0 00:00:02,035 --> 00:00:04,504 NARRATOR: You can find it in a rain forest, 1 00:00:04,504 --> 00:00:08,708 o n the frontiers, of mednical reseach, 2 00:00:08,708 --> 00:00:10,076 in the movies, 3 00:00:10,076 --> 00:00:14,814 and it's all over the world of wireless communications. 4 00:00:14,814 --> 00:00:17,384 One of nature's biggest design secrets 5 00:00:17,384 --> 00:00:19,219 has finally been revealed. 6 00:00:19,219 --> 00:00:21,721 My God! Of course, it's obvious. 7 00:00:21,721 --> 00:00:25,759 NARRATOR: lt's an odd-looking shape you may never have heard of, 8 00:00:25,759 --> 00:00:26,659 But it's everywhere around you, 9 00:00:26,659 --> 00:00:31,931 the jagged, repeating form called a fractal. 10 00:00:31,931 --> 00:00:32,866 They're all over in biology. 11 00:00:32,866 --> 00:00:36,970 They are solutions that natural selection has come up with 12 00:00:36,970 --> 00:00:40,273 over and over and over again. 13 00:00:40,273 --> 00:00:42,675 NARRATOR: Fractals are in our lungs, 14 00:00:42,675 --> 00:00:45,512 kidneys,, and blood vessels,. 15 00:00:45,512 --> 00:00:47,947 Flowers, plants, 16 00:00:47,947 --> 00:00:49,249 weather systems, 17 00:00:49,249 --> 00:00:50,717 the rhythms of the heart, 18 00:00:50,717 --> 00:00:53,953 the very essences of life. 19 00:00:53,953 --> 00:00:56,790 NARRATOR: But it took a maverick mathematician 20 00:00:56,790 --> 00:00:59,225 to figure out how they work. 21 00:00:59,225 --> 00:01:01,661 l don't play with formulas; l play with pictures, 22 00:01:01,661 --> 00:01:03,930 and that is what l've been doing all my life. 23 00:01:03,930 --> 00:01:08,168 NARRATOR: His was a bold challenge to centuries-old assumptions 24 00:01:08,168 --> 00:01:11,797 about the various forms that nature takes. 25 00:01:13,339 --> 00:01:16,810 The blinders came off and people could see forms 26 00:01:16,810 --> 00:01:22,339 that were always there, But formerly were invisible. 27 00:01:24,717 --> 00:01:27,654 NARRATOR: Making the invisible visible. 28 00:01:27,654 --> 00:01:30,490 Finding order in disorder. 29 00:01:30,490 --> 00:01:34,227 What mysteries can it help us unravel? 30 00:01:34,227 --> 00:01:36,596 Coming up next on NOVA. 31 00:01:36,596 --> 00:01:40,532 ''Hunting the Hidden Dimension.'' 32 00:01:46,272 --> 00:01:48,475 Captioning sponsored by EXX ONMOBlL, 33 00:01:48,475 --> 00:01:49,709 D AVlD H . KOCH , 34 00:01:49,709 --> 00:01:51,311 the HOWARD HUGHES MEDlCAL lNSTlTUTE, 35 00:01:51,311 --> 00:01:53,246 the CORPORATlON FOR PUBLlC BROADCASTlNG 36 00:01:53,246 --> 00:01:55,976 and VlEWERS LlKE YOU . 37 00:01:59,219 --> 00:02:03,781 Major funding for NOVA is provided by the following: 38 00:02:05,225 --> 00:02:08,820 Taking on the world's toughest energy challenges. 39 00:02:10,029 --> 00:02:11,826 And by: 40 00:02:15,135 --> 00:02:16,864 And. . . 41 00:02:24,911 --> 00:02:28,047 And by the Corporation for Public Broadcasting 42 00:02:28,047 --> 00:02:32,177 and by contriButions to your PBS station from: 43 00:02:56,976 --> 00:03:00,747 NARRATOR: ln 1978, at Boeing Aircraft in Seattle, 44 00:03:00,747 --> 00:03:05,151 engineers were designing experimental aircraft. 45 00:03:05,151 --> 00:03:07,120 Exotic things with two wings or two tails 46 00:03:07,120 --> 00:03:09,355 or two fuselages, just weird stuff. 47 00:03:09,355 --> 00:03:11,457 'Cause who knows, it might work. 48 00:03:11,457 --> 00:03:15,261 NARRATOR: A young computer scientist named Loren Carpenter 49 00:03:15,261 --> 00:03:16,763 was helping them visualize 50 00:03:16,763 --> 00:03:19,365 what the planes might look like in flight. 51 00:03:19,365 --> 00:03:21,568 CARPENTER: l would get the data from them 52 00:03:21,568 --> 00:03:23,369 and make pictures, uh, from various angles. 53 00:03:23,369 --> 00:03:26,306 But l wanted to be able to put a mountain behind them, 54 00:03:26,306 --> 00:03:29,275 because every Boeing publicity photo in existence 55 00:03:29,275 --> 00:03:30,510 has a mountain behind it. 56 00:03:30,510 --> 00:03:31,711 But there was, no way to do mountains,. 57 00:03:31,711 --> 00:03:33,379 Mountains had millions and millions of little triangles 58 00:03:33,379 --> 00:03:35,148 or polygons, or whatever you want to call it, 59 00:03:35,148 --> 00:03:37,383 and, uh, we had enough trouble with a hundred. 60 00:03:37,383 --> 00:03:39,285 Especially in those days when our machines were, uh, 61 00:03:39,285 --> 00:03:40,653 slower than the ones you have in your watch. 62 00:03:40,653 --> 00:03:44,657 NARRATOR: Carpenter didn't want to make just any mountains. 63 00:03:44,657 --> 00:03:49,596 He wanted to create a landscape the planes could fly through; 64 00:03:49,596 --> 00:03:51,364 But there was no way to do that 65 00:03:51,364 --> 00:03:54,867 with existing animation techniques. 66 00:03:54,867 --> 00:03:56,803 From the time movies began, 67 00:03:56,803 --> 00:03:59,806 animators had to draw each frame by hand-- 68 00:03:59,806 --> 00:04:03,176 thousands of them to make even a short cartoon. 69 00:04:03,176 --> 00:04:06,579 ( echoing ): That's why they call me Thumper. 70 00:04:06,579 --> 00:04:10,850 NARRATOR: But that was before Loren Carpenter stumbled across 71 00:04:10,850 --> 00:04:13,586 the work of a little-known mathematician 72 00:04:13,586 --> 00:04:15,588 named Benoit Mandelbrot. 73 00:04:15,588 --> 00:04:19,759 CARPENTER: ln 1978, l ran into this book at a bookstore, 74 00:04:19,759 --> 00:04:21,694 Fractals. Form, Chance, and Dlmenslon 75 00:04:21,694 --> 00:04:22,662 by Benoit Mandelbrot, 76 00:04:22,662 --> 00:04:24,664 and it has to do with the fractal geometry of nature. 77 00:04:24,664 --> 00:04:27,333 So l bought the book, took it home and read it-- 78 00:04:27,333 --> 00:04:29,535 cover to cover, every last little word, 79 00:04:29,535 --> 00:04:32,171 including the footnotes and references-- twice. 80 00:04:32,171 --> 00:04:36,242 NARRATOR: ln his book Mandelbrot said that many forms in nature 81 00:04:36,242 --> 00:04:40,013 can be described mathematically as ''fractals,'' 82 00:04:40,013 --> 00:04:42,749 a word he invented to define shapes 83 00:04:42,749 --> 00:04:44,817 that look jagged and broken. 84 00:04:44,817 --> 00:04:46,719 He said that you can create a fractal 85 00:04:46,719 --> 00:04:51,591 by taking a smooth-looking shape and breaking it into pieces, 86 00:04:51,591 --> 00:04:54,160 over and over again. 87 00:04:54,160 --> 00:04:58,798 Carpenter decided he'd try doing that on his computer. 88 00:04:58,798 --> 00:05:00,500 CARPENTER: Within in three days, 89 00:05:00,500 --> 00:05:02,368 l was producing pictures of mountains 90 00:05:02,368 --> 00:05:04,804 o n my computer at work. 91 00:05:04,804 --> 00:05:06,706 The method is dead-simple. 92 00:05:06,706 --> 00:05:09,609 You start with a landscape made out of very rough triangles, 93 00:05:09,609 --> 00:05:11,711 big ones, and then for each triangle, 94 00:05:11,711 --> 00:05:15,581 break it into four triangles, and then do that again, 95 00:05:15,581 --> 00:05:17,417 and again and again. . . 96 00:05:17,417 --> 00:05:18,785 NARRATOR: Endless repetition-- 97 00:05:18,785 --> 00:05:21,788 what mathematicians call ''iteration.'' 98 00:05:21,788 --> 00:05:26,418 lt's one of the keys to fractal geometry. 99 00:05:27,393 --> 00:05:28,961 CARPENTER: The pictures were stunning. 100 00:05:28,961 --> 00:05:30,496 They were just totally stunning. 101 00:05:30,496 --> 00:05:32,732 No one has ever seen anything like this, 102 00:05:32,732 --> 00:05:35,001 and l just opened a whole new door 103 00:05:35,001 --> 00:05:38,671 to a new world of making pictures. 104 00:05:38,671 --> 00:05:42,342 And it got the computer graphics community excited about fractals 105 00:05:42,342 --> 00:05:43,609 because suddenly, they were easy to do. 106 00:05:43,609 --> 00:05:47,477 And so people started doing them all over the place. 107 00:05:47,847 --> 00:05:52,552 NARRATOR: Carpenter soon left Boeing to join Lucasfilm, 108 00:05:52,552 --> 00:05:54,654 where, instead of making mountains, 109 00:05:54,654 --> 00:05:56,522 he created a whole new planet 110 00:05:56,522 --> 00:05:59,892 for Star Trek l l. The Wrath of Khan. 111 00:05:59,892 --> 00:06:04,664 lt was the first-ever completely computer-generated sequence 112 00:06:04,664 --> 00:06:06,265 in a feature film. . . 113 00:06:06,265 --> 00:06:09,428 Fascinating. 114 00:06:09,769 --> 00:06:16,834 NARRATOR: . . .made possible by the new mathematics of fractal geometry. 115 00:06:17,944 --> 00:06:22,248 Benoit Mandelhbrot, whose work had inspired that innovation, 116 00:06:22,248 --> 00:06:24,050 was someone who prided himself 117 00:06:24,050 --> 00:06:27,620 o n s,tandingg oUts,idne the mains,trea m. 118 00:06:27,620 --> 00:06:30,523 l can see things that nobody else suspects 119 00:06:30,523 --> 00:06:32,425 until l point out to them. 120 00:06:32,425 --> 00:06:33,593 ''Oh, of course, of course.'' 121 00:06:33,593 --> 00:06:34,761 But they haven't seen it before. 122 00:06:34,761 --> 00:06:38,331 NARRATOR: You can see it in the clouds, 123 00:06:38,331 --> 00:06:40,233 in the mountains, 124 00:06:40,233 --> 00:06:44,170 even inside the human body. 125 00:06:44,170 --> 00:06:46,939 The key to fractal geometry and the thing that evaded anyone 126 00:06:46,939 --> 00:06:48,908 until, really, Mandelbrot sort of said 127 00:06:48,908 --> 00:06:51,444 this is the way to look at things, is that. . . 128 00:06:51,444 --> 00:06:52,678 if you look on the surface, 129 00:06:52,678 --> 00:06:56,182 you see complexity, and it looks very non-mathematical. 130 00:06:56,182 --> 00:07:00,553 What Mandelbrot said was that think not of what you see 131 00:07:00,553 --> 00:07:04,557 But what it took to produce what you see. 132 00:07:04,557 --> 00:07:06,859 NARRATOR: lt takes endless repetition 133 00:07:06,859 --> 00:07:10,696 and that gives rise to one of the defining characteristics 134 00:07:10,696 --> 00:07:11,798 of a fractal, 135 00:07:11,798 --> 00:07:15,301 what mathematicians call ''self-similarity.'' 136 00:07:15,301 --> 00:07:19,806 The main idea is always, as you zoom in and zoom out, 137 00:07:19,806 --> 00:07:21,908 the objects look the same. 138 00:07:21,908 --> 00:07:23,810 lf you look at something at this scale. . . 139 00:07:23,810 --> 00:07:27,713 and then you pick a small piece of it and you zoom in, 140 00:07:27,713 --> 00:07:29,348 it looks very much the same. 141 00:07:29,348 --> 00:07:32,985 NARRATOR: The whole of the fractal looks just like a part, 142 00:07:32,985 --> 00:07:36,956 which looks just like the next smaller part. 143 00:07:36,956 --> 00:07:42,588 The similarity of the pattern just keeps on going. 144 00:07:43,763 --> 00:07:46,132 One of the most familiar examples 145 00:07:46,132 --> 00:07:48,468 of self-similarity is a tree. 146 00:07:48,468 --> 00:07:50,803 lf we look at each of the nodes, 147 00:07:50,803 --> 00:07:52,004 the branching nodes of this tree, 148 00:07:52,004 --> 00:07:55,107 what you'll actually see is that the pattern of branching 149 00:07:55,107 --> 00:07:57,376 is very similar throughout the tree. 150 00:07:57,376 --> 00:08:01,080 As we go from the base of the tree to higher up, 151 00:08:01,080 --> 00:08:03,382 you'll see we'll have mother branches 152 00:08:03,382 --> 00:08:06,419 and branching then into daughter branches. 153 00:08:06,419 --> 00:08:08,588 lf we take this one branch and node 154 00:08:08,588 --> 00:08:11,557 and then go up to a higher branch or node, 155 00:08:11,557 --> 00:08:13,926 what we'll actually find is, again, 156 00:08:13,926 --> 00:08:15,928 that the pattern of branching is similar. 157 00:08:15,928 --> 00:08:20,266 Again, this pattern of branching is repeated throughout the tree, 158 00:08:20,266 --> 00:08:23,870 all the way, ultimately, out to the tips, 159 00:08:23,870 --> 00:08:24,871 where the leaves are. 160 00:08:24,871 --> 00:08:28,674 NARRATOR: You see self-similarity in everything: 161 00:08:28,674 --> 00:08:30,643 from a stalk of broccoli 162 00:08:30,643 --> 00:08:33,646 to the surface of the moon, 163 00:08:33,646 --> 00:08:37,750 to the arteries that transport blood through our bodies. 164 00:08:37,750 --> 00:08:39,418 But Mandelhbrot's, fascination 165 00:08:39,418 --> 00:08:41,754 with these irregular-looking shapes 166 00:08:41,754 --> 00:08:43,055 put him squarely at odds 167 00:08:43,055 --> 00:08:45,825 with centuries of mathematical tradition. 168 00:08:45,825 --> 00:08:51,264 ln the whole of science, the whole of mathematics, 169 00:08:51,264 --> 00:08:53,199 a smoothness with everything. 170 00:08:53,199 --> 00:08:59,866 What l did was to open up roughness for investigation. 171 00:09:01,807 --> 00:09:04,277 DEVLl N : We used mathematics to build the pyramids, 172 00:09:04,277 --> 00:09:06,012 to construct the Parthenon. 173 00:09:06,012 --> 00:09:07,146 We use mathematics to study 174 00:09:07,146 --> 00:09:09,148 the regular motion of the planets 175 00:09:09,148 --> 00:09:11,250 and so forth. 176 00:09:11,250 --> 00:09:14,120 We became used to the fact that certain patterns 177 00:09:14,120 --> 00:09:16,622 were amenable to mathematics-- the architectural ones, 178 00:09:16,622 --> 00:09:19,559 largely the patterns of human-made structures, 179 00:09:19,559 --> 00:09:22,395 where we had straight lines and circles 180 00:09:22,395 --> 00:09:24,497 and the perfect geometric shapes. 181 00:09:24,497 --> 00:09:28,234 The basic assumption that underlies classical mathematics 182 00:09:28,234 --> 00:09:30,903 is that everything is extremely regular. 183 00:09:30,903 --> 00:09:34,907 l mean, you reduce everything to straight lines. 184 00:09:34,907 --> 00:09:37,376 Circles, triangles. 185 00:09:37,376 --> 00:09:38,611 Flat surfaces. 186 00:09:38,611 --> 00:09:39,178 Plramids, 187 00:09:39,178 --> 00:09:42,014 tetrahedrons, icosahedrons, dodecahedrons. 188 00:09:42,014 --> 00:09:42,682 Smooth edges. 189 00:09:42,682 --> 00:09:48,054 DEVLl N : Classical mathematics is really only well-suited to study 190 00:09:48,054 --> 00:09:49,589 the world that we've created, 191 00:09:49,589 --> 00:09:53,993 the things we've built using that classical mathematics. 192 00:09:53,993 --> 00:09:55,528 The patterns in nature, 193 00:09:55,528 --> 00:09:57,463 the things that were already there 194 00:09:57,463 --> 00:09:58,931 before we came onto the planet, 195 00:09:58,931 --> 00:10:02,368 the trees, the plants, the clouds, the weather systems, 196 00:10:02,368 --> 00:10:06,772 those were outside of mathematics. 197 00:10:06,772 --> 00:10:08,641 NARRATOR: Until the 1970s, 198 00:10:08,641 --> 00:10:12,645 when Benoit Mandelbrot introduced his new geometry. 199 00:10:12,645 --> 00:10:17,516 DEVLl N : Mandelbrot came along and said ''Hey, guys, all you need to do 200 00:10:17,516 --> 00:10:21,487 ''is look at these patterns of nature in the right way, 201 00:10:21,487 --> 00:10:23,589 ''and you can apply mathematics. 202 00:10:23,589 --> 00:10:26,626 ''There is an order beneath the seeming chaos. 203 00:10:26,626 --> 00:10:28,461 ''You can write down formulas 204 00:10:28,461 --> 00:10:31,330 ''that describe clouds, and flowers and plants. 205 00:10:31,330 --> 00:10:33,799 ''lt's just that they're different kinds of formulas, 206 00:10:33,799 --> 00:10:38,736 and they give you a different kind of geometry.'' 207 00:10:44,777 --> 00:10:49,715 The big question is why did it take till the 1970s 208 00:10:49,715 --> 00:10:51,884 before somebody wrote a book 209 00:10:51,884 --> 00:10:54,487 called The Fractal Geometry of Nature. 210 00:10:54,487 --> 00:10:56,489 lf they're all around us, 211 00:10:56,489 --> 00:10:58,157 why didn't we see them before? 212 00:10:58,157 --> 00:10:59,458 The answer seems to be 213 00:10:59,458 --> 00:11:02,528 well, people were seeing them before. 214 00:11:02,528 --> 00:11:08,768 People clearly recoggnized this, repeating quality in nature. 215 00:11:08,768 --> 00:11:12,204 NARRATOR: People like the great 19th century Japanese artist 216 00:11:12,204 --> 00:11:15,174 Katsushika Hokusai. 217 00:11:15,174 --> 00:11:19,545 lf you look well enough, you see a shadow of a cloud 218 00:11:19,545 --> 00:11:20,613 over Mount Fuji. 219 00:11:20,613 --> 00:11:26,552 The cloud is billows upon billows upon billows. 220 00:11:26,552 --> 00:11:27,787 TAYLOR: Hokusai, the great wave. 221 00:11:27,787 --> 00:11:30,122 You know, on top of the great wave, 222 00:11:30,122 --> 00:11:33,025 there's smaller waves. 223 00:11:33,025 --> 00:11:34,827 MANDELBROT: After my book 224 00:11:34,827 --> 00:11:36,729 mentioned that Hokusai was fractal, 225 00:11:36,729 --> 00:11:40,266 l got inundated with people saying, 226 00:11:40,266 --> 00:11:42,868 ''Now we understand Hokusai.'' 227 00:11:42,868 --> 00:11:45,598 Hokusai was drawing fractals. 228 00:11:47,540 --> 00:11:48,374 TAYLOR: Everybody thinks 229 00:11:48,374 --> 00:11:50,776 that mathematicians are very different from artists. 230 00:11:50,776 --> 00:11:52,578 l've come to realize that art 231 00:11:52,578 --> 00:11:54,680 is actually really close to mathematics, 232 00:11:54,680 --> 00:11:57,450 and that they're just using different language. 233 00:11:57,450 --> 00:12:01,287 And so, for Mandelbrot, it's not about equations. 234 00:12:01,287 --> 00:12:06,554 lt's about how do we explain this visual phenomenon. 235 00:12:08,961 --> 00:12:10,563 NARRATOR: Mandelbrot's fascination 236 00:12:10,563 --> 00:12:14,800 with the visual side of math began when he was a student. 237 00:12:14,800 --> 00:12:20,372 MANDELBROT: lt is only in January '44 that suddenly, 238 00:12:20,372 --> 00:12:22,074 l fell in love with mathematics-- 239 00:12:22,074 --> 00:12:23,576 and not mathematics in general-- 240 00:12:23,576 --> 00:12:29,315 with geometry in its most concrete, sensual form. 241 00:12:29,315 --> 00:12:31,350 That part of geometry which. . . 242 00:12:31,350 --> 00:12:35,654 in which mathematics and the eye meet. 243 00:12:35,654 --> 00:12:39,058 The professor was talking about algebra, 244 00:12:39,058 --> 00:12:44,296 But l began to see in my mind geometric pictures which fitted 245 00:12:44,296 --> 00:12:47,533 this algebra, and once you see these pictures, 246 00:12:47,533 --> 00:12:48,834 the answer become obvious. 247 00:12:48,834 --> 00:12:53,339 So, l discovered something which l had no clue before, 248 00:12:53,339 --> 00:12:57,576 that l knew how to transform in my mind instantly 249 00:12:57,576 --> 00:13:00,479 the formulas into pictures. 250 00:13:00,479 --> 00:13:02,848 NARRATOR: As a young man, 251 00:13:02,848 --> 00:13:06,152 Mandelbrot developed a strong sense of self-reliance, 252 00:13:06,152 --> 00:13:08,053 shaped in large part 253 00:13:08,053 --> 00:13:09,355 by his experience as a Jew 254 00:13:09,355 --> 00:13:13,392 living under Nazi occupation in France. 255 00:13:13,392 --> 00:13:15,227 For four years, 256 00:13:15,227 --> 00:13:17,530 he managed to evade the constant threat 257 00:13:17,530 --> 00:13:18,497 of arrest and deportation. 258 00:13:18,497 --> 00:13:21,934 MANDELBROT: There is nothing more, um, hardening, 259 00:13:21,934 --> 00:13:24,770 in a certain sense, than surviving a war. 260 00:13:24,770 --> 00:13:28,474 Even not a soldier, But as a hunted civilian. 261 00:13:28,474 --> 00:13:30,142 l knew. . . l knew how to act, 262 00:13:30,142 --> 00:13:33,913 and l didn't trust people's wisdom very much. 263 00:13:33,913 --> 00:13:38,918 NARRATOR: After the war, Mandelbrot got his Ph. D. 264 00:13:38,918 --> 00:13:41,487 He tried teaching at a French university, 265 00:13:41,487 --> 00:13:43,522 But he didn't seem to fit in. 266 00:13:43,522 --> 00:13:45,057 MANDELBROT: They say, well, 267 00:13:45,057 --> 00:13:46,959 l'm very gifted, But very misled, 268 00:13:46,959 --> 00:13:48,727 and l do things the wrong way. 269 00:13:48,727 --> 00:13:51,630 l was very much, um, a fish out of water. 270 00:13:51,630 --> 00:13:55,401 So l abandoned this job in France and took the gamble 271 00:13:55,401 --> 00:13:57,767 to go to lBM. 272 00:13:58,604 --> 00:14:00,231 NARRATOR: lt was 1958. 273 00:14:00,306 --> 00:14:03,969 The giant American corporation was pioneering a technology 274 00:14:04,043 --> 00:14:06,170 that would soon revolutionize 275 00:14:06,278 --> 00:14:08,371 the way we all live: 276 00:14:08,480 --> 00:14:10,812 the computer. 277 00:14:10,850 --> 00:14:14,320 lBM was looking for creative thinkers-- 278 00:14:14,320 --> 00:14:17,590 non-conformists, even rebels. 279 00:14:17,590 --> 00:14:20,893 People like Benoit Mandelhbrot. 280 00:14:20,893 --> 00:14:23,996 MANDELBROT: ln fact, they had cornered the market 281 00:14:23,996 --> 00:14:27,166 for a certain type of oddball. 282 00:14:27,166 --> 00:14:29,935 We never had the slightest feeling 283 00:14:29,935 --> 00:14:33,029 of being the estahblishment. 284 00:14:33,939 --> 00:14:36,909 NARRATOR: Mandelbrot's colleagues told the young mathematician 285 00:14:36,909 --> 00:14:39,678 about a problem of great concern to the company. 286 00:14:39,678 --> 00:14:44,416 lBM engineers were transmitting computer data over phone lines, 287 00:14:44,416 --> 00:14:48,554 But sometimes, the information was not getting through. 288 00:14:48,554 --> 00:14:50,756 MANDELBROT: They realized 289 00:14:50,756 --> 00:14:51,991 that every so often, 290 00:14:51,991 --> 00:14:56,428 the lines became, uh, extremely noisy. 291 00:14:56,428 --> 00:14:58,397 Errors occurred in large numbers. 292 00:14:58,397 --> 00:15:02,301 lt was indeed an extremely messy situation. 293 00:15:02,301 --> 00:15:06,138 NARRATOR: Mandelbrot graphed the noise data, 294 00:15:06,138 --> 00:15:07,973 and what he saw surprised him. 295 00:15:07,973 --> 00:15:12,745 Regardless of the timescale, the graph looked similar. 296 00:15:12,745 --> 00:15:14,713 One day, 297 00:15:14,713 --> 00:15:17,349 one hour, onesecond-- 298 00:15:17,349 --> 00:15:18,751 it didn't matter. 299 00:15:18,751 --> 00:15:21,754 lt looked about the same. 300 00:15:21,754 --> 00:15:24,356 lt turned out to be self-similar with a vengeance. 301 00:15:24,356 --> 00:15:26,725 NARRATOR: Mandelbrot was amazed. 302 00:15:26,725 --> 00:15:29,561 The strange pattern reminded him of something 303 00:15:29,561 --> 00:15:32,364 that had intrigued him as a young man-- 304 00:15:32,364 --> 00:15:33,899 a mathematical mystery 305 00:15:33,899 --> 00:15:37,102 that dated back nearly 100 years: 306 00:15:37,102 --> 00:15:40,503 the mystery of the monsters. 307 00:15:41,340 --> 00:15:45,044 The story really begins in the late 19th century. 308 00:15:45,044 --> 00:15:47,579 Mathematicians had written down a formal description 309 00:15:47,579 --> 00:15:48,647 of what a curve must be. 310 00:15:48,647 --> 00:15:52,151 But within that dnescription, there were these other things,, 311 00:15:52,151 --> 00:15:56,155 things that satisfied the formal definition of what a curve is, 312 00:15:56,155 --> 00:15:58,857 But were so weird that you could never draw them, 313 00:15:58,857 --> 00:16:00,592 or you couldn't even imagine drawing them. 314 00:16:00,592 --> 00:16:03,195 They were just regarded as monsters 315 00:16:03,195 --> 00:16:05,197 or things, beyond the realm. 316 00:16:05,197 --> 00:16:07,066 ABRAHAM: They're not lines. 317 00:16:07,066 --> 00:16:08,600 They're nothing like lines. 318 00:16:08,600 --> 00:16:09,735 They're not circles. 319 00:16:09,735 --> 00:16:12,169 They were, like, really, really weird. 320 00:16:12,938 --> 00:16:17,276 NARRATOR: The German mathematician Georg Cantor created the first 321 00:16:17,276 --> 00:16:19,211 of the monsters, in 18h8h3. 322 00:16:19,211 --> 00:16:22,448 RON EGLASH : He just took a straight line, and he said, 323 00:16:22,448 --> 00:16:24,516 ''l'm gonna break this line into thirds, 324 00:16:24,516 --> 00:16:26,418 and the middle third l'm gonna erase.'' 325 00:16:26,418 --> 00:16:29,154 So you're left with two lines at each end. 326 00:16:29,154 --> 00:16:31,056 And now l'm gonna take those two lines, 327 00:16:31,056 --> 00:16:34,159 take out the middle third, and we'll do it again. 328 00:16:34,159 --> 00:16:36,829 So he does that over and over again. 329 00:16:36,829 --> 00:16:38,364 Most people would think, 330 00:16:38,364 --> 00:16:39,965 well, if l've thrown everything away, 331 00:16:39,965 --> 00:16:42,267 eventually, there's nothing left. 332 00:16:42,267 --> 00:16:43,435 Not the case. 333 00:16:43,435 --> 00:16:45,104 There's not just one point left. 334 00:16:45,104 --> 00:16:46,572 There's not just two points left. 335 00:16:46,572 --> 00:16:49,641 There's infinitely many points left. 336 00:16:49,641 --> 00:16:52,444 NARRATOR: As you zoom in on the Cantor set, 337 00:16:52,444 --> 00:16:53,412 the pattern stays the same, 338 00:16:53,412 --> 00:16:58,984 much like the noise patterns that Mandelbrot had seen at lBM. 339 00:16:58,984 --> 00:17:02,121 Another strange shape was put forward 340 00:17:02,121 --> 00:17:07,259 by the Swedish mathematician Helge Von Koch. 341 00:17:07,259 --> 00:17:09,962 Kochs,would start with an equilateral triangle, 342 00:17:09,962 --> 00:17:12,931 one of the classical Euclidean geometric figures, 343 00:17:12,931 --> 00:17:13,932 and on each side. . . 344 00:17:13,932 --> 00:17:16,001 . . . l take a piece, and l substitute two pieces 345 00:17:16,001 --> 00:17:17,603 that are now longer than the original piece. 346 00:17:17,603 --> 00:17:19,972 And for each of those pieces, l substitute two pieces 347 00:17:19,972 --> 00:17:22,174 that are each longer than the original piece. 348 00:17:22,174 --> 00:17:23,609 Over and over again. 349 00:17:23,609 --> 00:17:24,943 You get the same shape, But now, 350 00:17:24,943 --> 00:17:27,613 each line has that little triangular bump on it. 351 00:17:27,613 --> 00:17:28,380 And l break it again, 352 00:17:28,380 --> 00:17:29,748 and l break it again, and l break it again, 353 00:17:29,748 --> 00:17:31,550 and each time l break it, the line gets longer. 354 00:17:31,550 --> 00:17:32,785 Every iteration, every cycle, 355 00:17:32,785 --> 00:17:35,811 he's adding on another little triangle. 356 00:17:36,688 --> 00:17:40,526 lmagine iterating that process of adding little bits, 357 00:17:40,526 --> 00:17:42,127 infinitely many times. 358 00:17:42,127 --> 00:17:44,863 What you end up with is something 359 00:17:44,863 --> 00:17:47,966 that's infinitely long. 360 00:17:47,966 --> 00:17:50,369 NARRATOR: The Koch Curve was a paradox. 361 00:17:50,369 --> 00:17:54,106 To the eye, the curve appears to be perfectly finite. 362 00:17:54,106 --> 00:17:57,776 But mathematicalll, it is, infinite, 363 00:17:57,776 --> 00:18:00,179 which means it cannot be measured. 364 00:18:00,179 --> 00:18:03,982 EGLASH : At the time they called it a pathological curve, 365 00:18:03,982 --> 00:18:06,485 because it made no sense, according to the way 366 00:18:06,485 --> 00:18:07,886 people were thinking about measurement, 367 00:18:07,886 --> 00:18:08,987 and Euclidean geometry and so on. 368 00:18:08,987 --> 00:18:12,624 NARRATOR: But the Koch Curve turned out to be crucial 369 00:18:12,624 --> 00:18:14,827 to a nagging measurement problem: 370 00:18:14,827 --> 00:18:17,162 the length of a coastline. 371 00:18:17,162 --> 00:18:22,401 ln the 1940s,, British scientist Lewis, Richardson had observed 372 00:18:22,401 --> 00:18:23,802 that there can be great variation 373 00:18:23,802 --> 00:18:26,872 between different measurements of a coastline. 374 00:18:26,872 --> 00:18:29,141 lt depends on how long your yardstick is 375 00:18:29,141 --> 00:18:30,442 and how much patience you have. 376 00:18:30,442 --> 00:18:32,578 lf you measure the coastline of Britain 377 00:18:32,578 --> 00:18:35,481 with a one-mile yardstick, you'd get so many yardsticks, 378 00:18:35,481 --> 00:18:36,882 which gives you so many miles. 379 00:18:36,882 --> 00:18:39,284 lf you measure it with a one-foot yardstick, 380 00:18:39,284 --> 00:18:40,953 it turns out that it's longer. 381 00:18:40,953 --> 00:18:43,288 And every time you use a shorter yardstick, 382 00:18:43,288 --> 00:18:44,123 you get a longer number. 383 00:18:44,123 --> 00:18:46,758 DEVLl N : Because you can always find finer indentations. 384 00:18:46,758 --> 00:18:50,596 NARRATOR: Mandelbrot saw that the finer and finer indentations 385 00:18:50,596 --> 00:18:53,966 in the Koch Curve were precisely what was needed 386 00:18:53,966 --> 00:18:56,201 to model coastlines. 387 00:18:56,201 --> 00:18:59,204 He wrote a very famous article in Sclence Magazlne called 388 00:18:59,204 --> 00:19:00,873 ''How Long ls the Coastline of Britain?'' 389 00:19:00,873 --> 00:19:05,777 NARRATOR: A coastline, in geometric terms, said Mandelbrot, is a fractal. 390 00:19:05,777 --> 00:19:08,547 And though he knew he couldn't measure its length, 391 00:19:08,547 --> 00:19:14,086 he suspected he could measure something else: its roughness. 392 00:19:14,086 --> 00:19:18,590 To do that required rethinking one of the basic concepts 393 00:19:18,590 --> 00:19:20,993 in math: dimension. 394 00:19:20,993 --> 00:19:23,729 What we would think of as normal geometry-- 395 00:19:23,729 --> 00:19:25,164 one dimension is the straight line, 396 00:19:25,164 --> 00:19:28,767 two dimensions is, say, the box that has surface area. 397 00:19:28,767 --> 00:19:31,537 NARRATOR: And three dimensions is a cube. 398 00:19:31,537 --> 00:19:33,572 But cold something have a dimension 399 00:19:33,572 --> 00:19:37,142 somewhere in between, say, two and three? 400 00:19:37,142 --> 00:19:41,880 Mandelbrot said, yes, fractals do. 401 00:19:41,880 --> 00:19:44,550 And the rougher they are, 402 00:19:44,550 --> 00:19:46,952 the higher their fractal dimension. 403 00:19:46,952 --> 00:19:48,820 DEVLl N : There are all of these 404 00:19:48,820 --> 00:19:51,023 technical terms, like fractal dimension, 405 00:19:51,023 --> 00:19:52,491 and self-similarity, 406 00:19:52,491 --> 00:19:55,861 But those are the nuts and bolts of the mathematics itself. 407 00:19:55,861 --> 00:20:00,566 What that fractal geometry does is give us a way of looking at-- 408 00:20:00,566 --> 00:20:03,635 in a way that's extremely precise-- 409 00:20:03,635 --> 00:20:07,765 the world in which we live, in particular, the living world. 410 00:20:11,143 --> 00:20:13,612 NARRATOR: Mandelbrot's fresh ways of thinking 411 00:20:13,612 --> 00:20:16,815 were made possible by his enthusiastic embrace 412 00:20:16,815 --> 00:20:17,883 of new tech nology. 413 00:20:17,883 --> 00:20:22,487 Computers made it easy for Mandelbrot to do iteration-- 414 00:20:22,487 --> 00:20:24,556 the endlessly repeating cycles of calculation 415 00:20:24,556 --> 00:20:28,026 that were demanded by the mathematical monsters. 416 00:20:28,026 --> 00:20:32,231 MANDELBROT: The computer was totally essential. 417 00:20:32,231 --> 00:20:34,299 Otherwise, it would have taken a very big, long effort. 418 00:20:34,299 --> 00:20:39,771 NARRATOR: Mandelbrot decided to zero in on yet another of the monsters-- 419 00:20:39,771 --> 00:20:42,307 a problem introduced during World War l 420 00:20:42,307 --> 00:20:47,346 by a young French mathematician named Gaston Julia. 421 00:20:47,346 --> 00:20:50,082 DEVLl N : Gaston Julia-- 422 00:20:50,082 --> 00:20:53,585 he was actually looking at what happens when you take 423 00:20:53,585 --> 00:20:54,386 a simple equation 424 00:20:54,386 --> 00:20:56,421 and you iterate it through a feedback loop. 425 00:20:56,421 --> 00:20:57,489 That means you take a number, 426 00:20:57,489 --> 00:21:00,359 you plug it into the formula, you get a number out. 427 00:21:00,359 --> 00:21:02,894 You take that number, back to the beginning, 428 00:21:02,894 --> 00:21:03,795 and you feed it into 429 00:21:03,795 --> 00:21:05,764 the same formula, get another number out. 430 00:21:05,764 --> 00:21:08,700 And you keep iterating that over and over again. 431 00:21:08,700 --> 00:21:10,502 And the question is, what happens 432 00:21:10,502 --> 00:21:12,938 when you iterate it lots of times. 433 00:21:12,938 --> 00:21:18,443 NARRATOR: The series of numbers you get is called a set-- the Julia set. 434 00:21:18,443 --> 00:21:20,512 But working by hand, 435 00:21:20,512 --> 00:21:21,713 you could never really know 436 00:21:21,713 --> 00:21:23,715 what the complete set looked like. 437 00:21:23,715 --> 00:21:25,951 ABRAHAM: There were attempts to draw it. 438 00:21:25,951 --> 00:21:27,953 Doing a bunch of arithmetic by hand 439 00:21:27,953 --> 00:21:29,755 and putting a point on graph paper. 440 00:21:29,755 --> 00:21:32,858 You would have to feed it back hundreds, thousands, 441 00:21:32,858 --> 00:21:34,293 millions of times. 442 00:21:34,293 --> 00:21:37,629 The development of that new kind of mathematics had to wait 443 00:21:37,629 --> 00:21:40,393 until fast computers were invented. 444 00:21:41,199 --> 00:21:43,835 NARRATOR: At lBM, Mandelbrot did something 445 00:21:43,835 --> 00:21:46,071 Julia could never do: 446 00:21:46,071 --> 00:21:50,509 use a computer to run the equations millions of times. 447 00:21:50,509 --> 00:21:52,177 He then turned the numbers 448 00:21:52,177 --> 00:21:55,981 from his Julia sets into points on a graph. 449 00:21:55,981 --> 00:22:01,653 MANDELBROT: My first step was to just draw mindlessly 450 00:22:01,653 --> 00:22:03,522 a large number of Julia sets. 451 00:22:03,522 --> 00:22:06,358 Not one picture, hundreds of pictures. 452 00:22:06,358 --> 00:22:10,562 NARRATOR: Those images led Mandelbrot to a breakthrough. 453 00:22:10,562 --> 00:22:14,299 ln 1980, he created an equation of his own, 454 00:22:14,299 --> 00:22:17,569 one that combined all of the Julia sets 455 00:22:17,569 --> 00:22:19,371 into a single image. 456 00:22:19,371 --> 00:22:22,140 When Mandelbrot iterated his equation, 457 00:22:22,140 --> 00:22:24,309 he got his own set of numbers. 458 00:22:24,309 --> 00:22:27,646 Graphed on a computer, it was a kind of road map 459 00:22:27,646 --> 00:22:31,083 of all the Julia sets and quickly became famous 460 00:22:31,083 --> 00:22:34,653 as the emblem of fractal geometry. . . 461 00:22:34,653 --> 00:22:37,850 the Mandelbrot set. 462 00:22:38,323 --> 00:22:41,793 They intersect at certain areas, and it's got like a, you know. . . 463 00:22:41,793 --> 00:22:44,596 And they have little curlicues built into them. 464 00:22:44,596 --> 00:22:46,798 Black beetle-like thing . 465 00:22:46,798 --> 00:22:48,433 Crawling across the floor. 466 00:22:48,433 --> 00:22:49,468 Seahorses. Dragons. 467 00:22:49,468 --> 00:22:51,803 Something similar to my hair, actually. 468 00:22:51,803 --> 00:22:54,397 (laughing ) 469 00:22:54,706 --> 00:22:56,608 NARRATOR: With this mysterious image, 470 00:22:56,608 --> 00:22:58,977 Mandelbrot was issuing a bold challenge 471 00:22:58,977 --> 00:23:03,311 to long-standing ideas about the limits of mathematics. 472 00:23:04,449 --> 00:23:07,519 The blinders came off, and people could see forms 473 00:23:07,519 --> 00:23:12,718 that were always there, But formerly were invisible. 474 00:23:13,191 --> 00:23:16,595 DEVLlN : The Mandelbrot set was a great example 475 00:23:16,595 --> 00:23:19,731 of what you could do in fractal geometry, 476 00:23:19,731 --> 00:23:22,267 just as the archetypical example 477 00:23:22,267 --> 00:23:25,964 of classical geometry is the circle. 478 00:23:30,409 --> 00:23:33,512 ABRAHAM: When you zoom in, you see them coming up again, 479 00:23:33,512 --> 00:23:35,147 so you see self-similarity. 480 00:23:35,147 --> 00:23:37,749 You see, by zooming in, you zoom, zoom, zoom, 481 00:23:37,749 --> 00:23:38,917 you're zooming in, you're zooming in, 482 00:23:38,917 --> 00:23:41,520 and pop, suddenly it seems like you're exactly 483 00:23:41,520 --> 00:23:42,921 where you were before, But you're not. 484 00:23:42,921 --> 00:23:45,290 lt's just that way down there, it has the same kind 485 00:23:45,290 --> 00:23:52,128 of structure as way up here, and the sameness can be grokked. 486 00:24:00,539 --> 00:24:02,336 NARRATOR: Mandelbrot's mesmerizing images 487 00:24:02,374 --> 00:24:06,276 launched a fad in the world of popular culture. 488 00:24:06,344 --> 00:24:08,005 MANDELBROT: Suddenly, this thing caught 489 00:24:08,113 --> 00:24:10,707 like. . . like a bush fire. 490 00:24:10,749 --> 00:24:13,309 Everybody wanted to have it. 491 00:24:21,827 --> 00:24:24,930 DEVLl N : l thought, this is something big going on. 492 00:24:24,930 --> 00:24:28,627 This was a cultural event of great proportions. 493 00:24:30,669 --> 00:24:33,538 NARRATOR: ln the late 1970s, Jhane Barnes 494 00:24:33,538 --> 00:24:37,474 had just launched a business designing men's clothing. 495 00:24:37,909 --> 00:24:39,911 J HANE BARNES: When l started my business in '76, 496 00:24:39,911 --> 00:24:43,348 l was doing fabrics the old-fashioned way, 497 00:24:43,348 --> 00:24:44,916 just on graph paper, 498 00:24:44,916 --> 00:24:47,219 weaving them on a little handloom. 499 00:24:47,219 --> 00:24:49,621 NARRATOR: But then, she discovered fractals 500 00:24:49,621 --> 00:24:52,691 and realized that the simple rules that made them 501 00:24:52,691 --> 00:24:55,760 could be used to create intricate clothing designs. 502 00:24:55,760 --> 00:24:59,564 BARNES: l thought, this is amazing, so that very simple concept, 503 00:24:59,564 --> 00:25:02,534 l said, ''Oh, l can make designs with that.'' 504 00:25:02,534 --> 00:25:05,303 But in the '80s,, l realll didn't know 505 00:25:05,303 --> 00:25:08,039 how to design a fractal, because there wasn't software. 506 00:25:08,039 --> 00:25:09,674 NARRATOR: So Barnes got help 507 00:25:09,674 --> 00:25:10,976 from two people who knew a lot 508 00:25:10,976 --> 00:25:14,379 about maths and computers,: Bill Jones, 509 00:25:14,379 --> 00:25:16,147 and Dana Cartwright. 510 00:25:16,147 --> 00:25:19,351 BARNES: l had Dana and Bill writing my software for me. 511 00:25:19,351 --> 00:25:23,588 They said, ''Oh, your work is very mathematical.'' 512 00:25:23,588 --> 00:25:24,656 And l was like, ''lt is? 513 00:25:24,656 --> 00:25:26,258 That's my weakest subject in school.'' 514 00:25:26,258 --> 00:25:29,694 We had a physicist and a mathematician 515 00:25:29,694 --> 00:25:31,062 and a textile designer. 516 00:25:31,062 --> 00:25:32,931 BARNES: We had so much to learn from each other. 517 00:25:32,931 --> 00:25:36,735 DANA CARTWRlGHT: l did not know what a warp and a weft is. 518 00:25:36,735 --> 00:25:37,802 You know, Jane. . . 519 00:25:37,802 --> 00:25:40,305 her ability with numbers is fairly restricted, 520 00:25:40,305 --> 00:25:42,274 if l can put that politely. 521 00:25:42,274 --> 00:25:44,009 All, um, the parameters here. . . 522 00:25:44,009 --> 00:25:47,112 BARNES: There was a way we were going to communicate. 523 00:25:47,112 --> 00:25:48,613 We were going to get together somehow, 524 00:25:48,613 --> 00:25:50,148 and it really did happen pretty quickly. 525 00:25:50,148 --> 00:25:54,152 The general fashion press thought ''Jane's a little nuts.'' 526 00:25:54,152 --> 00:25:57,222 They started calling me the Fashion Nerd, 527 00:25:57,222 --> 00:25:58,356 you know, But that was okay. 528 00:25:58,356 --> 00:26:00,792 That was okay with me because l was learning a lot. 529 00:26:00,792 --> 00:26:04,091 This was fun and very, very inspirational. 530 00:26:05,630 --> 00:26:10,363 l'm getting things that wouldn't be possible by hand. 531 00:26:11,603 --> 00:26:15,874 You know, sometimes when l think about things in my head 532 00:26:15,874 --> 00:26:19,444 and l say, ''You know, l just saw light coming 533 00:26:19,444 --> 00:26:20,912 ''through that screen door, 534 00:26:20,912 --> 00:26:23,081 ''and look at the moireing effects 535 00:26:23,081 --> 00:26:25,617 that are happening on the ground.'' 536 00:26:25,617 --> 00:26:26,718 Can l go draw that? 537 00:26:26,718 --> 00:26:31,289 No way, But l can describe that to my mathematician. 538 00:26:31,289 --> 00:26:32,223 This kind of reminds me of. . . 539 00:26:32,223 --> 00:26:37,162 He sends me back the generator, all ready for me to try, 540 00:26:37,162 --> 00:26:39,464 and l sit down at the computer and say, 541 00:26:39,464 --> 00:26:40,966 ''Well let's see what it's doing.'' 542 00:26:40,966 --> 00:26:44,202 And l have parameters that l can control. 543 00:26:44,202 --> 00:26:46,638 And l keep pushing, and l go, 544 00:26:46,638 --> 00:26:50,375 ''Well, this is not what l expected at all. . . 545 00:26:50,375 --> 00:26:54,038 um, at all, But it's cool.'' 546 00:26:56,414 --> 00:26:57,415 ( weapons blasting ) 547 00:26:57,415 --> 00:26:59,684 OBl-WAN KENOBl : Use the Force, Luke. 548 00:26:59,684 --> 00:27:01,386 ( weapons blasting ) 549 00:27:01,386 --> 00:27:04,689 NARRATOR: The same kinds of fractal design principles 550 00:27:04,689 --> 00:27:08,960 have completely transformed the magic of special effects. 551 00:27:08,960 --> 00:27:14,633 D AN PlPON l : This is a key moment from Star Wars. Eplsode l l l, 552 00:27:14,633 --> 00:27:16,868 where our two heroes have run out 553 00:27:16,868 --> 00:27:19,871 onto the end of this, giant mechanical arm 554 00:27:19,871 --> 00:27:24,509 and the lava splashes down onto the arm. 555 00:27:24,509 --> 00:27:26,044 My starting point here is to actually take 556 00:27:26,044 --> 00:27:29,848 the three-dimensional model and take essentially a jet 557 00:27:29,848 --> 00:27:33,318 and just shoot lava up into the air. 558 00:27:33,318 --> 00:27:34,719 This looks kind of boring. 559 00:27:34,719 --> 00:27:36,154 lt's doing roughly the right thing, 560 00:27:36,154 --> 00:27:39,924 But the motion has no kind of visual interest to it. 561 00:27:39,924 --> 00:27:41,393 Let's, look at what happens, here 562 00:27:41,393 --> 00:27:44,329 when l add the fractal swirl to it. 563 00:27:44,329 --> 00:27:47,132 Where this becomes fractal is, 564 00:27:47,132 --> 00:27:49,000 we take that same swirl pattern, 565 00:27:49,000 --> 00:27:51,736 we shrink it down and reapply it. 566 00:27:51,736 --> 00:27:55,240 We take that, we shrink it down again, we reapply it. 567 00:27:55,240 --> 00:27:57,609 We shrink it down again, we reapply it. 568 00:27:57,609 --> 00:27:59,944 And from here on, it's just a case 569 00:27:59,944 --> 00:28:02,847 of layering up more and more and more. 570 00:28:02,847 --> 00:28:04,015 l've used the same technique 571 00:28:04,015 --> 00:28:06,885 to create these additional lava streams. 572 00:28:06,885 --> 00:28:09,220 l then do it again here 573 00:28:09,220 --> 00:28:12,290 to get some just red hot embers. 574 00:28:12,290 --> 00:28:15,126 Then, we take all of those layers, and we add them up, 575 00:28:15,126 --> 00:28:18,063 and we get the final composite image. 576 00:28:18,063 --> 00:28:19,831 My hero lava in the foreground, 577 00:28:19,831 --> 00:28:22,400 the extra lava in the background. 578 00:28:22,400 --> 00:28:27,428 The embers, sparks, steam, smoke. 579 00:28:29,507 --> 00:28:31,304 (grunting ) 580 00:28:35,580 --> 00:28:37,649 NARRATOR: Designers and artists the world over 581 00:28:37,649 --> 00:28:41,086 have embraced the visual potential of fractals, 582 00:28:41,086 --> 00:28:44,556 But when the Mandelbrot set was first published, 583 00:28:44,556 --> 00:28:46,825 mathematicians, for the most part, 584 00:28:46,825 --> 00:28:48,727 reacted with scorn. 585 00:28:48,727 --> 00:28:51,696 ABRAHAM: ln the Mathematlcal lntelllgencer, 586 00:28:51,696 --> 00:28:55,133 which is a gossip sheet for professional mathematicians, 587 00:28:55,133 --> 00:28:57,068 there were article after article 588 00:28:57,068 --> 00:28:58,970 saying he wasn't a mathematician; 589 00:28:58,970 --> 00:29:01,740 he was a bad mathematician; it's not mathematics; 590 00:29:01,740 --> 00:29:03,508 fractal geometry is worthless. 591 00:29:03,508 --> 00:29:06,544 The eye had been banished out of science. 592 00:29:06,544 --> 00:29:09,247 The eye had been excommunicated. 593 00:29:09,247 --> 00:29:14,319 ABRAHAM: His colleagues, especially the really good ones, 594 00:29:14,319 --> 00:29:17,088 pure mathematicians that he respected, 595 00:29:17,088 --> 00:29:18,189 they turned against him. 596 00:29:18,189 --> 00:29:20,725 Because, see now, you get used to the world 597 00:29:20,725 --> 00:29:22,460 that you've created and that you live in, 598 00:29:22,460 --> 00:29:23,828 and mathematicians had become very used 599 00:29:23,828 --> 00:29:27,599 to this world of smooth curves that they could do things with. 600 00:29:27,599 --> 00:29:32,871 ABRAHAM: They were clinging to the old paradigm 601 00:29:32,871 --> 00:29:35,473 when Mandelbrot and a few people 602 00:29:35,473 --> 00:29:40,706 were way out there bringing in the new paradigm. 603 00:29:41,813 --> 00:29:46,050 And he used to call me up on the telephone late at night, 604 00:29:46,050 --> 00:29:49,320 because he was bothered, and we'd talk about it. 605 00:29:49,320 --> 00:29:50,855 Mandelbrot was saying, 606 00:29:50,855 --> 00:29:53,858 ''This is a branch of geometry just like Euclid.'' 607 00:29:53,858 --> 00:29:54,959 Well, that offended them. 608 00:29:54,959 --> 00:29:57,462 They said, ''No, this is an artifact 609 00:29:57,462 --> 00:30:00,659 of your stupid computing machine.'' 610 00:30:02,500 --> 00:30:05,503 MANDELBROT: l know very well that there is this line 611 00:30:05,503 --> 00:30:06,938 that fractals are pretty pictures, 612 00:30:06,938 --> 00:30:07,972 But are pretty useless. 613 00:30:07,972 --> 00:30:09,440 Well, it's a pretty jingle, 614 00:30:09,440 --> 00:30:11,305 But it's completely ridiculous. 615 00:30:11,843 --> 00:30:14,179 NARRATOR: Mandelbrot replied to his critics 616 00:30:14,179 --> 00:30:18,516 with his new book: The Fractal Geometry of Nature. 617 00:30:18,516 --> 00:30:20,218 lt was filled with examples 618 00:30:20,218 --> 00:30:23,021 of how his ideas could be useful to science. 619 00:30:23,021 --> 00:30:25,657 Mandelbrot argued that with fractals, 620 00:30:25,657 --> 00:30:28,726 he could precisely measure natural shapes 621 00:30:28,726 --> 00:30:31,529 and make calculations that could be applied 622 00:30:31,529 --> 00:30:33,998 to all kinds of formations, 623 00:30:33,998 --> 00:30:37,135 from the drainage patterns of rivers 624 00:30:37,135 --> 00:30:39,070 to the movements of clouds. 625 00:30:39,070 --> 00:30:42,307 DEVLl N : So this domain of growing, living systems, 626 00:30:42,307 --> 00:30:44,142 which l , along with most other mathematicians, 627 00:30:44,142 --> 00:30:46,744 had always regarded as pretty well off-limits 628 00:30:46,744 --> 00:30:49,681 for mathematics and certainly off-limits for geometry, 629 00:30:49,681 --> 00:30:51,282 suddenly was center stage. 630 00:30:51,282 --> 00:30:54,052 lt was Mandelbrot's book that convinced us 631 00:30:54,052 --> 00:30:56,120 that this wasn't just artwork. 632 00:30:56,120 --> 00:30:58,656 This was new science in the making. 633 00:30:58,656 --> 00:31:01,459 This was a completely new way of looking 634 00:31:01,459 --> 00:31:03,127 at the world in which we live 635 00:31:03,127 --> 00:31:05,930 that allowed us not just to look at it, 636 00:31:05,930 --> 00:31:07,065 not just to measure it, 637 00:31:07,065 --> 00:31:10,068 But to do mathematics and thereby understand it 638 00:31:10,068 --> 00:31:13,094 in a deeper way than we had before. 639 00:31:13,738 --> 00:31:16,641 As someone who's been working with fractals for 20 years, 640 00:31:16,641 --> 00:31:18,743 l'm not going to tell you fractals are cool. 641 00:31:18,743 --> 00:31:21,679 l'm going to tell you fractals are useful, 642 00:31:21,679 --> 00:31:24,215 and that's what's important to me. 643 00:31:24,215 --> 00:31:27,785 NARRATOR: ln the 1990s, a Boston radio astronomer 644 00:31:27,785 --> 00:31:30,955 named Nathan Cohen used fractal mathematics 645 00:31:30,955 --> 00:31:32,991 to make a technological breakthrough 646 00:31:32,991 --> 00:31:34,959 in electronic communication. 647 00:31:34,959 --> 00:31:35,593 (beeping ) 648 00:31:35,593 --> 00:31:39,464 Cohen had a hobby: he was a ham radio operator, 649 00:31:39,464 --> 00:31:40,932 But his landlord had a rule 650 00:31:40,932 --> 00:31:43,935 against rigging antennas on the building. 651 00:31:43,935 --> 00:31:46,738 NATHAN COHEN : l was at an astronomy conference in Hungary, 652 00:31:46,738 --> 00:31:48,473 and Dr. Mandelbrot was giving a talk 653 00:31:48,473 --> 00:31:51,276 about the large-scale structure of the universe 654 00:31:51,276 --> 00:31:56,915 and reporting how using fractals is a very good way 655 00:31:56,915 --> 00:31:58,349 of understanding that kind of structure, 656 00:31:58,349 --> 00:32:03,221 which really wowed the entire group of astronomers. 657 00:32:03,221 --> 00:32:05,123 He showed several different fractals 658 00:32:05,123 --> 00:32:08,092 that l , in my own mind, looked at and said, 659 00:32:08,092 --> 00:32:08,927 ''Oh, wouldn't it be funny 660 00:32:08,927 --> 00:32:11,362 ''if you made an antenna out of that shape? 661 00:32:11,362 --> 00:32:12,931 l wonder what it would do.'' 662 00:32:12,931 --> 00:32:15,199 NARRATOR: One of the first designs he tried 663 00:32:15,199 --> 00:32:18,903 was inspired by one of the 19th century ''monsters'': 664 00:32:18,903 --> 00:32:21,572 the snowflake of Helge von Koch. 665 00:32:21,572 --> 00:32:23,708 l thought back to the lecture and said, 666 00:32:23,708 --> 00:32:25,176 ''Well, l've got a piece of wire. 667 00:32:25,176 --> 00:32:28,111 What happens if l bend it?'' 668 00:32:29,180 --> 00:32:31,382 After l bent the wire, l hooked it up 669 00:32:31,382 --> 00:32:32,951 to the cable and my ham radio, 670 00:32:32,951 --> 00:32:35,486 and l was quite surprised to see that it worked 671 00:32:35,486 --> 00:32:37,355 the first time out of the box. 672 00:32:37,355 --> 00:32:38,790 lt worked very well, and l discovered 673 00:32:38,790 --> 00:32:40,925 that, much of a surprise to me, 674 00:32:40,925 --> 00:32:42,493 that l could actually make the antenna 675 00:32:42,493 --> 00:32:46,497 much smaller using the fractal design, 676 00:32:46,497 --> 00:32:48,366 so it was, frankly, an interesting way 677 00:32:48,366 --> 00:32:51,733 to beat a bad rap with the landlord. 678 00:32:52,503 --> 00:32:55,840 NARRATOR: Cohen's experiments soon led him to another discovery. 679 00:32:55,840 --> 00:32:59,844 Using a fractal design not only made antennas smaller, 680 00:32:59,844 --> 00:33:01,713 But enabled them to receive 681 00:33:01,713 --> 00:33:04,482 a much wider range of frequencies. 682 00:33:04,482 --> 00:33:07,452 COHEN : Using fractals, experimentally l came up 683 00:33:07,452 --> 00:33:09,153 with a very wideband antenna. 684 00:33:09,153 --> 00:33:10,188 And then l worked backwards 685 00:33:10,188 --> 00:33:12,056 and said, ''Why is it working this way? 686 00:33:12,056 --> 00:33:15,059 ''What is it about nature that requires you 687 00:33:15,059 --> 00:33:17,595 to use the fractal to get there?'' 688 00:33:17,595 --> 00:33:19,297 The result of that work was 689 00:33:19,297 --> 00:33:21,466 a mathematical theorem that showed 690 00:33:21,466 --> 00:33:23,301 if you want to get something 691 00:33:23,301 --> 00:33:25,069 that works as an antenna 692 00:33:25,069 --> 00:33:27,405 over a very wide range of frequencies, 693 00:33:27,405 --> 00:33:29,340 you need to have self-similarity. 694 00:33:29,340 --> 00:33:33,378 lt has to be fractal in its shape to make it work. 695 00:33:33,378 --> 00:33:36,447 Now, that was an exact solution. lt wasn't like, 696 00:33:36,447 --> 00:33:37,548 ''Oh, here's a way of doing it 697 00:33:37,548 --> 00:33:39,617 and there's a lot of other ways of doing it.'' 698 00:33:39,617 --> 00:33:41,119 lt turned out mathematically, 699 00:33:41,119 --> 00:33:43,821 we were able to demonstrate that was the only technique 700 00:33:43,821 --> 00:33:45,189 you would use to get there. 701 00:33:45,189 --> 00:33:46,224 ( cell phone ringing ) 702 00:33:46,224 --> 00:33:47,458 NARRATOR: Cohen made his discovery 703 00:33:47,458 --> 00:33:50,962 at a time when cell phone companies were facing a problem. 704 00:33:50,962 --> 00:33:54,932 They were offering new features to their customers, 705 00:33:54,932 --> 00:33:57,835 like Bluetooth, walkie-talkie, and Wi-Fi, 706 00:33:57,835 --> 00:34:01,439 But each of them ran on a separate frequency. 707 00:34:01,439 --> 00:34:03,207 COHEN : You need to be able to use all 708 00:34:03,207 --> 00:34:04,909 those different frequencies and have access to them 709 00:34:04,909 --> 00:34:09,480 without ten stubby antennas sticking out at the same time. 710 00:34:09,480 --> 00:34:10,415 The alternative option is 711 00:34:10,415 --> 00:34:12,417 you can let your cell phone look like a porcupine. 712 00:34:12,417 --> 00:34:15,648 But most people don't want to carry a round a porcupine. 713 00:34:17,388 --> 00:34:19,991 NARRATOR: Today, fractal antennas are used 714 00:34:19,991 --> 00:34:21,959 in tens of millions of cell phones, 715 00:34:21,959 --> 00:34:26,030 and other wireless communication devices all over the world. 716 00:34:26,030 --> 00:34:30,501 COHEN : We're going to see over the next ten to 15 to 20 years that 717 00:34:30,501 --> 00:34:32,136 you're going to have to use fractals 718 00:34:32,136 --> 00:34:34,539 because it's the only way to get, uh, cheaper costs 719 00:34:34,539 --> 00:34:36,974 and smaller size for all the complex 720 00:34:36,974 --> 00:34:39,101 telecommunication needs we're having. 721 00:34:41,779 --> 00:34:43,514 MANDELBROT: Once you realize that 722 00:34:43,514 --> 00:34:47,585 a shrewd engineer would use fractals in many, many contexts, 723 00:34:47,585 --> 00:34:52,123 you better understand why nature, which is shrewder, 724 00:34:52,123 --> 00:34:54,125 uses them in its ways. 725 00:34:54,125 --> 00:34:55,927 They're all over in biology. 726 00:34:55,927 --> 00:34:56,928 They're solutions 727 00:34:56,928 --> 00:34:59,464 that natural selection has come up with 728 00:34:59,464 --> 00:35:03,000 over and over and over and over again. 729 00:35:03,000 --> 00:35:04,502 NARRATOR: One powerful example: 730 00:35:04,502 --> 00:35:07,572 the rhythms of the heart. (beating ) 731 00:35:07,572 --> 00:35:10,808 Something that Boston cardiologist Ary Goldberger 732 00:35:10,808 --> 00:35:14,212 has been studying his entire professional life. 733 00:35:14,212 --> 00:35:16,380 ARY GOLDBERGER: The notion of sort of the human body 734 00:35:16,380 --> 00:35:18,916 as a machine goes back through the tradition 735 00:35:18,916 --> 00:35:20,751 of Newton and the machine like universe. 736 00:35:20,751 --> 00:35:21,986 So somehow we're, we're machines, 737 00:35:21,986 --> 00:35:24,889 we're mechanisms; the heartbeat is this timekeeper. 738 00:35:24,889 --> 00:35:27,592 Galileo was reported to have used 739 00:35:27,592 --> 00:35:29,260 his pulse to time 740 00:35:29,260 --> 00:35:31,262 the swinging of a pendular motion. 741 00:35:31,262 --> 00:35:35,700 So that all fit in with the idea that a normal heartbeat 742 00:35:35,700 --> 00:35:37,135 is like a metronome. 743 00:35:37,135 --> 00:35:39,837 NARRATOR: But when Goldberger and his colleagues 744 00:35:39,837 --> 00:35:42,540 analyzed data from thousands of people, 745 00:35:42,540 --> 00:35:45,543 they found the old theory was wrong. 746 00:35:45,543 --> 00:35:46,544 MADALENA D AMASlO COSTA: This is, um, 747 00:35:46,544 --> 00:35:50,848 where l show the heartbeat time series of a healthy subject. 748 00:35:50,848 --> 00:35:52,150 And as you can see, 749 00:35:52,150 --> 00:35:55,453 the heartbeat is not constant over time. 750 00:35:55,453 --> 00:35:57,522 lt fluctuates, and it fluctuates a lot. 751 00:35:57,522 --> 00:35:59,090 For example, in this case it fluctuates between 752 00:35:59,090 --> 00:36:03,356 60 beats per minute and 120 beats per minute. 753 00:36:03,761 --> 00:36:06,264 NARRATOR: The patterns looked familiar to Goldberger, 754 00:36:06,264 --> 00:36:09,634 who happened to have read Benoit Mandelhbrot's, book. 755 00:36:09,634 --> 00:36:12,403 GOLDBERGER: When you actually plotted out the intervals 756 00:36:12,403 --> 00:36:16,174 between heartbeats, what you saw was very close 757 00:36:16,174 --> 00:36:19,777 to the rough edges of the mountain ranges 758 00:36:19,777 --> 00:36:22,541 that were in Mandelbrot's book. 759 00:36:22,847 --> 00:36:25,483 You blow them up, uh, expand them, 760 00:36:25,483 --> 00:36:27,785 you actually see that there are more of these 761 00:36:27,785 --> 00:36:28,986 wrinkles upon wrinkles. 762 00:36:28,986 --> 00:36:31,355 The healthy heartbeat, it turned out, 763 00:36:31,355 --> 00:36:33,949 had this fractal architecture. 764 00:36:34,659 --> 00:36:36,227 People said, "This, isn't cardniology. 765 00:36:36,227 --> 00:36:38,796 Do cardiology if you want to get funded.'' 766 00:36:38,796 --> 00:36:42,200 But it turns, out it is, cardniology. 767 00:36:42,200 --> 00:36:45,770 NARRATOR: Goldberger found that the healthy heartbeat 768 00:36:45,770 --> 00:36:47,772 has a distinctive fractal pattern, 769 00:36:47,772 --> 00:36:51,609 a signature that one day may help cardiologists 770 00:36:51,609 --> 00:36:54,169 spot heart problems sooner. 771 00:36:56,781 --> 00:36:59,217 Please look a round the screen for me. 772 00:36:59,217 --> 00:37:01,953 All right, Cooper, we're going to do 773 00:37:01,953 --> 00:37:03,054 the calibration. 774 00:37:03,054 --> 00:37:05,523 NARRATOR: At the University of Oregon, 775 00:37:05,523 --> 00:37:08,492 Richard Taylor is using fractals to reveal 776 00:37:08,492 --> 00:37:12,730 the secrets of another part of the body: the eye. 777 00:37:12,730 --> 00:37:14,532 TAYLOR: What we want to do 778 00:37:14,532 --> 00:37:16,634 is see what is that eye doing 779 00:37:16,634 --> 00:37:20,838 that allows it to absorb so much visual information. 780 00:37:20,838 --> 00:37:24,442 And so that's what led us into the eye trajectories. 781 00:37:24,442 --> 00:37:27,878 Under the monitor is a little infrared camera, 782 00:37:27,878 --> 00:37:30,147 which will actually monitor 783 00:37:30,147 --> 00:37:31,282 where the eye is looking. 784 00:37:31,282 --> 00:37:33,851 And it actually records that data. 785 00:37:33,851 --> 00:37:36,587 And so what we get out is a trajectory 786 00:37:36,587 --> 00:37:38,623 of where the eye has been looking. 787 00:37:38,623 --> 00:37:41,025 Oh, it's interesting how they go around 788 00:37:41,025 --> 00:37:41,959 in the patterns. . . 789 00:37:41,959 --> 00:37:44,562 TAYLOR: And so the computer will get out this graph, 790 00:37:44,562 --> 00:37:47,431 and it will look, you know, have all of these various, 791 00:37:47,431 --> 00:37:49,500 uh, little structure in it. 792 00:37:49,500 --> 00:37:52,036 And it's that pattern that we zoom in-- 793 00:37:52,036 --> 00:37:53,971 we tell the computer to zoom in on-- 794 00:37:53,971 --> 00:37:56,107 and, and see the fractal dimension. 795 00:37:56,107 --> 00:38:01,178 NARRATOR: The tests show that the eye does not always look at things 796 00:38:01,178 --> 00:38:03,447 in an orderly or smooth way. 797 00:38:03,447 --> 00:38:06,183 lf we could understand more about how the eye 798 00:38:06,183 --> 00:38:08,486 takes in information, we could do 799 00:38:08,486 --> 00:38:10,521 a better job of designing the things 800 00:38:10,521 --> 00:38:12,990 that we really need to see. 801 00:38:12,990 --> 00:38:14,258 TAYLOR: A traffic light. 802 00:38:14,258 --> 00:38:16,093 You're looking at the traffic light. 803 00:38:16,093 --> 00:38:17,295 You've got traffic. 804 00:38:17,295 --> 00:38:18,095 You've got pedestrians. 805 00:38:18,095 --> 00:38:20,298 Your eye is looking all over the place 806 00:38:20,298 --> 00:38:22,926 trying to assess all of this information. 807 00:38:23,668 --> 00:38:27,872 People design aircraft cockpits,, rows, of dials, 808 00:38:27,872 --> 00:38:28,939 and things like that. 809 00:38:28,939 --> 00:38:31,809 lf your eye is darting around 810 00:38:31,809 --> 00:38:34,111 based on a fractal geometry, though, 811 00:38:34,111 --> 00:38:35,613 maybe that's not the best way. 812 00:38:35,613 --> 00:38:39,640 Maybe you don't want these things in a simple row. 813 00:38:40,284 --> 00:38:42,353 We're trying to work out the natural way 814 00:38:42,353 --> 00:38:45,156 that the eye wants to absorb the information. 815 00:38:45,156 --> 00:38:46,457 ls it going to be similar 816 00:38:46,457 --> 00:38:49,760 to a lot of these other subconscious processes? 817 00:38:49,760 --> 00:38:52,263 Bodymotion, when youre balancing , 818 00:38:52,263 --> 00:38:54,265 what are you actually doing there? 819 00:38:54,265 --> 00:38:56,967 lt's something subconscious, and it works. 820 00:38:56,967 --> 00:39:00,571 And you're stringing together big sways 821 00:39:00,571 --> 00:39:01,906 and small sways and smaller sways. 822 00:39:01,906 --> 00:39:04,208 Could those all be connected together 823 00:39:04,208 --> 00:39:08,546 to actually be doing a fractal pattern there? 824 00:39:08,546 --> 00:39:12,249 More and more physiological processes 825 00:39:12,249 --> 00:39:14,410 have been found to be fractal. 826 00:39:15,820 --> 00:39:18,856 NARRATOR: Not everyone in science is convinced 827 00:39:18,856 --> 00:39:23,094 of fractal geometry's potential for delivering new knowledge. 828 00:39:23,094 --> 00:39:25,529 Skeptics argue that it's done little 829 00:39:25,529 --> 00:39:27,598 to advance mathematical theory. 830 00:39:27,598 --> 00:39:31,135 But in Toronto, biophysicist Peter Burns, 831 00:39:31,135 --> 00:39:32,336 and his colleagues 832 00:39:32,336 --> 00:39:36,040 see fractals as a practical tool, a way to develop 833 00:39:36,040 --> 00:39:38,743 mathematical models that might help 834 00:39:38,743 --> 00:39:41,045 in diagnosing cancer earlier. 835 00:39:41,045 --> 00:39:44,782 Detecting very small tumors is one of the big challenges 836 00:39:44,782 --> 00:39:46,716 in medical imaging. 837 00:39:47,418 --> 00:39:48,953 NARRATOR: Burns, knew that one 838 00:39:48,953 --> 00:39:51,255 early sign of cancer is particularly 839 00:39:51,255 --> 00:39:53,724 difficult to see: a network 840 00:39:53,724 --> 00:39:57,194 of tiny blood vessels that forms with the tumor. 841 00:39:57,194 --> 00:39:59,163 Conventional imaging techniques, 842 00:39:59,163 --> 00:40:02,533 like ultrasound, aren't powerful enough to show them. 843 00:40:02,533 --> 00:40:04,735 BURNS: We need to be able to see structures which are 844 00:40:04,735 --> 00:40:08,239 just a few tenths of a millionths of a meter across. 845 00:40:08,239 --> 00:40:10,241 When it comes to a living patient, 846 00:40:10,241 --> 00:40:12,643 we don't have the tools to be able 847 00:40:12,643 --> 00:40:14,178 to see these tiny blood vessels. 848 00:40:14,178 --> 00:40:18,883 NARRATOR: But ultrasound does provide a very good picture 849 00:40:18,883 --> 00:40:21,352 of the overall movement of blood. 850 00:40:21,352 --> 00:40:23,721 "ls, there any way," Burns, wondered, 851 00:40:23,721 --> 00:40:26,957 ''that images of blood flow could reveal the hidden 852 00:40:26,957 --> 00:40:28,626 structure of the blood vessels?'' 853 00:40:28,626 --> 00:40:31,495 To find out, Burns, and his, colleagues, 854 00:40:31,495 --> 00:40:33,397 used fractal geometry 855 00:40:33,397 --> 00:40:35,065 to make a mathematical model. 856 00:40:35,065 --> 00:40:36,600 BURNS: lf you have a mathematical way 857 00:40:36,600 --> 00:40:38,903 of analyzing a structure, 858 00:40:38,903 --> 00:40:40,137 you can make a model. 859 00:40:40,137 --> 00:40:42,973 What fractals do is they give you some simple rules 860 00:40:42,973 --> 00:40:44,375 by which you can create models. 861 00:40:44,375 --> 00:40:47,845 And by changing some of the parameters of the model, 862 00:40:47,845 --> 00:40:50,211 we can change how the structure looks. 863 00:40:50,815 --> 00:40:51,916 NARRATOR: The model showed 864 00:40:51,916 --> 00:40:54,785 the flow of blood in a kidney, 865 00:40:54,785 --> 00:40:56,086 first through normal blood vessels, 866 00:40:56,086 --> 00:40:59,490 and then through vessels feeding a cancerous tumor. 867 00:40:59,490 --> 00:41:02,760 Burns, discovered that the two kinds, of networks, 868 00:41:02,760 --> 00:41:06,797 had very different fractal dimensions. 869 00:41:06,797 --> 00:41:08,566 lnstead of being neatly bifurcating, 870 00:41:08,566 --> 00:41:10,835 looking like a, a nice elm tree, 871 00:41:10,835 --> 00:41:15,039 the tumor vasculature is chaotic and tangled 872 00:41:15,039 --> 00:41:19,109 and disorganized, looking more like a mistletoe bush. 873 00:41:19,109 --> 00:41:21,245 NARRATOR: And the flow of blood 874 00:41:21,245 --> 00:41:23,881 through these tangled vessels looked very different 875 00:41:23,881 --> 00:41:26,450 than in a normal network-- a difference 876 00:41:26,450 --> 00:41:31,387 doctors might one day be able to detect with ultrasound. 877 00:41:31,722 --> 00:41:34,725 We always thought that we have to make medical images 878 00:41:34,725 --> 00:41:37,394 sharper and sharper, ever more precise, 879 00:41:37,394 --> 00:41:39,997 ever more microscopic in their resolution, 880 00:41:39,997 --> 00:41:42,199 to find out the information 881 00:41:42,199 --> 00:41:45,169 about the structure that's there. 882 00:41:45,169 --> 00:41:46,237 What's exciting about this 883 00:41:46,237 --> 00:41:47,905 is it's giving us microscopic information 884 00:41:47,905 --> 00:41:52,740 without us actually having to look through a microscope. 885 00:41:53,677 --> 00:41:56,947 We think that this fractal approach may be helpful 886 00:41:56,947 --> 00:42:00,150 in distinguishing benign from malignant lesions 887 00:42:00,150 --> 00:42:03,347 in a way that hasn't been possible up to now. 888 00:42:03,821 --> 00:42:06,957 NARRATOR: lt may take years before fractals 889 00:42:06,957 --> 00:42:09,059 can help doctors predict cancer. 890 00:42:09,059 --> 00:42:12,263 But they are already offering clues, to one of biology's, 891 00:42:12,263 --> 00:42:13,998 more tantalizing mysteries: 892 00:42:13,998 --> 00:42:18,669 why big animals use energy more efficiently 893 00:42:18,669 --> 00:42:20,569 than little ones. 894 00:42:21,372 --> 00:42:23,407 That's a question that fascinates 895 00:42:23,407 --> 00:42:24,441 biologists James Brown 896 00:42:24,441 --> 00:42:27,511 and Brian Enquist and physicist Geoffrey West. 897 00:42:27,511 --> 00:42:29,880 GEOFFREY WEST: There is an extraordinary 898 00:42:29,880 --> 00:42:33,951 economy of scale as you increase in size. 899 00:42:33,951 --> 00:42:36,921 NARRATOR: An elephant, for example, 900 00:42:36,921 --> 00:42:40,624 is 200,000 times heavier than a mouse, 901 00:42:40,624 --> 00:42:44,128 But uses only about 10,000 times more energy 902 00:42:44,128 --> 00:42:46,931 in the form of calories it consumes. 903 00:42:46,931 --> 00:42:51,502 WEST: The bigger you are, you actually need less energy 904 00:42:51,502 --> 00:42:55,739 per gram of tissue to stay alive. 905 00:42:55,739 --> 00:42:58,970 That is an amazing fact. 906 00:42:59,410 --> 00:43:01,412 NARRATOR: And even more amazing is the fact 907 00:43:01,412 --> 00:43:05,482 that this relationship between the mass and energy use 908 00:43:05,482 --> 00:43:08,319 of any living thing is governed by a strict 909 00:43:08,319 --> 00:43:10,354 mathematical formula. 910 00:43:10,354 --> 00:43:12,590 JAMES BROW N: So far as, we know, that law 911 00:43:12,590 --> 00:43:18,462 is universal, or almost universal, across all of life. 912 00:43:18,462 --> 00:43:21,332 So it operates from the tiniest bacteria 913 00:43:21,332 --> 00:43:25,393 to whales and Sequoia trees. 914 00:43:26,303 --> 00:43:27,638 NARRATOR: But even though this, law 915 00:43:27,638 --> 00:43:29,840 had been discovered back in the 1930s, 916 00:43:29,840 --> 00:43:32,610 no one had been able to explain it. 917 00:43:32,610 --> 00:43:34,712 BROWN: We had this, idea that 918 00:43:34,712 --> 00:43:37,615 it probably had something to do with how resources 919 00:43:37,615 --> 00:43:40,317 are distributed within the bodies of organisms 920 00:43:40,317 --> 00:43:41,952 as they varied in size. 921 00:43:41,952 --> 00:43:44,388 We took this big leap and said, 922 00:43:44,388 --> 00:43:46,790 ''All of life in some way 923 00:43:46,790 --> 00:43:50,961 ''is sustained by these underlying 924 00:43:50,961 --> 00:43:55,432 ''networks that are transporting oxygen, 925 00:43:55,432 --> 00:43:59,837 resources, metabolites that are feeding cells.'' 926 00:43:59,837 --> 00:44:01,171 Circulatory systems 927 00:44:01,171 --> 00:44:02,506 and respiratory systems 928 00:44:02,506 --> 00:44:04,108 and renal systems 929 00:44:04,108 --> 00:44:05,876 and neural systems. 930 00:44:05,876 --> 00:44:10,080 lt was obvious that fractals were staring us in the face. 931 00:44:10,080 --> 00:44:13,917 NARRATOR: lf all these biological networks are fractal, 932 00:44:13,917 --> 00:44:17,554 it means they obey some simple mathematical rules, 933 00:44:17,554 --> 00:44:21,058 which can lead to new insights into how they work. 934 00:44:21,058 --> 00:44:23,160 BROWN : lf you think about it for a minute, 935 00:44:23,160 --> 00:44:25,162 it would be incredibly inefficient 936 00:44:25,162 --> 00:44:26,196 to have a set of blueprints 937 00:44:26,196 --> 00:44:29,933 for every single stage of increasing size. 938 00:44:29,933 --> 00:44:31,969 But if yoU have a fractal code, 939 00:44:31,969 --> 00:44:34,672 a code that says when to branch 940 00:44:34,672 --> 00:44:38,175 as you get bigger and bigger, then, uh, 941 00:44:38,175 --> 00:44:40,678 a very simple genetic code can produce 942 00:44:40,678 --> 00:44:44,011 what looks like a complicated organism. 943 00:44:44,748 --> 00:44:49,086 Evolution by natural selection has hit upon a design 944 00:44:49,086 --> 00:44:53,284 that appears to give the most bang for the buck. 945 00:44:55,993 --> 00:44:59,554 NARRATOR: ln 1997, West Brown 946 00:44:59,630 --> 00:45:01,723 and Enquist announced their controversial theory 947 00:45:01,799 --> 00:45:06,031 that fractals hold the key to the mysterious relationship 948 00:45:06,170 --> 00:45:08,798 between mass and energy use in animals. 949 00:45:08,872 --> 00:45:13,707 Now, they are putting their theory to a bold new test: 950 00:45:13,811 --> 00:45:15,108 an experiment to help determine 951 00:45:15,179 --> 00:45:17,170 if the fractal structure of a single tree 952 00:45:17,247 --> 00:45:21,445 can predict how an entire rain forest works. 953 00:45:21,652 --> 00:45:24,348 Measurements of its trunk. . . 954 00:45:28,125 --> 00:45:31,528 NARRATOR: Enquist has traveled to Costa Rica-- 955 00:45:31,528 --> 00:45:33,263 to Guanacaste province, 956 00:45:33,263 --> 00:45:36,027 in the northwestern part of the country. 957 00:45:37,434 --> 00:45:41,371 The government has set aside more than 300,000 acres 958 00:45:41,371 --> 00:45:46,832 in Guanacaste as a conservation area. 959 00:45:47,911 --> 00:45:50,547 This rain forest, like others around the world, 960 00:45:50,547 --> 00:45:53,917 plays a vital role in regulating the earth's climate, 961 00:45:53,917 --> 00:45:57,853 by removing carbon dioxide from the atmosphere. 962 00:45:58,288 --> 00:45:59,857 lf you look at the forest, it basically breathes. 963 00:45:59,857 --> 00:46:03,594 And if we understand the total amount of carbon dioxide, 964 00:46:03,594 --> 00:46:06,563 that's coming into, uh, these trees within this forest 965 00:46:06,563 --> 00:46:09,900 we can then better understand how, uh, 966 00:46:09,900 --> 00:46:12,536 this forest then ultimately regulates the total amount 967 00:46:12,536 --> 00:46:15,105 of carbon dioxide in our atmosphere. 968 00:46:15,105 --> 00:46:18,942 NARRATOR: With carbon dioxide levels around the world rising, 969 00:46:18,942 --> 00:46:23,080 how much CO2 can rain forests like this one absorb, 970 00:46:23,080 --> 00:46:26,583 and how important is their role in protecting us 971 00:46:26,583 --> 00:46:30,053 from further global warming? 972 00:46:30,053 --> 00:46:32,289 Enquist and a team of US scientists 973 00:46:32,289 --> 00:46:36,994 think that fractal geometry may help answer these questions. 974 00:46:36,994 --> 00:46:37,628 . . .baseline. 975 00:46:37,628 --> 00:46:39,830 Let's, try to get the height of the tree measured. 976 00:46:39,830 --> 00:46:42,599 NARRATOR: They are going to start by doing 977 00:46:42,599 --> 00:46:43,534 just about the last thing 978 00:46:43,534 --> 00:46:48,639 you'd think a scientist would do here: cut down a balsa tree. 979 00:46:48,639 --> 00:46:50,073 lt's dying anyway, 980 00:46:50,073 --> 00:46:52,910 and they have the permission of the authorities. 981 00:46:52,910 --> 00:46:53,610 So Christina, 982 00:46:53,610 --> 00:46:55,612 as soon as you know the height of that tree, 983 00:46:55,612 --> 00:46:58,882 we can actually figure out the approximate angle 984 00:46:58,882 --> 00:47:00,350 that we need to take it down on. 985 00:47:00,350 --> 00:47:03,821 NARRATOR: Hooking a guide line on a high branch 986 00:47:03,821 --> 00:47:06,957 helps insure the tree will land where they want it to. 987 00:47:06,957 --> 00:47:08,826 Yay! 988 00:47:08,826 --> 00:47:10,594 Good work. 989 00:47:10,594 --> 00:47:12,095 Very good. 990 00:47:12,095 --> 00:47:13,864 Very nice. 991 00:47:13,864 --> 00:47:17,391 ( mechanical whirring ) 992 00:47:34,585 --> 00:47:36,053 Nice. 993 00:47:36,053 --> 00:47:37,754 Well done. 994 00:47:37,754 --> 00:47:39,523 Jose, perfectol Estaw bien ? 995 00:47:39,523 --> 00:47:42,526 NARRATOR: Enquist and his colleagues 996 00:47:42,526 --> 00:47:44,127 then measure the width and length 997 00:47:44,127 --> 00:47:49,566 of the branches to quantify the tree's fractal structure. 998 00:47:49,566 --> 00:47:51,227 Eight. 999 00:47:52,069 --> 00:47:53,559 10.06. 1000 00:47:54,938 --> 00:47:56,838 No, that's eight. 1001 00:47:59,843 --> 00:48:01,333 6.3. .03. 1002 00:48:02,412 --> 00:48:03,847 6.0. 1003 00:48:03,847 --> 00:48:04,248 Eight. 1004 00:48:04,248 --> 00:48:05,772 Seven on the nose. 1005 00:48:05,983 --> 00:48:09,419 NARRATOR: They also measure how much carbon a single leaf contains, 1006 00:48:09,419 --> 00:48:11,822 which should allow them to figure out 1007 00:48:11,822 --> 00:48:14,124 what the whole tree can absorb. 1008 00:48:14,124 --> 00:48:16,326 So if we know the amount of carbon dioxide 1009 00:48:16,326 --> 00:48:18,629 that one leaf is able to take in, 1010 00:48:18,629 --> 00:48:21,498 then hopefully using the fractal branching rule 1011 00:48:21,498 --> 00:48:23,800 we can know how much carbon dioxide 1012 00:48:23,800 --> 00:48:25,569 the entire tree is taking in. 1013 00:48:25,569 --> 00:48:28,272 NARRATOR: Their next step is to move 1014 00:48:28,272 --> 00:48:32,003 from the tree to the whole forest. 1015 00:48:34,711 --> 00:48:37,581 All right, this is good. 1016 00:48:37,581 --> 00:48:39,249 1 3.2. 1017 00:48:39,249 --> 00:48:39,983 3.3. 1018 00:48:39,983 --> 00:48:41,752 ENQU lST: We're going to census this forest. 1019 00:48:41,752 --> 00:48:43,820 We're going to be measuring 1020 00:48:43,820 --> 00:48:45,289 the diameter at the base of the tree, 1021 00:48:45,289 --> 00:48:48,125 ranging all the way from the largest trees down 1022 00:48:48,125 --> 00:48:49,626 to the smallest trees. 1023 00:48:49,626 --> 00:48:53,897 And in that way we can then sample the distriBution 1024 00:48:53,897 --> 00:48:55,999 of sizes, within the forest. 1025 00:48:55,999 --> 00:48:59,457 lt's 61 .8 centimeters. 1026 00:48:59,870 --> 00:49:04,241 Even though the forest may appear random and chaotic, 1027 00:49:04,241 --> 00:49:06,643 the team believes it actually has a structure-- 1028 00:49:06,643 --> 00:49:09,947 one that amazingly is almost identical 1029 00:49:09,947 --> 00:49:14,441 to the fractal structure of the tree they have just cut down. 1030 00:49:14,851 --> 00:49:18,188 BROWN: The beautiful thing is, 1031 00:49:18,188 --> 00:49:21,024 that the distribution of the sizes 1032 00:49:21,024 --> 00:49:23,493 of individual trees in the forest 1033 00:49:23,493 --> 00:49:26,663 appears to exactly match the distribution 1034 00:49:26,663 --> 00:49:29,967 of the sizes, of individnual branches, 1035 00:49:29,967 --> 00:49:33,003 within a single tree. 1036 00:49:33,003 --> 00:49:34,805 NARRATOR: lf they're correct, 1037 00:49:34,805 --> 00:49:38,275 studying a single tree will make it easier 1038 00:49:38,275 --> 00:49:40,444 to predict how much carbon dioxide 1039 00:49:40,444 --> 00:49:44,346 an entire forest can absorb. 1040 00:49:45,916 --> 00:49:47,284 When they finish here, 1041 00:49:47,284 --> 00:49:50,053 they take their measurements back to base camp, 1042 00:49:50,053 --> 00:49:52,856 where they'll see if their ideas hold up. 1043 00:49:52,856 --> 00:49:55,125 So is this the. . . this is the tree plot, right? 1044 00:49:55,125 --> 00:49:56,259 The cool thing is that, 1045 00:49:56,259 --> 00:49:59,763 if you look at the tree, you see the same pattern 1046 00:49:59,763 --> 00:50:02,065 amongst the branches as we see amongst the trunks 1047 00:50:02,065 --> 00:50:03,834 in the forest. Very nice. 1048 00:50:03,834 --> 00:50:06,069 NARRATOR: Just as, they'd predicted, 1049 00:50:06,069 --> 00:50:08,638 the relative number of big and small trees 1050 00:50:08,638 --> 00:50:11,008 closely matches the relative number 1051 00:50:11,008 --> 00:50:13,343 of big and small branches,. 1052 00:50:13,343 --> 00:50:16,413 ENQUlST: lt's actually phenomenal that it is parallel. 1053 00:50:16,413 --> 00:50:18,982 The slope of that line for the tree appears 1054 00:50:18,982 --> 00:50:22,085 to be the same for the forest as well. 1055 00:50:22,085 --> 00:50:23,720 So l guess it was worth cutting up the tree. 1056 00:50:23,720 --> 00:50:26,188 lt was definitely worth cutting up the tree. 1057 00:50:27,224 --> 00:50:30,394 NARRATOR: So far, the measurements from the field 1058 00:50:30,394 --> 00:50:32,362 appear to support the scientist's theory 1059 00:50:32,362 --> 00:50:35,832 that a single tree can help scientists assess 1060 00:50:35,832 --> 00:50:38,201 how much this rain forest is helping 1061 00:50:38,201 --> 00:50:40,771 to slow down global warming. 1062 00:50:40,771 --> 00:50:43,707 By anallzing the fractal patterns, within the forest, 1063 00:50:43,707 --> 00:50:45,842 that then enables us to do something 1064 00:50:45,842 --> 00:50:48,645 that we haven't really been able to do before. 1065 00:50:48,645 --> 00:50:50,447 Have then a mathematical basis 1066 00:50:50,447 --> 00:50:53,517 to then predict how the forest as a whole 1067 00:50:53,517 --> 00:50:55,986 takes in carbon dioxide and, ultimately, 1068 00:50:55,986 --> 00:50:58,688 that's important for understanding 1069 00:50:58,688 --> 00:51:01,748 what may happen with global climate change. 1070 00:51:03,226 --> 00:51:04,494 NARRATOR: For generations, 1071 00:51:04,494 --> 00:51:07,397 scientists believed that the wildness of nature 1072 00:51:07,397 --> 00:51:09,966 could not be defined by mathematics. 1073 00:51:09,966 --> 00:51:14,538 But fractal geometry is, leading to a whole new understanding , 1074 00:51:14,538 --> 00:51:16,773 revealing an underlying order 1075 00:51:16,773 --> 00:51:21,311 governed by simple mathematical rules. 1076 00:51:21,311 --> 00:51:24,414 What l thought of in my hikes through forests, 1077 00:51:24,414 --> 00:51:26,116 that, you know, it's just a bunch of trees 1078 00:51:26,116 --> 00:51:29,219 of different sizes, big ones here, small ones there, 1079 00:51:29,219 --> 00:51:32,956 looking like it's sort of some arbitrary chaotic mess 1080 00:51:32,956 --> 00:51:36,827 actually has an extraordinary structure. 1081 00:51:36,827 --> 00:51:39,729 NARRATOR: A structure that can be mapped out 1082 00:51:39,729 --> 00:51:43,631 and measured using fractal geometry. 1083 00:51:45,402 --> 00:51:49,005 ENQUlST: What's absolutely amazing is that you can 1084 00:51:49,005 --> 00:51:51,942 translate what you see in the natural world 1085 00:51:51,942 --> 00:51:53,110 in the language of mathematics. 1086 00:51:53,110 --> 00:51:56,443 And l can't think of anything more beautiful than that. 1087 00:52:01,518 --> 00:52:03,286 Math is our one and only strategy 1088 00:52:03,286 --> 00:52:07,124 for understanding the complexity of nature. 1089 00:52:07,124 --> 00:52:10,827 Now, fractal geometry has given us 1090 00:52:10,827 --> 00:52:13,130 a much larger vocabulary. 1091 00:52:13,130 --> 00:52:14,998 And with the larger vocabulary 1092 00:52:14,998 --> 00:52:18,764 we can read more of the book of nature. 1093 00:52:49,966 --> 00:52:51,535 On NOVA's ''Hldden Dlmenslon'' Web slte, 1094 00:52:51,535 --> 00:52:56,006 explore the Mandelbrot set, see a gallery of fractal lmages 1095 00:52:56,006 --> 00:52:57,240 and much more. 1096 00:52:57,240 --> 00:53:00,107 Flnd lt on pbs. org. 1097 00:53:02,078 --> 00:53:06,742 Major funding for NOVA is provided by the following: 1098 00:53:07,784 --> 00:53:11,584 Taking on the world's toughest energy challenges. 1099 00:53:12,589 --> 00:53:14,216 And by: 1100 00:53:17,961 --> 00:53:19,519 And. . . 1101 00:53:27,904 --> 00:53:31,041 And by the Corporation for Public Broadcasting 1102 00:53:31,041 --> 00:53:35,102 and by contriButions to your PBS station from: 1103 00:53:37,480 --> 00:53:41,416 Captioned by Media Access Group at WGBH access.wgbh.org 1104 00:54:20,357 --> 00:54:23,087 NOVA is a production of: