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