1 00:00:59,166 --> 00:01:01,491 Very strange. 2 00:01:13,230 --> 00:01:16,783 Huh, that's an odd-looking creature 3 00:01:27,953 --> 00:01:31,720 What kind of a crazy place is this? 4 00:01:45,520 --> 00:01:49,141 Well, what do you know? Square roots! 5 00:01:51,264 --> 00:01:59,776 Pi is equal to 3.141592653589747 etc. etc. etc. 6 00:02:01,545 --> 00:02:04,706 Hello? (echo) 7 00:02:06,103 --> 00:02:07,891 Hello, Donald. 8 00:02:08,061 --> 00:02:11,402 That's me! Where am I?! 9 00:02:11,402 --> 00:02:13,838 Mathmagic land. 10 00:02:13,838 --> 00:02:17,411 Mathmagic land? Never heard of it. 11 00:02:17,642 --> 00:02:20,879 It's the land of great adventure. 12 00:02:20,879 --> 00:02:23,048 Well, who are you? 13 00:02:23,048 --> 00:02:27,075 I'm a spirit, the true spirit of adventure. 14 00:02:27,075 --> 00:02:29,788 That's for me! What's next? 15 00:02:29,788 --> 00:02:32,974 A journey through the wonderland of mathematics. 16 00:02:32,974 --> 00:02:36,561 Mathematics? That's for eggheads! 17 00:02:36,561 --> 00:02:39,141 Eggheads? Now hold on, Donald. 18 00:02:39,141 --> 00:02:41,654 You like music don't you? 19 00:02:41,792 --> 00:02:42,726 Yeah. 20 00:02:42,757 --> 00:02:46,171 Well, without eggheads, there would be no music. 21 00:02:46,263 --> 00:02:47,129 Bah. 22 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. 23 00:02:55,009 --> 00:02:56,225 Pythagoras? 24 00:02:56,225 --> 00:02:58,513 The father of mathematics and music. 25 00:02:58,513 --> 00:03:00,435 Mathematics and music? 26 00:03:00,435 --> 00:03:04,622 Ahh, you'll find mathematics in the darndest places. 27 00:03:05,268 --> 00:03:06,445 Watch 28 00:03:07,198 --> 00:03:08,747 First we'll need a string 29 00:03:08,747 --> 00:03:09,701 Hey! 30 00:03:10,055 --> 00:03:13,142 Stretch it good and tight; pluck it! 31 00:03:14,049 --> 00:03:17,176 Now divide in half. Pluck again. 32 00:03:18,007 --> 00:03:21,952 You see? It's the same tone, one octave higher. 33 00:03:21,952 --> 00:03:24,507 Now divide the next section. 34 00:03:24,568 --> 00:03:26,639 And the next. 35 00:03:26,639 --> 00:03:31,690 Pythagoras discovered the octave had a ratio of two to one. 36 00:03:31,690 --> 00:03:39,475 With simple fractions, he got this [major triad] 37 00:03:39,475 --> 00:03:47,668 And from this harmony in numbers, developed the musical scale of today. [major scale] 38 00:03:49,299 --> 00:03:54,789 By golly, you DO find mathematics in the darndest places. 39 00:03:55,251 --> 00:03:57,737 You can imagine how excited Pythagoras was, 40 00:03:57,737 --> 00:04:03,384 when he shared his findings with his pals and fraternity of eggheads, known as the Pythagoreans. 41 00:04:03,877 --> 00:04:08,571 They used to be meet in secret to discuss their mathematical discoveries. 42 00:04:08,571 --> 00:04:11,340 Only members were allowed to attend. 43 00:04:11,340 --> 00:04:15,502 They had a secret emblem, the pentagram. 44 00:04:18,887 --> 00:04:22,129 Let's see what the topic is for today. 45 00:04:22,467 --> 00:04:26,229 [major scale] 46 00:04:45,662 --> 00:04:48,231 [music] 47 00:04:48,985 --> 00:04:50,981 What's going on? 48 00:04:50,981 --> 00:04:53,396 Shh! It's a jam session. 49 00:04:55,350 --> 00:04:57,498 Gimme something with a beat! 50 00:04:57,498 --> 00:04:58,600 Shhh! 51 00:04:59,308 --> 00:05:02,277 [percussion] 52 00:05:31,033 --> 00:05:34,041 So from these eggheads, the Pythagoreans, 53 00:05:34,041 --> 00:05:36,102 with their mathematical formula 54 00:05:36,102 --> 00:05:38,756 came the basis of our music of today. 55 00:05:38,756 --> 00:05:42,179 [big band] 56 00:07:06,487 --> 00:07:09,377 Pythag, old boy, put her there 57 00:07:12,838 --> 00:07:17,707 Now I'll be a goshdarned egghead [laugh] 58 00:07:17,707 --> 00:07:24,252 It was our old friend Pythagoras who discovered that the pentagram was full of mathemagic. 59 00:07:26,867 --> 00:07:31,703 The two shorter lines combined exactly equal the third 60 00:07:31,703 --> 00:07:36,288 and this line shows the magic proportions of the famous golden section 61 00:07:37,027 --> 00:07:40,558 The second and third lines exactly equal the fourth 62 00:07:40,558 --> 00:07:43,799 Once again we have the golden section 63 00:07:44,829 --> 00:07:47,224 But this is only the beginning 64 00:07:47,224 --> 00:07:49,239 Hidden within the pentagram 65 00:07:49,239 --> 00:07:52,345 is a secret for creating a golden rectangle 66 00:07:52,345 --> 00:07:58,056 which the Greeks admired for its beautiful proportions and magic qualities 67 00:07:58,056 --> 00:08:02,268 The star contains the golden rectangle many times over 68 00:08:29,283 --> 00:08:31,675 It's a most remarkable shape 69 00:08:31,675 --> 00:08:35,269 It can mathematically reproduce itself indefinitely 70 00:08:40,161 --> 00:08:43,964 All these rectangles have exactly the same proportions 71 00:08:51,272 --> 00:08:54,317 This figure also contains a magic spiral 72 00:08:54,317 --> 00:08:59,256 that repeats the proportions of the golden section into infinity 73 00:09:00,410 --> 00:09:05,556 To the Greeks, the golden rectangle represented a mathematical law of beauty 74 00:09:05,556 --> 00:09:08,986 We find it in their classical architecture 75 00:09:08,986 --> 00:09:13,203 The Parthenon, perhaps one of the most famous of early Greek buildings, 76 00:09:13,203 --> 00:09:15,708 contains many golden rectangles. 77 00:09:38,477 --> 00:09:42,332 These same golden proportions are also found in their sculpture. 78 00:09:59,101 --> 00:10:00,885 In the centuries that followed 79 00:10:00,885 --> 00:10:06,665 the golden rectangle dominated the idea of beauty in architecture throughout the Western world. 80 00:10:06,665 --> 00:10:10,718 The Cathedral of Notre Dame is an outstanding example. 81 00:10:12,579 --> 00:10:15,622 The Renaissance painters knew this secret well. 82 00:10:21,299 --> 00:10:25,960 Today, the golden rectangle is very much a part of our modern world. 83 00:10:30,544 --> 00:10:34,457 Modern painters have rediscovered the magic of these proportions. 84 00:10:38,226 --> 00:10:42,179 Indeed, this ideal proportion is to be found in life itself. 85 00:10:42,379 --> 00:10:44,829 Boy, oh boy, oh boy! 86 00:10:45,198 --> 00:10:50,429 This is mathematics? I like mathematical figures like that. 87 00:10:50,429 --> 00:10:52,428 Ah, ah, ah, Donald. 88 00:10:52,428 --> 00:10:53,747 Let me try it! 89 00:10:53,747 --> 00:10:54,661 No, no. 90 00:10:54,661 --> 00:10:56,355 Ideal proportion 91 00:10:56,878 --> 00:10:58,182 Not quite 92 00:10:59,520 --> 00:11:01,992 Uh uh. No, I'm afraid not. 93 00:11:04,654 --> 00:11:07,310 Well, we can't all be mathematically perfect. 94 00:11:07,310 --> 00:11:08,460 Oh yeah? 95 00:11:11,229 --> 00:11:13,830 There, I knew I could do it. 96 00:11:14,015 --> 00:11:16,468 Now that you're all pent up in a pentagon 97 00:11:16,468 --> 00:11:20,422 let's see how nature uses the same mathematical form. 98 00:11:20,683 --> 00:11:21,906 The petunia 99 00:11:25,121 --> 00:11:26,637 The star jasmine 100 00:11:31,099 --> 00:11:32,508 The starfish 101 00:11:37,123 --> 00:11:38,529 The wax flower 102 00:11:43,760 --> 00:11:46,812 There are literally thousands of members in good standing 103 00:11:46,812 --> 00:11:50,407 in nature's Pythagorean society of the star. 104 00:11:57,868 --> 00:12:00,915 All nature's works have a mathematical logic 105 00:12:00,915 --> 00:12:02,938 and her patterns are limitless. 106 00:12:27,338 --> 00:12:30,004 The magic proportions of the golden section 107 00:12:30,004 --> 00:12:33,411 are often found in the spirals of nature's designs. 108 00:12:49,150 --> 00:12:54,090 The profusion of mathematical forms brings to mind the words of Pythagoras: 109 00:12:54,090 --> 00:12:58,878 "Everything is arranged according to number and mathematical shape." 110 00:12:59,555 --> 00:13:02,355 Yes, there is mathematics in music, 111 00:13:02,355 --> 00:13:05,027 in art, in just about everything. 112 00:13:05,166 --> 00:13:09,483 And as as the Greeks had guessed, the rules are always the same. 113 00:13:36,683 --> 00:13:40,004 Well, Donald, did you enjoy your geometrical journey? 114 00:13:40,004 --> 00:13:45,218 Gee, Mr. Spirit, there's a lot more to mathematics than two times two! 115 00:13:45,218 --> 00:13:46,439 That's right, Donald 116 00:13:46,439 --> 00:13:49,109 And you can find mathematics in games, too! 117 00:13:49,109 --> 00:13:51,152 Games! Oh, boy! 118 00:13:51,629 --> 00:13:54,671 Let's begin with a game that's played on squares. 119 00:13:54,671 --> 00:13:55,782 Checkers? 120 00:13:55,782 --> 00:13:57,058 No, chess. 121 00:13:57,058 --> 00:13:58,223 Chess?! 122 00:13:58,223 --> 00:14:01,337 A mathematical contest between two minds. 123 00:14:01,352 --> 00:14:05,805 It's a game that has been enjoyed for centuries by kings and commoners. 124 00:14:05,805 --> 00:14:10,269 In fact, Louis Carroll, a famous mathematician with a literary mind, 125 00:14:10,269 --> 00:14:15,785 used chess as a setting for his classic tale, Through the Looking Glass. 126 00:14:16,201 --> 00:14:22,071 Alice found herself face to face with a none-too-friendly group of chess pieces. 127 00:14:22,071 --> 00:14:24,938 Good heavens, what's this? 128 00:14:25,123 --> 00:14:28,896 Upon my soul, it appears to be a lost pawn! 129 00:14:29,050 --> 00:14:31,873 I'm no pawn, I'm Donald Duck! 130 00:14:31,920 --> 00:14:34,227 He says he's Donald Duck! 131 00:14:34,381 --> 00:14:36,162 Preposterous! 132 00:14:36,270 --> 00:14:38,266 Or, it could be an Alice. 133 00:14:38,266 --> 00:14:39,291 Alice?! 134 00:14:39,291 --> 00:14:42,127 No, no no. It's a lost pawn. 135 00:14:42,666 --> 00:14:46,578 Lost pawn? Stop that pawn! 136 00:14:46,578 --> 00:14:50,643 Ow, Mr. Spirit! Help, help, help! 137 00:15:10,736 --> 00:15:13,032 Whew, that was close! 138 00:15:13,032 --> 00:15:16,341 Now you can look at this game from a safer perspective. 139 00:15:22,264 --> 00:15:25,048 Chess is a game of calculated strategy, 140 00:15:25,048 --> 00:15:27,552 and since the board is geometrical, 141 00:15:27,552 --> 00:15:29,737 the moves are mathematical. 142 00:15:53,152 --> 00:15:55,836 Checkmate, and the game is over. 143 00:15:55,836 --> 00:15:59,574 That's very interesting. What's next? 144 00:15:59,574 --> 00:16:03,321 Practically all games are played on geometrical areas. 145 00:16:03,321 --> 00:16:05,506 The baseball field is a diamond. 146 00:16:05,506 --> 00:16:06,703 Oh boy! 147 00:16:12,795 --> 00:16:16,008 And without mathematics, we couldn't even keep score. 148 00:16:16,008 --> 00:16:20,077 Football is played on a rectangle divided by yard lines. 149 00:16:22,030 --> 00:16:26,606 Basketball is a game of circles, spheres and rectangles. 150 00:16:30,729 --> 00:16:33,353 Even hopscotch has its multiple squares. 151 00:16:44,153 --> 00:16:45,432 What's next? 152 00:16:45,971 --> 00:16:47,134 Tiddlywinks? 153 00:16:48,027 --> 00:16:53,291 No, a mathematical game played on a field of two perfect squares 154 00:16:53,291 --> 00:16:55,683 using three perfect spheres 155 00:16:55,683 --> 00:16:57,629 and a lot of diamonds. 156 00:16:57,629 --> 00:16:59,716 In other words, billiards. 157 00:16:59,716 --> 00:17:02,609 Oh boy! That's for me! 158 00:17:02,609 --> 00:17:04,726 You know the game, don't you Donald? 159 00:17:04,726 --> 00:17:09,641 Of course, the cue ball has to hit the other two balls 160 00:17:09,641 --> 00:17:10,646 like this! 161 00:17:16,061 --> 00:17:20,142 Now let's see how an expert at three-cushion billiards uses his head. 162 00:17:22,280 --> 00:17:23,807 Three-cushion? 163 00:17:23,807 --> 00:17:27,664 Yes. The cue ball not only has to hit both the other balls, 164 00:17:27,664 --> 00:17:31,703 but it must contact at least three cushions before it hits the final ball. 165 00:17:42,734 --> 00:17:44,545 One, two, three 166 00:17:58,314 --> 00:18:00,180 One, two, three 167 00:18:10,042 --> 00:18:13,572 It takes an expert to make several shots in succession 168 00:18:13,572 --> 00:18:16,501 One, two, three, four 169 00:18:18,301 --> 00:18:19,558 five, six. 170 00:18:21,697 --> 00:18:24,493 Wow! That was a lucky shot! 171 00:18:24,908 --> 00:18:27,209 Luck? No. It's skill. 172 00:18:27,870 --> 00:18:30,637 For this game, you have to know all the angles. 173 00:18:52,067 --> 00:18:56,844 One, two, three, four, five, six, seven. 174 00:18:57,606 --> 00:19:00,843 That's amazing! How does he do it? 175 00:19:00,843 --> 00:19:02,912 First, there's technique. 176 00:19:02,912 --> 00:19:06,663 He's striking the cue ball low, so it'll spin backwards. 177 00:19:11,017 --> 00:19:14,860 Hitting the ball on the right side will make it hug the rail. 178 00:19:14,860 --> 00:19:17,466 These trick shots take a lot of practice. 179 00:19:18,712 --> 00:19:21,225 Hahaha! He missed that time! 180 00:19:21,225 --> 00:19:22,545 One, two... 181 00:19:26,130 --> 00:19:26,996 three. 182 00:19:29,596 --> 00:19:32,155 What's so mathematical about that? 183 00:19:32,155 --> 00:19:35,037 Oh, this game takes precise calculation. 184 00:19:35,037 --> 00:19:37,594 He figures out each shot in his head. 185 00:19:38,671 --> 00:19:42,700 He could play it like this, but it calls for quite a bit of luck. 186 00:19:42,700 --> 00:19:44,538 There is a better choice. 187 00:19:44,538 --> 00:19:49,360 For this, he uses the diamond markings on the rail as a mathematical guide. 188 00:19:49,360 --> 00:19:53,355 First, he figures the natural angle for hitting the object balls. 189 00:19:53,355 --> 00:19:57,674 And then he finds that his cue ball must bounce off the number three diamond. 190 00:19:57,674 --> 00:20:01,797 Next, he gets ready for the shot and he needs a number for his cue position. 191 00:20:01,797 --> 00:20:04,936 This calls for a different set of numbers. 192 00:20:05,198 --> 00:20:07,459 Very confusing, isn't it? 193 00:20:07,844 --> 00:20:09,889 Not when you get the hang of it. 194 00:20:09,889 --> 00:20:12,734 You see, the cue position is four. 195 00:20:12,734 --> 00:20:14,789 Now, a simple subtraction. 196 00:20:14,789 --> 00:20:16,578 Three from four is one. 197 00:20:16,578 --> 00:20:19,903 So, if he shoots for the first diamond, he should make it. 198 00:20:19,903 --> 00:20:22,459 It's called "playing the diamond system". 199 00:20:27,536 --> 00:20:29,440 Natural angle, 2. 200 00:20:29,440 --> 00:20:32,953 Cue position, one and a half, two, two and a half, three, 201 00:20:33,230 --> 00:20:34,413 three and a half. 202 00:20:34,413 --> 00:20:37,012 Two from three and a half is one and a half. 203 00:20:37,012 --> 00:20:41,333 So, shoot halfway between the first and second diamonds. 204 00:20:44,795 --> 00:20:47,212 There's nothing to it! Let me try! 205 00:20:51,335 --> 00:20:52,921 Let's see now. 206 00:20:53,167 --> 00:20:58,485 If I shoot it here, and it bounces there, and uh, no there. 207 00:20:58,808 --> 00:21:00,519 If I shoot it here... 208 00:21:00,519 --> 00:21:04,169 Four and half minus three, three and a half plus four... 209 00:21:04,169 --> 00:21:05,865 Add it to two... 210 00:21:07,711 --> 00:21:09,821 And dividing it.. and... 211 00:21:09,821 --> 00:21:11,713 I guess I should shoot about here. 212 00:21:11,713 --> 00:21:15,570 No, no, Donald. There's no guesswork to mathematics. 213 00:21:15,570 --> 00:21:17,062 It's quite simple. 214 00:21:17,062 --> 00:21:20,167 Natural angle for the hit: two. 215 00:21:20,167 --> 00:21:22,857 Cue position: three and a half. 216 00:21:22,857 --> 00:21:25,696 How much is three and a half minus two? 217 00:21:25,696 --> 00:21:27,991 Uhhh... one and a half! 218 00:21:38,714 --> 00:21:41,852 Hey! It works! Oh boy! 219 00:21:41,852 --> 00:21:43,594 It's a cinch! 220 00:21:44,610 --> 00:21:48,097 If i hit it here, add three and a half plus four 221 00:21:48,097 --> 00:21:50,803 Four and a half minus three... [???] 222 00:21:51,942 --> 00:21:54,655 You're making it tough for yourself, Donald. 223 00:22:05,808 --> 00:22:09,444 How do you like that for mathematics, Mr. Spirit? 224 00:22:09,444 --> 00:22:14,461 Wonderful, Donald. And now you're ready for the most exciting game of all. 225 00:22:14,461 --> 00:22:16,080 Oh, boy! 226 00:22:16,695 --> 00:22:19,874 And the playing field for this game is in the mind. 227 00:22:20,982 --> 00:22:24,665 Uh oh, look at the condition of your mind! 228 00:22:24,665 --> 00:22:33,437 Antiquated ideas, bungling, false concepts, superstitions, confusion! 229 00:22:33,437 --> 00:22:36,091 To think straight, we'll have to clean house. 230 00:22:48,091 --> 00:22:50,461 There, that's more like it. 231 00:22:50,461 --> 00:22:53,044 A nice clean sweep. 232 00:22:53,044 --> 00:22:56,334 This game is played with circles and triangles. 233 00:22:56,334 --> 00:22:58,588 Think of a perfect circle. 234 00:23:02,465 --> 00:23:07,917 A perfect circle. Perfect. Circle. 235 00:23:08,748 --> 00:23:11,695 Perfect. Ahhhhh. 236 00:23:11,695 --> 00:23:15,057 Put a triangle inside and turn it. 237 00:23:15,057 --> 00:23:18,895 Now spin the circle, and what have you got? 238 00:23:19,956 --> 00:23:21,156 A ball! 239 00:23:21,713 --> 00:23:23,576 Yes, a sphere. 240 00:23:23,576 --> 00:23:27,819 The shape of things is first discovered in the mind. 241 00:23:27,819 --> 00:23:30,325 Slice off the top and we have a... 242 00:23:33,309 --> 00:23:35,745 A magnifying glass! 243 00:23:35,745 --> 00:23:37,131 That's right. 244 00:23:37,131 --> 00:23:39,901 A lens is a section of a sphere. 245 00:23:39,901 --> 00:23:43,383 All optical instruments are created through mathematics. 246 00:23:46,783 --> 00:23:51,399 You see, there's a lot more to mathematics than just numbers and equations. 247 00:23:52,429 --> 00:23:55,246 Let's get back to our circle and triangle. 248 00:23:59,000 --> 00:24:01,298 Roll it and we have a... 249 00:24:01,298 --> 00:24:02,568 A... a wheel! 250 00:24:10,337 --> 00:24:14,443 The circle has been the basis for many of man's important inventions. 251 00:24:20,197 --> 00:24:23,552 The mind can create the most amazing things. 252 00:24:24,537 --> 00:24:27,475 If we spin the triangle, we have a... 253 00:24:27,475 --> 00:24:28,174 Cone! 254 00:24:28,174 --> 00:24:29,531 Slice the cone. 255 00:24:29,531 --> 00:24:30,466 giggle 256 00:24:30,466 --> 00:24:33,938 The cone is full of useful mathematical shapes. 257 00:24:35,815 --> 00:24:38,955 Slice it again. Slice it several times. 258 00:24:42,047 --> 00:24:46,738 The orbits of all planets and satellites can be found in the cone. 259 00:24:46,738 --> 00:24:50,555 No matter how you slice it, it's always mathematics. 260 00:24:50,555 --> 00:24:54,710 A slice like this gives us the reflector of a search light. 261 00:24:55,541 --> 00:24:59,356 A slice like this, the mirror of a giant telescope. 262 00:25:01,480 --> 00:25:04,680 A line on a cone, and we have a drill. 263 00:25:08,449 --> 00:25:10,157 And the spring. 264 00:25:14,342 --> 00:25:15,447 Now you're ticking. 265 00:25:27,063 --> 00:25:28,412 Number, please? 266 00:25:40,643 --> 00:25:45,338 The mind is the birthplace for all of man's scientific achievements. 267 00:26:00,754 --> 00:26:04,763 The mind knows no limits when used properly. 268 00:26:04,763 --> 00:26:06,855 Think of a pentagram, Donald. 269 00:26:09,532 --> 00:26:11,526 Now, put another inside. 270 00:26:11,526 --> 00:26:13,981 A third. And a fourth. 271 00:26:14,535 --> 00:26:18,069 No pencil is sharp enough to draw as fine as you can think 272 00:26:18,069 --> 00:26:22,093 and no paper large enough to hold your imagination. 273 00:26:22,093 --> 00:26:26,566 In fact, it is only in the mind that we can conceive infinity. 274 00:26:28,259 --> 00:26:33,040 Mathematical thinking has opened the doors to the exciting adventures of science. 275 00:26:35,086 --> 00:26:37,265 I'll be dog-darned! 276 00:26:37,265 --> 00:26:40,030 I've never seen so many doors before. 277 00:26:40,030 --> 00:26:43,023 Each discovery leads to many others. 278 00:26:43,023 --> 00:26:44,804 An endless chain. 279 00:26:45,127 --> 00:26:48,897 Hey! Hey! Whatsa matter with these doors? 280 00:26:48,897 --> 00:26:52,775 Hey! These doors won't open! They're locked! 281 00:26:52,775 --> 00:26:54,724 Of course they're locked. 282 00:26:54,724 --> 00:26:56,852 These are the doors of the future, 283 00:26:56,852 --> 00:26:58,568 and the key is... 284 00:26:58,568 --> 00:27:00,220 Mathematics! 285 00:27:00,328 --> 00:27:02,844 Right. Mathematics. 286 00:27:02,844 --> 00:27:07,517 The boundless treasures of science are locked behind those doors. 287 00:27:07,517 --> 00:27:13,698 In time, they will be opened by the curious and inquiring minds of future generations. 288 00:27:15,467 --> 00:27:17,628 In the words of Galileo: 289 00:27:17,628 --> 00:27:25,813 "Mathematics is the alphabet with which God has written the universe."