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