountains on my computer at work
04:32the method is dead simple you start with a landscape made out of very rough triangles big ones and then for each triangle
04:40break it into into four triangles and then do that again and again and again and again and again
04:45endless repetition what mathematicians call iteration it's one of the keys to fractal geometry
04:53the pictures were stunning they were just totally stunning no one has ever seen anything like this
05:01and I just opened a whole new door to the new world of making pictures
05:05and it got the computer graphics community excited about fractals because suddenly they were easy to do
05:12and so people started doing them all over the place
05:16Carpenter soon left Boeing to join Lucasfilm
05:20where instead of making mountains he created a whole new planet for Star Trek 2 The Wrath of Khan
05:27it was the first ever completely computer generated sequence in a feature film
05:34Fascinating
05:38made possible by the new mathematics of fractal geometry
05:43Benoit Mandelbrot whose work had inspired that innovation was someone who prided himself on standing outside the mainstream
05:54I can see things that nobody else suspects until I point out to them
06:00oh of course of course but they haven't seen before
06:03you can see it in the clouds in the mountains even inside the human body
06:10the key to fractal geometry and the thing that evaded anyone until really Mandelbrot sort of said this is the way to look at things
06:18is that if you look on the surface you see complexity and it looks very non mathematical
06:24what Mandelbrot said was that think not of what you see but what it took to produce what you see
06:33it takes endless repetition and that gives rise to one of the defining characteristics of a fractal
06:39what mathematicians call self similarity
06:44the main idea is always as you zoom in and zoom out the object looks the same
06:50if you look at something at this scale and then you pick a small piece of it and you zoom in
06:55it looks very much the same
06:58the whole of the fractal looks just like a part
07:01which looks just like the next smaller part
07:04the similarity of the pattern just keeps on going
07:09one of the most familiar examples of self similarity is a tree
07:16if we look at each of the nodes the branching nodes of this tree
07:20what you'll actually see is that the pattern of branching is very similar throughout the tree
07:25as we go from the base of the tree to higher up
07:29you'll see we'll have mother branches then branching then into daughter branches
07:34if we take this one branch and node and then go up to a higher branch or node
07:40what we'll actually find is again that the pattern of branching is similar
07:45again this pattern of branching is repeated throughout the tree
07:49all the way ultimately out to the tips where the leaves are
07:54you see self similarity in everything from a stalk of broccoli
07:59to the surface of the moon
08:01to the arteries that transport blood through our bodies
08:05but Mandelbrot's fascination with these irregular looking shapes
08:10put him squarely at odds with centuries of mathematical tradition
08:14in the whole of science
08:17the whole of mathematics
08:19the smoothness was everything
08:21what I did was to open up roughness for investigation
08:28we use mathematics to build the pyramids to construct the path and on
08:34we use mathematics to study the regular motion of the planets and so forth
08:38we became used to the fact that certain patterns were amenable to mathematics
08:44the architectural ones but largely the patterns of human made structures
08:48where we had straight lines and circles and the perfect geometric shapes
08:54the basic assumption that underlies classical mathematics
08:57is that everything is extremely regular
08:59I mean you reduce everything to straight lines
09:02circles, triangles, flat surfaces
09:06pyramids, tetrahedron, acosahedron, nodecahedron, smooth edges
09:11classical mathematics is really only well suited to study the world that we've created
09:18the things we've built using that classical mathematics
09:20the patterns in nature
09:23the things that were already there before we came onto the planet
09:27the trees, the plants, the clouds, the weather systems
09:30those were outside of mathematics
09:32until the 1970s
09:37when Benoit Mandelbrot introduced his new geometry
09:41Mandelbrot came along and said
09:43hey guys
09:44all you need to do is look at these patterns of nature
09:48in the right way
09:49and you can apply mathematics
09:51there is an order beneath the seeming chaos
09:54you can write down formulas
09:56that describe clouds and flowers and plants
09:59it's just that they're different kinds of formulas
10:02and they give you a different kind of geometry
10:14the big question is
10:16why did it take till the 1970s
10:18before somebody wrote a book called
10:21the fractal geometry of nature
10:23if they're all around us
10:24why didn't we see them before
10:26the answer seems to be
10:27well people were seeing them before
10:30people clearly recognize this repeating quality in nature
10:36people like the great 19th century Japanese artist
10:40Katsushka Hokusai
10:42if you look well enough
10:45you see a shadow of a cloud over Mount Fuji
10:48the cloud is billows upon billows upon billows
10:52Hokusai with the great wave
10:56the great wave
10:57you know on top of the great wave
10:58the smaller waves
10:59after my book mentioned that Hokusai was fractal
11:05I got inundated with people saying
11:07now we understand Hokusai
11:09Hokusai was growing fractals
11:13everybody thinks that mathematicians are very different from artists
11:19I've come to realize that art is actually really close to mathematics
11:23and that they're just using different language
11:26and so for Mandelbrot it's not about equations
11:30it's about how do we explain this visual phenomenon
11:37Mandelbrot's fascination with the visual side of math began when he was a student
11:42it is only in January 44 that suddenly I fell in love with mathematics
11:49and not mathematics in general
11:51with the geometry in its most concrete sensual form
11:57that that part of geometry which in which mathematics and the eye meet
12:03the professor was talking about algebra
12:07but I began to see in my mind
12:10geometric pictures which fitted this algebra
12:13and once you see these pictures the answers become obvious
12:17so I discovered something which I had no clue before
12:21that I knew how to transform in my mind instantly
12:25the formulas into pictures
12:28as a young man Mandelbrot developed a strong sense of self-reliance
12:34shaped in large part by his experience as a Jew living under Nazi occupation in France
12:40for four years he managed to evade the constant threat of arrest and deportation
12:47there is nothing more hardening in a certain sense than surviving a war
12:53even not as a soldier but as a hunted civilian
12:56I knew I knew how to act and I didn't trust people's wisdom very much
13:03after the war Mandelbrot got his PhD
13:07he tried teaching at a French university but he didn't seem to fit in
13:11they say well I'm very gifted by very misled and I do things the wrong way
13:17I was very much a fish out of water
13:20so I abandoned this job in France and took the gamble to go to IBM
13:25it was 1958
13:28the giant American corporation was pioneering a technology that would soon revolutionize the way we all live
13:35the computer
13:40IBM was looking for creative thinkers
13:43nonconformists even rebels
13:46people like Benoit Mandelbrot
13:49in fact they had cornered the market for a certain type of oddball
13:55we never had the slightest feeling of being the establishment
14:03Mandelbrot's colleagues told the young mathematician about a problem of great concern to the company
14:08IBM engineers were transmitting computer data over phone lines
14:13but sometimes the information was not getting through
14:17they realized that every so often the lines became extremely noisy
14:23error occurred in large numbers
14:26it was indeed an extremely messy situation
14:31Mandelbrot graphed the noise data
14:34and what he saw surprised him
14:36regardless of the time scale
14:38the graph looked similar
14:41one day
14:42one hour
14:43one second
14:44it didn't matter
14:46it looked about the same
14:48it turned out to be so similar to the vengeance
14:52Mandelbrot was amazed
14:55the strange pattern reminded him of something that had intrigued him as a young man
14:59a mathematical mystery that dated back nearly a hundred years
15:05the mystery of the monsters
15:07the story really begins in the late 19th century
15:12mathematicians had written down a formal description of what a curve must be
15:17but within that description
15:19there were these other things
15:21things that satisfied the formal definition of what a curve is
15:24but were so weird that you could never draw them
15:27or you couldn't even imagine drawing them
15:29they were just regarded as monsters or things beyond the realm
15:33they're not lines
15:36they're nothing like lines
15:37they're not circles
15:38they were like really really weird
15:40the German mathematician Georg Cantor
15:44created the first of the monsters in 1883
15:48he just took a straight line
15:50and he said
15:51I'm gonna break this line into thirds
15:53and the middle third I'm gonna erase
15:55so you're left with two lines at each end
15:57and now I'm gonna take those two lines
15:59take out the middle third
16:01and we'll do it again
16:02so he does that over and over again
16:04most people would think
16:06well if I've thrown everything away
16:08eventually there's nothing left
16:09not the case
16:11there's not just one point left
16:13there's not just two points left
16:15there's infinitely many points left
16:17as you zoom in on the Cantor set
16:20the pattern stays the same
16:22much like the noise patterns that Mandelbrot had seen at IBM
16:26another strange shape was put forward by the Swedish mathematician Helge von Koch
16:35Koch said where's the word
16:36you start with an equilateral triangle
16:38one of the classical Euclidean geometric figures
16:41and on each side
16:42I take a piece and I substitute two pieces
16:44that are now longer than the original piece
16:46and for each of those pieces
16:47I substitute two pieces that are each longer than the original piece
16:50over and over again
16:51you get the same shape
16:52but now each line has that little triangular bump on it
16:55and I break it again
16:56and I break it again
16:57and I break it again
16:58and each time I break it the line gets longer
16:59every iteration
17:00every cycle
17:01he's adding on another little triangle
17:03imagine iterating that process of adding little bits
17:08infinitely many times
17:10what you end up with is something that's infinitely long
17:17the Koch curve was a paradox
17:19to the eye the curve appears to be perfectly finite
17:23but mathematically it is infinite
17:26which means it cannot be measured
17:29at the time they called it a pathological curve
17:32because it made no sense according to the way people were thinking about
17:35measurement and Euclidean geometry and so on
17:38but the Koch curve turned out to be crucial to a nagging measurement problem
17:43the length of a coastline
17:46in the 1940s
17:48British scientist Lewis Richardson had observed that there can be great variation
17:52between different measurements of a coastline
17:55it depends on how long your yardstick is
17:57how much patience you have
17:59if you measure the coastline of Britain with a one mile yardstick
18:02you get so many yardsticks which gives you so many miles
18:05if you measure it with a one foot yardstick
18:07it turns out that it's longer
18:08and every time you use a shorter yardstick you get a longer number
18:11because you can always find finer indentations
18:14Mandelbrot saw that the finer and finer indentations in the Koch curve
18:20were precisely what was needed to model coastlines
18:23he wrote a very famous article in science magazine called
18:27how long is the coastline of Britain
18:29a coastline in geometric terms said Mandelbrot is a fractal
18:34and though he knew he couldn't measure its length
18:37he suspected he could measure something else
18:40its roughness
18:42to do that required rethinking one of the basic concepts in math
18:47dimension
18:48what we would think of as normal geometry
18:51one dimension is the straight line
18:53two dimensions is say the box that has surface area
18:57and three dimensions is a cube
18:59but could something have a dimension somewhere in between
19:03say two and three
19:05Mandelbrot said yes
19:07fractals do
19:09and the rougher they are
19:12the higher their fractal dimension
19:14there are all of these technical terms like fractal dimension
19:19and self-similarity
19:21but those are the nuts and bolts of the mathematics itself
19:24what that fractal geometry does
19:26is give us a way of looking at
19:29in a way that's extremely precise
19:31the world in which we live in particular the living world
19:39Mandelbrot's fresh ways of thinking were made possible
19:42by his enthusiastic embrace of new technology
19:46computers made it easy for Mandelbrot to do iteration
19:50the endlessly repeating cycles of calculation
19:53that were demanded by the mathematical monsters
19:57the computer was totally essential
20:00otherwise have taken a very big long effort
20:03Mandelbrot decided to zero in on yet another of the monsters
20:08a problem introduced during World War I
20:11by a young French mathematician named Gaston Julia
20:17Gaston Julia
20:18he was actually looking at what happens when you take a simple equation
20:22and you iterate it through a feedback loop
20:24that means you take a number
20:26you plug it into the formula
20:27you get a number out
20:28you take that number back to the beginning
20:31and you feed it into the same formula
20:32get another number out
20:33and you keep iterating that over and over again
20:36and the question is
20:38what happens when you iterate it lots of times
20:41the series of numbers you get is called a set
20:44the Julia set
20:46but working by hand
20:48you could never really know what the complete set looked like
20:51there were attempts to draw it
20:54doing a bunch of arithmetic by hand
20:56and putting a point on graph paper
20:57you would have to feed it back
20:59hundreds thousands millions of times
21:02the development of that new kind of mathematics
21:05had to wait until fast computers were invented
21:08at IBM Mandelbrot did something Julia could never do
21:13use a computer to run the equations millions of times
21:17he then turned the numbers from his Julia sets
21:22into points on a graph
21:24my first step was to just draw mindlessly
21:29a large number of Julia sets
21:31not one picture
21:32hundreds of pictures
21:35those images led Mandelbrot to a breakthrough
21:39in 1980 he created an equation of his own
21:42one that combined all of the Julia sets
21:45into a single image
21:47when Mandelbrot iterated his equation
21:50he got his own set of numbers
21:53graphed on a computer
21:54it was a kind of road map of all the Julia sets
21:57and quickly became famous
21:59as the emblem of fractal geometry
22:03the Mandelbrot set
22:06they intersect at certain areas
22:08and it's got like a, you know
22:10and they have little
22:11curlicues built into them
22:12black
22:13beetle-like thing
22:14crawling across the floor
22:16seahorses
22:17dragons
22:18something similar to my hair actually
22:23with this mysterious image
22:25Mandelbrot was issuing a bold challenge
22:27to long-standing ideas about the limits of mathematics
22:32the blinders came off
22:34and people could see forms
22:36that were always there
22:38but formerly were invisible
22:42the Mandelbrot set was a great example
22:45of what you could do in fractal geometry
22:47just as
22:48the archetypical example
22:50of classical geometry
22:51is the circle
23:00when you zoom in you see them coming up again
23:01so you see self similarity
23:03you see by zooming in
23:04you zoom zoom zoom
23:05you're zooming in
23:06you're zooming in
23:07and pop
23:08suddenly it seems like
23:09you're exactly where you were before
23:10but you're not
23:11it's just that way down there
23:12it has the same kind of structure as
23:14way up here
23:15and the sameness
23:17can be grokked
23:28Mandelbrot's mesmerizing images
23:30launched a fad
23:31in the world of popular culture
23:33suddenly this thing caught
23:35like
23:36like bushfire
23:37everybody wanted to have it
23:39I thought this is something big going on
23:53this was a cultural event of great proportions
23:59in the late 1970s
24:01Jane Barnes had just launched a business designing men's clothing
24:05when I started my business in 76
24:08I was doing fabrics the old-fashioned way
24:10just
24:11on graph paper
24:12weaving them on a little hand loom
24:15but then she discovered fractals
24:18and realized that the simple rules that made them
24:21could be used to create intricate clothing designs
24:24I thought this is amazing
24:26so that very simple concept
24:28I said oh I can make designs with that
24:30but in the 80s I really didn't know
24:33how to design a fractal because there wasn't software
24:36so Barnes got help from two people who knew a lot about math and computers
24:41Bill Jones and Dana Cartwright
24:44I had Dana and Bill writing my software for me
24:48they said oh your work is very mathematical
24:51and I was like it is that's my weakest subject in school
24:54we had a physicist and a mathematician
24:57and a textile designer
24:59we had so much to learn from each other
25:01I did not know what a warp and a weft is
25:04you know Jane her ability with numbers is fairly restricted
25:08if I can put that politely
25:10all the parameters here
25:12there was a way we were going to communicate
25:14we were going to get together somehow
25:16and it really did happen pretty quickly
25:18the general fashion press thought Jane's a little nuts
25:22they started calling me the fashion nerd
25:24you know but that was okay that was okay with me
25:27because I was learning a lot this was fun
25:29and very very inspirational
25:31I'm getting things that wouldn't be possible by hand
25:38you know sometimes when I think about things in my head
25:43and I say you know I just saw light coming through that screen door
25:48and look at the moraying effects that are happening on the ground
25:53can I go draw that?
25:56no way
25:57but I can describe that to my mathematician
25:59this is kind of reminds me of
26:00he sends me back the generator all ready for me to try
26:05and I sit down at the computer and say well let's see what it's doing
26:09and I have parameters that I can control
26:12and I keep pushing and I go
26:15well this is not what I expected at all
26:18um at all but it's cool
26:30the same kinds of fractal design principles
26:32have completely transformed the magic of special effects
26:38this is a key moment from Star Wars Episode 3
26:42where our two heroes have run out onto the end of this giant mechanical arm
26:48and the lava splashes down onto the arm
26:52my starting point here is to actually take the three dimensional model
26:56and take essentially a jet and just shoot lava up into the air
27:01this looks kind of boring
27:03it's doing roughly the right thing
27:04but the motion has no kind of visual interest to it
27:07let's look at what happens here when I add the fractal swirl to it
27:11where this becomes fractal is
27:14we take that same swirl pattern
27:16we shrink it down
27:18and reapply it
27:20we take that we shrink it down again
27:22we reapply it
27:23we shrink it down again
27:24we reapply it
27:25and from here on it's just a case of
27:28layering up more and more and more
27:30I've used the same technique to create these additional lava streams
27:34I then do it again here to get some just red hot embers
27:39then we take all of those layers and we add them up
27:43and we get the final composite image
27:46my hero lava in the foreground
27:48the extra lava in the background
27:50the embers, sparks, steam, smoke
27:54designers and artists the world over have embraced the visual potential of fractals
28:09but when the Mandelbrot set was first published
28:13mathematicians for the most part reacted with scorn
28:17in the mathematical intelligencer
28:20which is a gossip sheet for professional mathematicians
28:23there were article after article
28:25saying he wasn't a mathematician
28:27he was a bad mathematician
28:28it's not mathematics
28:29fractal geometry is worthless
28:32the eye had been banished out of science
28:34the eye had been excommunicated
28:37his colleagues especially the really good ones
28:42pure mathematicians that he respected
28:44they turned against him
28:46because you know
28:47you get used to the world that you've created
28:49and that you live in
28:50and mathematicians become very used to this world
28:53of smooth curves that they could do things with
28:55they were clinging to the old paradigm
29:00when the Mandelbrot and a few people
29:03were way out there bringing in the new paradigm
29:06he used to call me up on the telephone late at night
29:14because he was bothered and we'd talk about it
29:17Mandelbrot was saying
29:18this is a branch of geometry just like Euclid
29:21well they offended them
29:23they said no
29:24this is an artifact of your stupid computing machine
29:27I know very well that there's this line that fractals are pretty pictures but they're pretty useless
29:36well it's a pretty jingle but it's completely ridiculous
29:40Mandelbrot replied to his critics with his new book
29:43the fractal geometry of nature
29:46it was filled with examples of how his ideas could be useful to science
29:50Mandelbrot argued that with fractals he could precisely measure natural shapes
29:56and make calculations that could be applied to all kinds of formations
30:01from the drainage patterns of rivers
30:04to the movements of clouds
30:07so this domain of growing living systems
30:10which I along with most other mathematicians
30:12had always regarded as pretty well off limits for mathematics
30:15and certainly off limits for geometry
30:17suddenly was center stage
30:19it was Mandelbrot's book that convinced us
30:22that this wasn't just artwork
30:24this was new science in the making
30:27this was a completely new way of looking at the world in which we live
30:31that allowed us not just to look at it
30:34not just to measure it
30:35but to do mathematics
30:37and thereby understand it in a deeper way than we had before
30:42as someone who's been working with fractals for 20 years
30:45I'm not going to tell you fractals are cool
30:47I'm going to tell you fractals are useful
30:49and that's what's important to me
30:52in the 1990s a Boston radio astronomer named Nathan Cohen
30:57used fractal mathematics to make a technological breakthrough in electronic communication
31:02Cohen had a hobby
31:05he was a ham radio operator
31:07but his landlord had a rule against rigging antennas on the building
31:11I was in an astronomy conference in Hungary
31:14and Dr. Mandelbrot was giving a talk about the large scale structure of the universe
31:20and reporting how using fractals is a very good way of understanding that kind of structure
31:27which really wowed the entire group of astronomers
31:31he showed several different fractals that I in my own mind looked at and said
31:36oh wouldn't it be funny if you made an antenna out of that shape
31:39I wonder what it would do
31:41one of the first designs he tried was inspired by one of the 19th century monsters
31:47the snowflake of Helga von Koch
31:50I thought back to the lecture and said
31:52well I've got a piece of wire what happens if I bend it?
31:57after I bent the wire I hooked it up to the cable and my ham radio
32:01and I was quite surprised to see that it worked the first time out of the box
32:05it worked very well
32:07and I discovered that
32:08much of a surprise to me that I could actually make the antenna much smaller
32:11using the fractal design
32:13so it was frankly an interesting way to beat a bad rap with the landlord
32:20Cohen's experiments soon led him to another discovery
32:24using a fractal design not only made antennas smaller
32:27but enabled them to receive a much wider range of frequencies
32:32using fractals experimentally I came up with a very wide band antenna
32:37and then I worked backwards and said why is it working this way
32:40what is it about nature that requires you to use the fractal to get there
32:45the result of that work was a mathematical theorem that showed
32:49if you want to get something that works as an antenna over a very wide range of frequencies
32:54you need to have self similarity it has to be fractal in its shape to make it work
33:01now that was an exact solution it wasn't like oh here's a way of doing it
33:05and there's a lot of other ways of doing it
33:07it turned out mathematically we were able to demonstrate that was the only technique you could use to get there
33:12Cohen made his discovery at a time when cell phone companies were facing a problem
33:18they were offering new features to their customers like Bluetooth walkie-talkie and Wi-Fi
33:25but each of them ran on a separate frequency
33:29you need to be able to use all those different frequencies and have access to them
33:33without ten stubby antennas sticking out at the same time
33:37the alternative option is you can let your cell phone look like a porcupine
33:40but most people don't want to carry around a porcupine
33:43today fractal antennas are used in tens of millions of cell phones and other wireless communication devices all over the world
33:54we're going to see over the next 10 to 15 to 20 years that you're going to have to use fractals
34:00because it's the only way to get cheaper cost and smaller size for all the complex telecommunication needs we're having
34:06Once you realize that a Schroeder engineer would use fractals in many many contexts you better understand why nature which is Schroeder uses them in its ways
34:21They're all over in biology
34:23they're solutions that natural selection has come up with over and over and over and over again
34:31One powerful example, the rhythms of the heart
34:35something that Boston cardiologist Erie Goldberger has been studying his entire professional life
34:42The notion of sort of the human body as a machine goes back through the tradition of Newton and the machine-like universe
34:49so somehow we're machines, we're mechanisms, the heartbeat is this timekeeper
34:53Galileo was reported to have used his pulse to time the swinging of a pendular motion
35:00so that all fit in with the idea that the normal heartbeat is like a metronome
35:06But when Goldberger and his colleagues analyzed data from thousands of people, they found the old theory was wrong
35:14This is where I showed the heartbeat time series of a healthy subject
35:19and as you can see, the heartbeat is not constant over time
35:23It fluctuates and it fluctuates a lot, for example, in this case it fluctuates between 60 beats per minute and 120 beats per minute
35:31The patterns looked familiar to Goldberger, who happened to have read Benoit Mandelbrot's book
35:38When you actually plotted out the intervals between heartbeats, what you saw was very close to the rough edges of the mountain ranges that were in Mandelbrot's book
35:51You blow them up, expand them, you actually see that there are more of these wrinkles upon wrinkles
35:57The healthy heartbeat, it turned out, had this fractal architecture
36:02People said, this isn't cardiology, we do cardiology if you want to get funded
36:07But it turns out it is cardiology
36:10Goldberger found that the healthy heartbeat has a distinctive fractal pattern
36:16A signature that one day may help cardiologists spot heart problems sooner
36:21Please look around the screen for me
36:28Alright Cooper, we're going to do the calibration
36:32At the University of Oregon, Richard Taylor is using fractals to reveal the secrets of another part of the body
36:39The eye
36:41What we want to do is see what is that eye doing that allows it to absorb so much visual information
36:49And so that's what led us into the eye trajectories
36:53Under the monitor is a little infrared camera
36:56Which will actually monitor where the eye is looking
36:59And it actually records that data
37:02And so what we get out is a trajectory of where the eye has been looking
37:07It's interesting how they go around the pattern
37:10And so the computer will get out this graph and it will look, you know, have all of these various little structure in it
37:17And it's that pattern that we zoom in, we tell the computer to zoom in on and see the fractal dimension
37:24The tests show that the eye does not always look at things in an orderly or smooth way
37:31If we could understand more about how the eye takes in information
37:35We could do a better job of designing the things that we really need to see
37:40A traffic light
37:42A traffic light
37:43You're looking at the traffic light
37:44You've got traffic
37:45You've got pedestrians
37:46Your eye is looking all over the place trying to assess all of this information
37:52People design aircraft cockpits
37:54Rows of dials and things like that
37:56If your eye is darting all around based on a fractal geometry though
38:01Maybe that's not the best way
38:03Maybe you don't want these things in a simple row
38:06We're trying to work out the natural way that the eye wants to absorb the information
38:13Is it going to be similar to a lot of these other subconscious processes?
38:17Body motion
38:18When you're balancing, what are you actually doing there?
38:22It's something subconscious and it works
38:24And you're stringing together big sways and small sways and smaller sways
38:29Could those all be connected together to actually be doing a fractal pattern there?
38:37More and more physiological processes have been found to be fractal
38:45Not everyone in science is convinced of fractal geometry's potential for delivering new knowledge
38:51Skeptics argue that it's done little to advance mathematical theory
38:55But in Toronto, biophysicist Peter Burns and his colleagues see fractals as a practical tool
39:02A way to develop mathematical models that might help in diagnosing cancer earlier
39:08Detecting very small tumors is one of the big challenges in medical imaging
39:13Burns knew that one early sign of cancer is particularly difficult to see
39:19A network of tiny blood vessels that forms with the tumor
39:24Conventional imaging techniques like ultrasound aren't powerful enough to show them
39:30We need to be able to see structures which are just a few tenths of a millionth of a meter across
39:35When it comes to a living patient, we don't have the tools to be able to see these tiny blood vessels
39:42But ultrasound does provide a very good picture of the overall movement of blood
39:49Is there any way, Burns wondered, that images of blood flow could reveal the hidden structure of the blood vessels?
39:56To find out, Burns and his colleagues used fractal geometry to make a mathematical model
40:03If you have a mathematical way of analyzing a structure, you can make a model
40:07What fractals do is they give you some simple rules by which you can create models
40:13And by changing some of the parameters of the model, we can change how the structure looks
40:19The model showed the flow of blood in the kidney
40:22First through normal blood vessels, and then through vessels feeding a cancerous tumor
40:28Burns discovered that the two kinds of networks had very different fractal dimensions
40:34Instead of being neatly bifurcating, looking like a nice elm tree
40:39The tumor vasculature is chaotic and tangled and disorganized, looking more like a mistletoe bush
40:47And the flow of blood through these tangled vessels look very different than in a normal network
40:53A difference doctors might one day be able to detect with ultrasound
40:59We've always thought that we have to make medical images sharper and sharper
41:04Ever more precise, ever more microscopic in their resolution
41:08To find out the information about the structure that's there
41:13What's exciting about this is it's giving us microscopic information
41:16Without us actually having to look through a microscope
41:19You can see it rises up as the bubbles
41:22We think that this fractal approach may be helpful in distinguishing benign from malignant lesions
41:28In a way that hasn't been possible up to now
41:33It may take years before fractals can help doctors predict cancer
41:37But they are already offering clues to one of biology's more tantalizing mysteries
41:42Why big animals use energy more efficiently than little ones?
41:47That's a question that fascinates biologists James Brown and Brian Enquist
41:53And physicist Jeffrey West
41:55There is an extraordinary economy of scale as you increase in size
42:01An elephant, for example, is 200,000 times heavier than a mouse
42:08But uses only about 10,000 times more energy in the form of calories it consumes
42:14The bigger you are, you actually need less energy per gram of tissue to stay alive
42:23That is an amazing fact
42:26This is a very tiny number
42:28And even more amazing is the fact that this relationship between the mass and energy use of any living thing
42:34Is governed by a strict mathematical formula
42:39So far as we know, that law is universal or almost universal across all of life
42:46So it operates from the tiniest bacteria to whales and sequoia trees
42:54But even though this law had been discovered back in the 1930s
42:58No one had been able to explain it
43:01We had this idea that it probably had something to do with how resources are distributed
43:06Within the bodies of organisms as they varied in size
43:10We took this big leap and said
43:12All of life in some way is sustained by these underlying networks that are transporting oxygen, resources, metabolites that are feeding cells
43:27Circulatory systems and respiratory systems and renal systems and neural systems
43:33It was obvious that fractals were staring us in the face
43:37If all these biological networks are fractal
43:42It means they obey some simple mathematical rules
43:46Which can lead to new insights into how they work
43:49If you think about it for a minute
43:51It would be incredibly inefficient to have a set of blueprints for every single stage of increasing size
43:58But if you have a fractal code, a code that says when to branch as you get bigger and bigger
44:04Then a very simple genetic code can produce what looks like a complicated organism
44:11Evolution by natural selection has hit upon a design that appears to give the most bang for the buck
44:20In 1997, West Brown and Enquist announced their controversial theory
44:29That fractals hold the key to the mysterious relationship between mass and energy use in animals
44:36Now they are putting their theory to a bold new test
44:39An experiment to help determine if the fractal structure of a single tree
44:44Can predict how an entire rain forest works
44:48Measurements of its trunk
44:49Measurements of its trunk
44:57Enquist has traveled to Costa Rica
44:59To Juana-Caste Province in the northwestern part of the country
45:03The government has set aside more than 300,000 acres in Juana-Caste as a conservation area
45:12This rain forest, like others around the world, plays a vital role in regulating the Earth's climate
45:22By removing carbon dioxide from the atmosphere
45:25If you look at the forest, it basically breathes
45:28And if we understand the total amount of carbon dioxide that's coming into these trees within this forest
45:35We can then better understand how this forest then ultimately regulates the total amount of carbon dioxide
45:41In our atmosphere
45:44With carbon dioxide levels around the world rising
45:47How much CO2 can rainforests like this one absorb?
45:51And how important is their role in protecting us from further global warming?
45:58Enquist and a team of U.S. scientists think that fractal geometry may help answer these questions
46:04Let's try to get the height of the tree measured here
46:08They are going to start by doing just about the last thing you'd think a scientist would do here
46:14Cut down a balsa tree
46:16It's dying anyway, and they have the permission of the authorities
46:20So Christine, as soon as you know the height of that tree
46:23We can actually figure out the approximate angle that we need to take it down on
46:29Hooking a guideline on a high branch helps ensure the tree will land where they want it to
46:35Good work
46:39Very good, very nice
46:41That's great!
46:53Very good, very nice
47:02Enquist and his colleagues then measure the width and length of the branches to quantify
47:15the trees fractal structure they also measure how much carbon a single leaf contains which
47:39should allow them to figure out what the whole tree can absorb so if we know the
47:44amount of carbon dioxide that one leaf is able to take in then hopefully using the
47:49fractal branching rule we can know how much carbon dioxide the entire tree is taking in
47:54their next step is to move from the tree to the whole forest all right this is good 13.2 3.3 are
48:09going to census this forest we're going to be measuring the diameter at the base of the trees
48:14ranging all the way from the largest trees down to the smallest trees and in that way
48:19we can then sample the distribution of sizes within the forest even though the forest may
48:31appear random and chaotic the team believes it actually has a structure one that amazingly is
48:38almost identical to the fractal structure of the tree they have just cut down the beautiful thing
48:45is that the distribution of the sizes of individual trees in the forest appears to exactly match the
48:54distribution of the sizes of individual branches within a single tree if they're correct studying a
49:04single tree will make it easier to predict how much carbon dioxide an entire forest can absorb
49:11when they finish here they take their measurements back to base camp where they'll see if their ideas hold up
49:20so is this the this is the tree plot right the the cool thing is that if you look at the tree you see
49:27the same pattern amongst the branches as we see amongst the trunks of the forest very nice just as they
49:34predicted the relative number of big and small trees closely matches the relative number of big and small
49:41branches it's actually phenomenal that it is parallel there the slope that line for the tree appears to be the
49:48same for the forest as well so I guess it was worth cutting up the tree it was definitely worth cutting up the
49:54tree so far the measurements from the field appear to support the scientists theory that a single tree can
50:03help scientists assess how much this rainforest is helping to slow down global warming by analyzing the
50:11fractal patterns within the forest that then enables us to do something that we haven't really been able to
50:16do before have then a mathematical basis to predict how the forest as a whole takes in carbon dioxide and
50:24ultimately that's important for understanding what may happen with global climate change
50:29for generations scientists believed that the wildness of nature could not be defined by mathematics but fractal
50:40geometry is leading to a whole new understanding revealing an underlying order governed by simple
50:47mathematical rules what I thought of in my hikes through forests that you know just a bunch of trees
50:55of different sizes big ones here small ones they're looking like it's sort of some arbitrary chaotic
51:01mess actually has an extraordinary structure a structure that can be mapped out and measured using fractal geometry
51:11what's absolutely amazing is that you can translate what you see in the natural world in the language of
51:21mathematics and I can't think of anything more beautiful than that math is our one and only strategy for understanding the complexity of nature
51:34now fractal geometry has given us a much larger vocabulary and with larger vocabulary we can read more of the book of nature
51:46about the world and that impression they can see what happens in a world that if you are even 뭐
51:51so far a source of nature is an ordinary language the world becomes moreischen to the name of nature the image tree
51:52we've set up their cinqigtailing nature to the world at the surface Landex and colors and colors to the art getting
51:55to any of the appropriate names of however many people work there and for many people work there's quite a sign that
51:57I'm bringing up now for you to ask the concept of reflecting to it's really important toLuc.
51:58There's about the concept of Picture.
52:01On NOVA's Hidden Dimension website, explore the Mandelbrot set, see a gallery of fractal
52:23images and much more.
52:25Find it on PBS.org.
52:31I'll see you next time.
53:01Bye.
53:02Bye.