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