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00:00 (upbeat music)
00:02 (upbeat music)
00:05 (upbeat music)
00:07 (upbeat music)
00:13 (upbeat music)
00:20 (upbeat music)
00:27 (upbeat music)
00:32 (upbeat music)
00:34 - That is strange.
01:01 Huh, that's an odd looking creature.
01:04 (upbeat music)
01:07 What kind of a crazy place is this?
01:18 (upbeat music)
01:22 (upbeat music)
01:25 (mumbles)
01:35 - Pi is equal to 3.141592653589747 etc, etc, etc.
01:43 - Huh?
01:48 Hello, hello, hello.
01:51 (mumbles)
01:53 - Hello, Donald.
01:56 - That's me.
01:57 Where am I?
01:59 - Mathemagic Land.
02:00 - Mathemagic Land?
02:03 (mumbles)
02:05 - It's the land of great adventure.
02:08 - Now, who are you?
02:11 - I'm a spirit, the true spirit of adventure.
02:15 - That's for me.
02:16 What's next?
02:17 - A journey through the wonderland of mathematics.
02:21 - Mathematics?
02:22 That's for eggheads.
02:24 - Eggheads?
02:25 Now hold on, Donald.
02:27 You like music, don't you?
02:29 - Yep.
02:30 - Well, without eggheads, there would be no music.
02:34 - Oh.
02:34 - Come on, let's go to ancient Greece,
02:38 to the time of Pythagoras,
02:40 the master egghead of them all.
02:43 - Pythagoras?
02:44 - The father of mathematics and music.
02:46 - Mathematics and music?
02:48 Ah, you'll find mathematics in the darndest places.
02:53 Watch.
02:54 First, we'll need a string.
02:58 Stretch it good and tight, plunk it.
03:01 Now divide in half, plunk again.
03:04 You see?
03:07 It's the same tone, one octave higher.
03:10 Now divide the next section.
03:12 And the next.
03:14 Pythagoras discovered the octave had a ratio of two to one.
03:19 With simple fractions, he got this.
03:22 And from this harmony in numbers,
03:29 developed the musical scale of today.
03:32 - By Donald, you do find mathematics
03:40 in the darndest places.
03:43 You can imagine how excited Pythagoras was
03:45 when he shared his findings with his pals,
03:47 a fraternity of eggheads known as the Pythagoreans.
03:51 - They used to meet in secret
03:53 to discuss their mathematical discoveries.
03:56 Only members were allowed to attend.
03:59 They had a secret emblem, the pentagram.
04:02 Let's see what the topic is for today.
04:10 (gentle music)
04:12 (harmonica music)
04:22 (upbeat music)
04:36 - What's going on?
04:38 - Shh, it's a jam session.
04:41 - Give us something with a beat.
04:45 - Shh.
04:46 (upbeat music)
04:50 (upbeat music)
05:19 - So from these eggheads, the Pythagoreans,
05:22 with their mathematical formula,
05:24 came the basis of our music of today.
05:27 (upbeat music)
05:29 (harmonica music)
05:36 (upbeat music)
05:39 (gentle music)
06:01 (upbeat music)
06:03 (dramatic music)
06:24 (dramatic music)
06:27 (gentle music)
06:38 (upbeat music)
06:52 (gentle music)
06:55 (gentle music)
07:00 (upbeat music)
07:07 (gentle music)
07:14 - The two shorter lines combined exactly equal the third,
07:18 and this line shows the magic proportions
07:21 of the famous golden section.
07:24 The second and third lines exactly equal the fourth.
07:28 Once again, we have the golden section.
07:31 But this is only the beginning.
07:34 Hidden within the pentagram is a secret
07:37 for creating a golden rectangle,
07:39 which the Greeks admired
07:41 for its beautiful proportions and magic qualities.
07:45 The star contains the golden rectangle many times over.
07:49 (upbeat music)
07:51 (gentle music)
07:54 It's a most remarkable shape.
08:18 It can mathematically reproduce itself indefinitely.
08:22 (upbeat music)
08:25 All these rectangles have exactly the same proportions.
08:31 (upbeat music)
08:34 This figure also contains a magic spiral
08:41 that repeats the proportions of the golden section
08:44 into infinity.
08:47 To the Greeks, the golden rectangle
08:49 represented a mathematical law of beauty.
08:52 We find it in their classical architecture.
08:55 The Parthenon, perhaps one of the most famous
08:58 of early Greek buildings, contains many golden rectangles.
09:02 (gentle music)
09:05 (gentle music)
09:08 These same golden proportions
09:27 are also found in their sculpture.
09:30 (gentle music)
09:34 (gentle music)
09:36 In the centuries that followed,
09:49 the golden rectangle dominated the idea
09:51 of beauty and architecture throughout the Western world.
09:55 The Cathedral of Notre Dame is an outstanding example.
10:00 The Renaissance painters knew this secret well.
10:04 (gentle music)
10:06 Today, the golden rectangle
10:11 is very much a part of our modern world.
10:14 (gentle music)
10:16 Modern painters have rediscovered
10:20 the magic of these proportions.
10:23 (gentle music)
10:26 Indeed, this ideal proportion is to be found in life itself.
10:30 - Boy, oh, boy, oh, boy.
10:32 This is mathematics?
10:35 I'm not mathematical figures like that.
10:38 - Uh-uh-uh, Donald.
10:40 - Let me try it.
10:41 - No, no.
10:42 - Ideal proportion.
10:44 - Not quite.
10:45 Uh-uh.
10:48 No, I'm afraid not.
10:50 (gentle music)
10:52 Well, we can't all be mathematically perfect.
10:55 - Oh, yeah?
10:56 (gentle music)
10:59 Yeah, I do want to do it.
11:01 - Now that you're all pent up in a pentagon,
11:04 let's see how nature uses this same mathematical form.
11:08 The petunia.
11:09 (gentle music)
11:12 The star jasmine.
11:14 (gentle music)
11:17 The starfish.
11:20 (gentle music)
11:24 The wax flower.
11:26 (gentle music)
11:29 There are literally thousands of members in good standing
11:34 in nature's Pythagorean society of the star.
11:38 (gentle music)
11:41 All nature's works have a mathematical logic
11:48 and her patterns are limitless.
11:51 (gentle music)
11:54 The magic proportions of the golden section
12:18 are often found in the spirals of nature's designs.
12:22 (gentle music)
12:25 The profusion of mathematical forms
12:39 brings to mind the words of Pythagoras.
12:42 Everything is arranged according to number
12:45 and mathematical shape.
12:47 Yes, there is mathematics in music,
12:50 in art, in just about everything.
12:53 And as the Greeks had guessed,
12:55 the rules are always the same.
12:57 (gentle music)
13:00 (gentle music)
13:12 (gentle music)
13:15 Well, Donald, did you enjoy your geometrical journey?
13:27 Gee, Mr. Spirit, there's a lot more to mathematics
13:31 than two times two.
13:32 That's right, Donald,
13:33 and you can find mathematics in games too.
13:36 Games? Oh boy.
13:38 Let's begin with a game that's played on squares.
13:42 Checkers?
13:43 No, chess.
13:44 Chess?
13:45 A mathematical contest between two minds.
13:48 It's a game that has been enjoyed for centuries
13:51 by kings and commoners.
13:53 In fact, Lewis Carroll,
13:54 a famous mathematician with a literary mind,
13:57 used chess as a setting for his classic tale,
14:01 Through the Looking Glass.
14:03 Alice found herself face to face
14:06 with a none too friendly group of chess pieces.
14:09 Good heavens, what's this?
14:12 Upon my soul, it appears to be a lost pawn.
14:16 I'm not a pawn, I'm Donald Duck.
14:19 He says he's Donald Duck.
14:21 Preposterous.
14:23 Or it could be an Alice.
14:25 Alice?
14:26 No, no, no, it's a lost pawn.
14:29 Lost pawn?
14:32 Stop that pawn.
14:33 This is spirit.
14:36 Phew, that was close.
14:59 Now you can look at this game from a safer perspective.
15:04 Chess is a game of calculated strategy,
15:12 and since the board is geometrical,
15:15 the moves are mathematical.
15:18 [music]
15:41 Checkmate, and the game is over.
15:44 That's very interesting.
15:46 What's next?
15:47 Practically all games are played on geometrical areas.
15:51 The baseball field is a diamond.
15:54 [music]
16:00 And without mathematics, we couldn't even keep score.
16:03 Football is played on a rectangle divided by yard lines.
16:08 Basketball is a game of circles, spheres, and rectangles.
16:15 [music]
16:18 Even hopscotch has its multiple squares.
16:21 [music]
16:31 What's next?
16:33 Dead or Alive?
16:35 No, a mathematical game played on a field of two perfect squares
16:41 using three perfect spheres
16:43 and a lot of diamonds.
16:45 In other words, billiards.
16:47 Oh boy, that's for me.
16:50 You know the game, don't you, Donald?
16:52 Of course.
16:53 The cue ball has to hit the other two balls, like this.
16:58 [music]
17:03 Now let's see how an expert at three-cushion billiards uses his head.
17:07 [music]
17:10 Three-cushion?
17:11 Yes. The cue ball not only has to hit both the other balls,
17:15 but it must contact at least three cushions before it hits the final ball.
17:19 [music]
17:30 One, two, three.
17:32 [music]
17:46 One, two, three.
17:48 [music]
17:58 It takes an expert to make several shots in succession.
18:01 One, two, three, four, five, six.
18:08 Wow! That was a good shot.
18:12 Luck? No, it's skill.
18:15 For this game, you have to know all the angles.
18:18 [music]
18:40 One, two, three, four, five, six, seven.
18:45 That's amazing! How does he do it?
18:48 First, there's technique.
18:50 He's striking the cue ball low, so it'll spin backwards.
18:54 [music]
18:59 Hitting the ball on the right side will make it hug the rail.
19:02 These trick shots take a lot of practice.
19:05 [music]
19:06 Ha, ha, he missed it! Ha, ha, ha!
19:09 One, two, three.
19:15 [music]
19:17 What's the best way to do about that?
19:20 Oh, this game takes precise calculation.
19:23 He figures out each shot in his head.
19:26 He could play it like this, but it calls for quite a bit of luck.
19:30 There is a better choice.
19:32 For this, he uses the diamond markings on the rail as a mathematical guide.
19:37 First, he figures the natural angle for hitting the object balls.
19:41 And then he finds that his cue ball must bounce off the number three diamond.
19:45 Next, he gets ready for the shot, and he needs a number for his cue position.
19:50 This calls for a different set of numbers.
19:53 Very confusing, isn't it?
19:56 Not when you get the hang of it.
19:58 You see, the cue position is four.
20:01 Now a simple subtraction.
20:03 Three from four is one.
20:05 That's for the first diamond. He should make it.
20:08 It's called playing the diamond system.
20:10 [music]
20:15 Natural angle two.
20:17 Cue position.
20:18 One and a half, two, two and a half, three.
20:21 Three and a half.
20:22 Two from three and a half is one and a half.
20:25 So, shoot halfway between the first and second diamonds.
20:29 [music]
20:32 There's nothing to it. Double draw.
20:35 [music]
20:39 Let's see, though.
20:41 If I set it here, it'll bounce there and go here.
20:46 If I set it here, four and a half minus three.
20:50 Three and a half plus four added to two.
20:53 And divided into two.
20:57 I guess I should shoot about here.
20:59 No, no, Donald.
21:01 There's no guesswork to mathematics.
21:03 It's quite simple.
21:05 Natural angle for the hit.
21:07 Two.
21:08 Cue position.
21:09 Three and a half.
21:10 How much is three and a half minus two?
21:13 Uh, one and a half.
21:16 [music]
21:26 Hey, it works!
21:28 Oh, boy!
21:29 It's a switch!
21:31 If I set it here, three and a half plus four to four and a half minus three.
21:37 [mumbling]
21:40 You're making it tough for yourself, Donald.
21:42 [music]
21:53 How do you like that for mathematics, Mr. Stewart?
21:57 Wonderful, Donald.
21:58 And now you're ready for the most exciting game of all.
22:02 Oh, boy!
22:04 And the playing field for this game is in the mind.
22:08 Oh, look at the condition of your mind.
22:12 Antiquated ideas.
22:14 Bungling.
22:15 False concepts.
22:17 Superstitions.
22:19 Confusion.
22:21 To think straight, we'll have to clean house.
22:23 [music]
22:36 There.
22:37 That's more like it.
22:38 A nice clean sweep.
22:41 This game is played with circles and triangles.
22:44 Think of a perfect circle.
22:46 [music]
22:50 A perfect circle.
22:53 Perfect circle.
22:56 Perfect.
22:58 Ah.
22:59 Put a triangle inside and turn it.
23:02 Now spin the circle.
23:04 And what have you got?
23:07 A ball!
23:09 Yes, a sphere.
23:11 The shape of things is first discovered in the mind.
23:15 Slice off the top and we have a...
23:21 A magnifying glass.
23:23 That's right.
23:24 A lens is a section of a sphere.
23:27 All optical instruments are created through mathematics.
23:30 [music]
23:34 You see, there's a lot more to mathematics than just numbers and equations.
23:40 Let's get back to our circle and triangle.
23:47 Roll it and we have a...
23:49 A wheel.
23:51 [music]
23:58 The circle has been the basis for many of man's important inventions.
24:02 [music]
24:08 The mind can create the most amazing things.
24:12 If we spin the triangle, we have a...
24:15 Cone.
24:16 Slice the cone.
24:18 The cone is full of useful mathematical shapes.
24:23 Slice it again.
24:25 Slice it several times.
24:30 The orbits of all planets and satellites can be found in the cone.
24:34 No matter how you slice it, it's always mathematics.
24:38 A slice like this gives us the reflector of a searchlight.
24:43 A slice like this, the mirror of a giant telescope.
24:49 A line on a cone and we have a drill.
24:56 And a spring.
25:02 Now you're ticking.
25:05 [music]
25:15 Number, please.
25:16 [music]
25:20 [music]
25:25 [music]
25:29 The mind is the birthplace for all of man's scientific achievements.
25:34 [music]
25:49 The mind knows no limits when used properly.
25:53 Think of a pentagram, Donald.
25:58 There's another inside.
25:59 A third.
26:00 And a fourth.
26:02 No pencil is sharp enough to draw as fine as you can think.
26:06 And no paper large enough to hold your imagination.
26:10 In fact, it is only in the mind that we can conceive infinity.
26:16 Mathematical thinking has opened the doors to the exciting adventures of science.
26:23 I'm a do-dood.
26:25 I've never seen so many doors before.
26:28 Each discovery leads to many others.
26:31 An endless chain.
26:33 Hey, hey!
26:35 What's the matter with these doors?
26:37 Hey!
26:38 These doors won't open.
26:39 They're locked.
26:41 Of course they are locked.
26:43 These are the doors of the future.
26:45 And the key is...
26:47 Mathematics.
26:48 Right.
26:49 Mathematics.
26:51 The boundless treasures of science are locked behind those doors.
26:55 In time, they will be opened by the curious and inquiring minds of future generations.
27:03 In the words of Galileo,
27:05 "Mathematics is the alphabet with which God has written the universe."
27:11 [music]