WEBVTT 00:00:59.166 --> 00:01:01.491 Very strange. 00:01:13.230 --> 00:01:16.783 Huh, that's an odd-looking creature 00:01:27.953 --> 00:01:31.720 What kind of a crazy place is this? 00:01:45.520 --> 00:01:49.141 Well, what do you know? Square roots! 00:01:51.264 --> 00:01:59.776 Pi is equal to 3.141592653589747 etc. etc. etc. 00:02:01.545 --> 00:02:04.706 Hello? (echo) 00:02:06.103 --> 00:02:07.891 Hello, Donald. 00:02:08.061 --> 00:02:11.402 That's me! Where am I?! 00:02:11.402 --> 00:02:13.838 Mathmagic land. 00:02:13.838 --> 00:02:17.411 Mathmagic land? Never heard of it. 00:02:17.642 --> 00:02:20.879 It's the land of great adventure. 00:02:20.879 --> 00:02:23.048 Well, who are you? 00:02:23.048 --> 00:02:27.075 I'm a spirit, the true spirit of adventure. 00:02:27.075 --> 00:02:29.788 That's for me! What's next? 00:02:29.788 --> 00:02:32.974 A journey through the wonderland of mathematics. 00:02:32.974 --> 00:02:36.561 Mathematics? That's for eggheads! 00:02:36.561 --> 00:02:39.141 Eggheads? Now hold on, Donald. 00:02:39.141 --> 00:02:41.654 You like music don't you? 00:02:41.792 --> 00:02:42.726 Yeah. 00:02:42.757 --> 00:02:46.171 Well, without eggheads, there would be no music. 00:02:46.263 --> 00:02:47.129 Bah. 00:02:47.668 --> 00:02:55.009 Come on, let's go to ancient Greece, to the time of Pythagoras, the master egghead of them all. 00:02:55.009 --> 00:02:56.225 Pythagoras? 00:02:56.225 --> 00:02:58.513 The father of mathematics and music. 00:02:58.513 --> 00:03:00.435 Mathematics and music? 00:03:00.435 --> 00:03:04.622 Ahh, you'll find mathematics in the darndest places. 00:03:05.268 --> 00:03:06.445 Watch 00:03:07.198 --> 00:03:08.747 First we'll need a string 00:03:08.747 --> 00:03:09.701 Hey! 00:03:10.055 --> 00:03:13.142 Stretch it good and tight; pluck it! 00:03:14.049 --> 00:03:17.176 Now divide in half. Pluck again. 00:03:18.007 --> 00:03:21.952 You see? It's the same tone, one octave higher. 00:03:21.952 --> 00:03:24.507 Now divide the next section. 00:03:24.568 --> 00:03:26.639 And the next. 00:03:26.639 --> 00:03:31.690 Pythagoras discovered the octave had a ratio of two to one. 00:03:31.690 --> 00:03:39.475 With simple fractions, he got this [major triad] 00:03:39.475 --> 00:03:47.668 And from this harmony in numbers, developed the musical scale of today. [major scale] 00:03:49.299 --> 00:03:54.789 By golly, you DO find mathematics in the darndest places. 00:03:55.251 --> 00:03:57.737 You can imagine how excited Pythagoras was, 00:03:57.737 --> 00:04:03.384 when he shared his findings with his pals and fraternity of eggheads, known as the Pythagoreans. 00:04:03.877 --> 00:04:08.571 They used to be meet in secret to discuss their mathematical discoveries. 00:04:08.571 --> 00:04:11.340 Only members were allowed to attend. 00:04:11.340 --> 00:04:15.502 They had a secret emblem, the pentagram. 00:04:18.887 --> 00:04:22.129 Let's see what the topic is for today. 00:04:22.467 --> 00:04:26.229 [major scale] 00:04:45.662 --> 00:04:48.231 [music] 00:04:48.985 --> 00:04:50.981 What's going on? 00:04:50.981 --> 00:04:53.396 Shh! It's a jam session. 00:04:55.350 --> 00:04:57.498 Gimme something with a beat! 00:04:57.498 --> 00:04:58.600 Shhh! 00:04:59.308 --> 00:05:02.277 [percussion] 00:05:31.033 --> 00:05:34.041 So from these eggheads, the Pythagoreans, 00:05:34.041 --> 00:05:36.102 with their mathematical formula 00:05:36.102 --> 00:05:38.756 came the basis of our music of today. 00:05:38.756 --> 00:05:42.179 [big band] 00:07:06.487 --> 00:07:09.377 Pythag, old boy, put her there 00:07:12.838 --> 00:07:17.707 Now I'll be a goshdarned egghead [laugh] 00:07:17.707 --> 00:07:24.252 It was our old friend Pythagoras who discovered that the pentagram was full of mathemagic. 00:07:26.867 --> 00:07:31.703 The two shorter lines combined exactly equal the third 00:07:31.703 --> 00:07:36.288 and this line shows the magic proportions of the famous golden section 00:07:37.027 --> 00:07:40.558 The second and third lines exactly equal the fourth 00:07:40.558 --> 00:07:43.799 Once again we have the golden section 00:07:44.829 --> 00:07:47.224 But this is only the beginning 00:07:47.224 --> 00:07:49.239 Hidden within the pentagram 00:07:49.239 --> 00:07:52.345 is a secret for creating a golden rectangle 00:07:52.345 --> 00:07:58.056 which the Greeks admired for its beautiful proportions and magic qualities 00:07:58.056 --> 00:08:02.268 The star contains the golden rectangle many times over 00:08:29.283 --> 00:08:31.675 It's a most remarkable shape 00:08:31.675 --> 00:08:35.269 It can mathematically reproduce itself indefinitely 00:08:40.161 --> 00:08:43.964 All these rectangles have exactly the same proportions 00:08:51.272 --> 00:08:54.317 This figure also contains a magic spiral 00:08:54.317 --> 00:08:59.256 that repeats the proportions of the golden section into infinity 00:09:00.410 --> 00:09:05.556 To the Greeks, the golden rectangle represented a mathematical law of beauty 00:09:05.556 --> 00:09:08.986 We find it in their classical architecture 00:09:08.986 --> 00:09:13.203 The Parthenon, perhaps one of the most famous of early Greek buildings, 00:09:13.203 --> 00:09:15.708 contains many golden rectangles. 00:09:38.477 --> 00:09:42.332 These same golden proportions are also found in their sculpture. 00:09:59.101 --> 00:10:00.885 In the centuries that followed 00:10:00.885 --> 00:10:06.665 the golden rectangle dominated the idea of beauty in architecture throughout the Western world. 00:10:06.665 --> 00:10:10.718 The Cathedral of Notre Dame is an outstanding example. 00:10:12.579 --> 00:10:15.622 The Renaissance painters knew this secret well. 00:10:21.299 --> 00:10:25.960 Today, the golden rectangle is very much a part of our modern world. 00:10:30.544 --> 00:10:34.457 Modern painters have rediscovered the magic of these proportions. 00:10:38.226 --> 00:10:42.179 Indeed, this ideal proportion is to be found in life itself. 00:10:42.379 --> 00:10:44.829 Boy, oh boy, oh boy! 00:10:45.198 --> 00:10:50.429 This is mathematics? I like mathematical figures like that. 00:10:50.429 --> 00:10:52.428 Ah, ah, ah, Donald. 00:10:52.428 --> 00:10:53.747 Let me try it! 00:10:53.747 --> 00:10:54.661 No, no. 00:10:54.661 --> 00:10:56.355 Ideal proportion 00:10:56.878 --> 00:10:58.182 Not quite 00:10:59.520 --> 00:11:01.992 Uh uh. No, I'm afraid not. 00:11:04.654 --> 00:11:07.310 Well, we can't all be mathematically perfect. 00:11:07.310 --> 00:11:08.460 Oh yeah? 00:11:11.229 --> 00:11:13.830 There, I knew I could do it. 00:11:14.015 --> 00:11:16.468 Now that you're all pent up in a pentagon 00:11:16.468 --> 00:11:20.422 let's see how nature uses the same mathematical form. 00:11:20.683 --> 00:11:21.906 The petunia 00:11:25.121 --> 00:11:26.637 The star jasmine 00:11:31.099 --> 00:11:32.508 The starfish 00:11:37.123 --> 00:11:38.529 The wax flower 00:11:43.760 --> 00:11:46.812 There are literally thousands of members in good standing 00:11:46.812 --> 00:11:50.407 in nature's Pythagorean society of the star. 00:11:57.868 --> 00:12:00.915 All nature's works have a mathematical logic 00:12:00.915 --> 00:12:02.938 and her patterns are limitless. 00:12:27.338 --> 00:12:30.004 The magic proportions of the golden section 00:12:30.004 --> 00:12:33.411 are often found in the spirals of nature's designs. 00:12:49.150 --> 00:12:54.090 The profusion of mathematical forms brings to mind the words of Pythagoras: 00:12:54.090 --> 00:12:58.878 "Everything is arranged according to number and mathematical shape." 00:12:59.555 --> 00:13:02.355 Yes, there is mathematics in music, 00:13:02.355 --> 00:13:05.027 in art, in just about everything. 00:13:05.166 --> 00:13:09.483 And as as the Greeks had guessed, the rules are always the same. 00:13:36.683 --> 00:13:40.004 Well, Donald, did you enjoy your geometrical journey? 00:13:40.004 --> 00:13:45.218 Gee, Mr. Spirit, there's a lot more to mathematics than two times two! 00:13:45.218 --> 00:13:46.439 That's right, Donald 00:13:46.439 --> 00:13:49.109 And you can find mathematics in games, too! 00:13:49.109 --> 00:13:51.152 Games! Oh, boy! 00:13:51.629 --> 00:13:54.671 Let's begin with a game that's played on squares. 00:13:54.671 --> 00:13:55.782 Checkers? 00:13:55.782 --> 00:13:57.058 No, chess. 00:13:57.058 --> 00:13:58.223 Chess?! 00:13:58.223 --> 00:14:01.337 A mathematical contest between two minds. 00:14:01.352 --> 00:14:05.805 It's a game that has been enjoyed for centuries by kings and commoners. 00:14:05.805 --> 00:14:10.269 In fact, Louis Carroll, a famous mathematician with a literary mind, 00:14:10.269 --> 00:14:15.785 used chess as a setting for his classic tale, Through the Looking Glass. 00:14:16.201 --> 00:14:22.071 Alice found herself face to face with a none-too-friendly group of chess pieces. 00:14:22.071 --> 00:14:24.938 Good heavens, what's this? 00:14:25.123 --> 00:14:28.896 Upon my soul, it appears to be a lost pawn! 00:14:29.050 --> 00:14:31.873 I'm no pawn, I'm Donald Duck! 00:14:31.920 --> 00:14:34.227 He says he's Donald Duck! 00:14:34.381 --> 00:14:36.162 Preposterous! 00:14:36.270 --> 00:14:38.266 Or, it could be an Alice. 00:14:38.266 --> 00:14:39.291 Alice?! 00:14:39.291 --> 00:14:42.127 No, no no. It's a lost pawn. 00:14:42.666 --> 00:14:46.578 Lost pawn? Stop that pawn! 00:14:46.578 --> 00:14:50.643 Ow, Mr. Spirit! Help, help, help! 00:15:10.736 --> 00:15:13.032 Whew, that was close! 00:15:13.032 --> 00:15:16.341 Now you can look at this game from a safer perspective. 00:15:22.264 --> 00:15:25.048 Chess is a game of calculated strategy, 00:15:25.048 --> 00:15:27.552 and since the board is geometrical, 00:15:27.552 --> 00:15:29.737 the moves are mathematical. 00:15:53.152 --> 00:15:55.836 Checkmate, and the game is over. 00:15:55.836 --> 00:15:59.574 That's very interesting. What's next? 00:15:59.574 --> 00:16:03.321 Practically all games are played on geometrical areas. 00:16:03.321 --> 00:16:05.506 The baseball field is a diamond. 00:16:05.506 --> 00:16:06.703 Oh boy! 00:16:12.795 --> 00:16:16.008 And without mathematics, we couldn't even keep score. 00:16:16.008 --> 00:16:20.077 Football is played on a rectangle divided by yard lines. 00:16:22.030 --> 00:16:26.606 Basketball is a game of circles, spheres and rectangles. 00:16:30.729 --> 00:16:33.353 Even hopscotch has its multiple squares. 00:16:44.153 --> 00:16:45.432 What's next? 00:16:45.971 --> 00:16:47.134 Tiddlywinks? 00:16:48.027 --> 00:16:53.291 No, a mathematical game played on a field of two perfect squares 00:16:53.291 --> 00:16:55.683 using three perfect spheres 00:16:55.683 --> 00:16:57.629 and a lot of diamonds. 00:16:57.629 --> 00:16:59.716 In other words, billiards. 00:16:59.716 --> 00:17:02.609 Oh boy! That's for me! 00:17:02.609 --> 00:17:04.726 You know the game, don't you Donald? 00:17:04.726 --> 00:17:09.641 Of course, the cue ball has to hit the other two balls 00:17:09.641 --> 00:17:10.646 like this! 00:17:16.061 --> 00:17:20.142 Now let's see how an expert at three-cushion billiards uses his head. 00:17:22.280 --> 00:17:23.807 Three-cushion? 00:17:23.807 --> 00:17:27.664 Yes. The cue ball not only has to hit both the other balls, 00:17:27.664 --> 00:17:31.703 but it must contact at least three cushions before it hits the final ball. 00:17:42.734 --> 00:17:44.545 One, two, three 00:17:58.314 --> 00:18:00.180 One, two, three 00:18:10.042 --> 00:18:13.572 It takes an expert to make several shots in succession 00:18:13.572 --> 00:18:16.501 One, two, three, four 00:18:18.301 --> 00:18:19.558 five, six. 00:18:21.697 --> 00:18:24.493 Wow! That was a lucky shot! 00:18:24.908 --> 00:18:27.209 Luck? No. It's skill. 00:18:27.870 --> 00:18:30.637 For this game, you have to know all the angles. 00:18:52.067 --> 00:18:56.844 One, two, three, four, five, six, seven. 00:18:57.606 --> 00:19:00.843 That's amazing! How does he do it? 00:19:00.843 --> 00:19:02.912 First, there's technique. 00:19:02.912 --> 00:19:06.663 He's striking the cue ball low, so it'll spin backwards. 00:19:11.017 --> 00:19:14.860 Hitting the ball on the right side will make it hug the rail. 00:19:14.860 --> 00:19:17.466 These trick shots take a lot of practice. 00:19:18.712 --> 00:19:21.225 Hahaha! He missed that time! 00:19:21.225 --> 00:19:22.545 One, two... 00:19:26.130 --> 00:19:26.996 three. 00:19:29.596 --> 00:19:32.155 What's so mathematical about that? 00:19:32.155 --> 00:19:35.037 Oh, this game takes precise calculation. 00:19:35.037 --> 00:19:37.594 He figures out each shot in his head. 00:19:38.671 --> 00:19:42.700 He could play it like this, but it calls for quite a bit of luck. 00:19:42.700 --> 00:19:44.538 There is a better choice. 00:19:44.538 --> 00:19:49.360 For this, he uses the diamond markings on the rail as a mathematical guide. 00:19:49.360 --> 00:19:53.355 First, he figures the natural angle for hitting the object balls. 00:19:53.355 --> 00:19:57.674 And then he finds that his cue ball must bounce off the number three diamond. 00:19:57.674 --> 00:20:01.797 Next, he gets ready for the shot and he needs a number for his cue position. 00:20:01.797 --> 00:20:04.936 This calls for a different set of numbers. 00:20:05.198 --> 00:20:07.459 Very confusing, isn't it? 00:20:07.844 --> 00:20:09.889 Not when you get the hang of it. 00:20:09.889 --> 00:20:12.734 You see, the cue position is four. 00:20:12.734 --> 00:20:14.789 Now, a simple subtraction. 00:20:14.789 --> 00:20:16.578 Three from four is one. 00:20:16.578 --> 00:20:19.903 So, if he shoots for the first diamond, he should make it. 00:20:19.903 --> 00:20:22.459 It's called "playing the diamond system". 00:20:27.536 --> 00:20:29.440 Natural angle, 2. 00:20:29.440 --> 00:20:32.953 Cue position, one and a half, two, two and a half, three, 00:20:33.230 --> 00:20:34.413 three and a half. 00:20:34.413 --> 00:20:37.012 Two from three and a half is one and a half. 00:20:37.012 --> 00:20:41.333 So, shoot halfway between the first and second diamonds. 00:20:44.795 --> 00:20:47.212 There's nothing to it! Let me try! 00:20:51.335 --> 00:20:52.921 Let's see now. 00:20:53.167 --> 00:20:58.485 If I shoot it here, and it bounces there, and uh, no there. 00:20:58.808 --> 00:21:00.519 If I shoot it here... 00:21:00.519 --> 00:21:04.169 Four and half minus three, three and a half plus four... 00:21:04.169 --> 00:21:05.865 Add it to two... 00:21:07.711 --> 00:21:09.821 And dividing it.. and... 00:21:09.821 --> 00:21:11.713 I guess I should shoot about here. 00:21:11.713 --> 00:21:15.570 No, no, Donald. There's no guesswork to mathematics. 00:21:15.570 --> 00:21:17.062 It's quite simple. 00:21:17.062 --> 00:21:20.167 Natural angle for the hit: two. 00:21:20.167 --> 00:21:22.857 Cue position: three and a half. 00:21:22.857 --> 00:21:25.696 How much is three and a half minus two? 00:21:25.696 --> 00:21:27.991 Uhhh... one and a half! 00:21:38.714 --> 00:21:41.852 Hey! It works! Oh boy! 00:21:41.852 --> 00:21:43.594 It's a cinch! 00:21:44.610 --> 00:21:48.097 If i hit it here, add three and a half plus four 00:21:48.097 --> 00:21:50.803 Four and a half minus three... [???] 00:21:51.942 --> 00:21:54.655 You're making it tough for yourself, Donald. 00:22:05.808 --> 00:22:09.444 How do you like that for mathematics, Mr. Spirit? 00:22:09.444 --> 00:22:14.461 Wonderful, Donald. And now you're ready for the most exciting game of all. 00:22:14.461 --> 00:22:16.080 Oh, boy! 00:22:16.695 --> 00:22:19.874 And the playing field for this game is in the mind. 00:22:20.982 --> 00:22:24.665 Uh oh, look at the condition of your mind! 00:22:24.665 --> 00:22:33.437 Antiquated ideas, bungling, false concepts, superstitions, confusion! 00:22:33.437 --> 00:22:36.091 To think straight, we'll have to clean house. 00:22:48.091 --> 00:22:50.461 There, that's more like it. 00:22:50.461 --> 00:22:53.044 A nice clean sweep. 00:22:53.044 --> 00:22:56.334 This game is played with circles and triangles. 00:22:56.334 --> 00:22:58.588 Think of a perfect circle. 00:23:02.465 --> 00:23:07.917 A perfect circle. Perfect. Circle. 00:23:08.748 --> 00:23:11.695 Perfect. Ahhhhh. 00:23:11.695 --> 00:23:15.057 Put a triangle inside and turn it. 00:23:15.057 --> 00:23:18.895 Now spin the circle, and what have you got? 00:23:19.956 --> 00:23:21.156 A ball! 00:23:21.713 --> 00:23:23.576 Yes, a sphere. 00:23:23.576 --> 00:23:27.819 The shape of things is first discovered in the mind. 00:23:27.819 --> 00:23:30.325 Slice off the top and we have a... 00:23:33.309 --> 00:23:35.745 A magnifying glass! 00:23:35.745 --> 00:23:37.131 That's right. 00:23:37.131 --> 00:23:39.901 A lens is a section of a sphere. 00:23:39.901 --> 00:23:43.383 All optical instruments are created through mathematics. 00:23:46.783 --> 00:23:51.399 You see, there's a lot more to mathematics than just numbers and equations. 00:23:52.429 --> 00:23:55.246 Let's get back to our circle and triangle. 00:23:59.000 --> 00:24:01.298 Roll it and we have a... 00:24:01.298 --> 00:24:02.568 A... a wheel! 00:24:10.337 --> 00:24:14.443 The circle has been the basis for many of man's important inventions. 00:24:20.197 --> 00:24:23.552 The mind can create the most amazing things. 00:24:24.537 --> 00:24:27.475 If we spin the triangle, we have a... 00:24:27.475 --> 00:24:28.174 Cone! 00:24:28.174 --> 00:24:29.531 Slice the cone. 00:24:29.531 --> 00:24:30.466 giggle 00:24:30.466 --> 00:24:33.938 The cone is full of useful mathematical shapes. 00:24:35.815 --> 00:24:38.955 Slice it again. Slice it several times. 00:24:42.047 --> 00:24:46.738 The orbits of all planets and satellites can be found in the cone. 00:24:46.738 --> 00:24:50.555 No matter how you slice it, it's always mathematics. 00:24:50.555 --> 00:24:54.710 A slice like this gives us the reflector of a search light. 00:24:55.541 --> 00:24:59.356 A slice like this, the mirror of a giant telescope. 00:25:01.480 --> 00:25:04.680 A line on a cone, and we have a drill. 00:25:08.449 --> 00:25:10.157 And the spring. 00:25:14.342 --> 00:25:15.447 Now you're ticking. 00:25:27.063 --> 00:25:28.412 Number, please? 00:25:40.643 --> 00:25:45.338 The mind is the birthplace for all of man's scientific achievements. 00:26:00.754 --> 00:26:04.763 The mind knows no limits when used properly. 00:26:04.763 --> 00:26:06.855 Think of a pentagram, Donald. 00:26:09.532 --> 00:26:11.526 Now, put another inside. 00:26:11.526 --> 00:26:13.981 A third. And a fourth. 00:26:14.535 --> 00:26:18.069 No pencil is sharp enough to draw as fine as you can think 00:26:18.069 --> 00:26:22.093 and no paper large enough to hold your imagination. 00:26:22.093 --> 00:26:26.566 In fact, it is only in the mind that we can conceive infinity. 00:26:28.259 --> 00:26:33.040 Mathematical thinking has opened the doors to the exciting adventures of science. 00:26:35.086 --> 00:26:37.265 I'll be dog-darned! 00:26:37.265 --> 00:26:40.030 I've never seen so many doors before. 00:26:40.030 --> 00:26:43.023 Each discovery leads to many others. 00:26:43.023 --> 00:26:44.804 An endless chain. 00:26:45.127 --> 00:26:48.897 Hey! Hey! Whatsa matter with these doors? 00:26:48.897 --> 00:26:52.775 Hey! These doors won't open! They're locked! 00:26:52.775 --> 00:26:54.724 Of course they're locked. 00:26:54.724 --> 00:26:56.852 These are the doors of the future, 00:26:56.852 --> 00:26:58.568 and the key is... 00:26:58.568 --> 00:27:00.220 Mathematics! 00:27:00.328 --> 00:27:02.844 Right. Mathematics. 00:27:02.844 --> 00:27:07.517 The boundless treasures of science are locked behind those doors. 00:27:07.517 --> 00:27:13.698 In time, they will be opened by the curious and inquiring minds of future generations. 00:27:15.467 --> 00:27:17.628 In the words of Galileo: 00:27:17.628 --> 00:27:25.813 "Mathematics is the alphabet with which God has written the universe."