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00:00 [MUSIC]
00:10 [MUSIC]
00:20 [MUSIC]
00:30 [MUSIC]
00:40 [MUSIC]
00:50 [MUSIC]
00:59 >> That is strange.
01:01 [MUSIC]
01:13 >> That's an odd looking creature.
01:16 [MUSIC]
01:28 >> What kind of a crazy place is this?
01:32 [MUSIC]
01:45 >> But what is all this?
01:48 [MUSIC]
01:51 >> Pi is equal to 3.141592653589747 etc, etc, etc.
01:59 >> Huh?
02:01 >> Hello, hello, hello.
02:05 >> Hello Donald.
02:07 >> Touch me.
02:09 Where am I?
02:11 >> Mathemagic Land.
02:13 >> Mathemagic Land?
02:15 Go ahead, go on.
02:17 >> It's the land of great adventure.
02:20 >> Now, who are you?
02:22 >> I'm a spirit, the true spirit of adventure.
02:26 >> Touch for me. What's next?
02:29 >> A journey through the wonderland of mathematics.
02:32 >> Mathematics? Touch for eggheads.
02:36 >> Eggheads? 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, let's go to ancient Greece, to the time of Pythagoras, 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.
03:05 Watch.
03:07 First we'll need a string.
03:10 Stretch it good and tight, plunk it.
03:14 Now divide in half. Plunk again.
03:18 You see? It's the same tone, one octave higher.
03:22 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.
03:40 And from this harmony in numbers, developed the musical scale of today.
03:49 >> By Donald, you do find mathematics in the darndest places.
03:55 >> You can imagine how excited Pythagoras was when he shared his findings with his pals, a fraternity of eggheads, known as the Pythagoreans.
04:04 >> They used to meet in secret to discuss their mathematical discoveries.
04:09 When members were allowed to attend, they had a secret emblem, the pentagram.
04:19 Let's see what the topic is for today.
04:22 [ Music ]
04:49 >> What's going on?
04:51 >> Shh! It's a jam session.
04:56 >> Tell us what the music beat.
04:58 >> Shh!
05:01 >> Shh!
05:03 [ Music ]
05:31 >> So from these eggheads, the Pythagoreans, with their mathematical formula, came the basis of our music of today.
05:39 [ Music ]
06:08 [ Music ]
06:37 [ Music ]
07:06 >> Psyduck boy, put it down.
07:13 Well, I'll be a post-mortem, then.
07:18 >> It was our old friend Pythagoras who discovered that the pentagram was full of mathemagic.
07:27 The two shorter lines combined exactly equal the third, and this line shows the magic proportions of the famous golden section.
07:37 The second and third lines exactly equal the fourth.
07:41 Once again, we have the golden section.
07:45 But this is only the beginning. Hidden within the pentagram is a secret for creating a golden rectangle, which the Greeks admired for its beautiful proportions and magic qualities.
07:58 The star contains the golden rectangle many times over.
08:02 [ Music ]
08:29 It's a most remarkable shape. It can mathematically reproduce itself indefinitely.
08:40 All these rectangles have exactly the same proportions.
08:51 This figure also contains a magic spiral 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, contains many golden rectangles.
09:38 These same golden proportions are also found in their sculpture.
09:59 In the centuries that followed, the golden rectangle dominated the idea 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.
10:21 Today, the golden rectangle is very much a part of our modern world.
10:30 Modern painters have rediscovered the magic of these proportions.
10:38 Indeed, this ideal proportion is to be found in life itself.
10:42 Boy, oh, boy, oh, boy. This is mathematics. I'm not mathematical 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.
10:59 Ha, ha. I'm afraid not.
11:04 Well, we can't all be mathematically perfect.
11:07 Oh, yeah?
11:11 Yeah, I do like to do it.
11:14 Now that you're all pent up in a pentagon, let's see how nature uses this same mathematical form.
11:20 The petunia.
11:25 The star jasmine.
11:31 The starfish.
11:37 The wax flower.
11:43 There are literally thousands of members in good standing in nature's Pythagorean society of the star.
11:57 All nature's works have a mathematical logic, and her patterns are limitless.
12:14 [Music]
12:27 The magic proportions of the golden section are often found in the spirals of nature's designs.
12:37 [Music]
12:49 The profusion of mathematical forms brings to mind the words of Pythagoras.
12:54 Everything is arranged according to number and mathematical shape.
12:59 Yes, there is mathematics in music, in art, in just about everything.
13:05 And as the Greeks had guessed, the rules are always the same.
13:09 [Music]
13:37 Donald, did you enjoy your geometrical journey?
13:40 Gee, Mr. Spirit, there's a lot more to mathematics than two times two.
13:45 That's right, Donald, and you can find mathematics in games, too.
13:49 Games? Oh, boy.
13:51 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 by kings and commoners.
14:05 In fact, Lewis Carroll, a famous mathematician with a literary mind,
14:10 used chess as a setting for his classic tale, Through the Looking-Glass.
14:16 Alice found herself face to face with a none too friendly group of chess pieces.
14:22 Good heavens, what's this?
14:25 Upon my soul, it appears to be a lost pawn.
14:29 I'm no pawn, I'm Donald Duck.
14:32 He says he's Donald Duck.
14:34 Preposterous.
14:36 Or it could be an Alice.
14:38 Alice?
14:39 No, no, no, it's a lost pawn.
14:43 Lost pawn? Stop that pawn.
14:47 Mr. Spirit!
14:49 [Music]
15:00 [Music]
15:09 Oh, no.
15:10 Phew, that was close.
15:12 Now you can look at this game from a safer perspective.
15:16 [Music]
15:22 Chess is a game of calculated strategy.
15:25 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.
15:58 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 Oh.
16:16 Football is played on a rectangle divided by yard lines.
16:19 [Music]
16:22 Basketball is a game of circles, spheres, and rectangles.
16:25 [Music]
16:31 Even hopscotch has its multiple squares.
16:34 [Music]
16:44 What's next?
16:46 Tether the Reds?
16:48 No.
16:49 A mathematical game played on a field of two perfect squares
16:53 using three perfect spheres and a lot of diamonds.
16:57 In other words, billiards.
16:59 Oh, boy.
17:01 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 [Music]
17:16 Now let's see how an expert at three-cushion billiards uses his head.
17:19 [Music]
17:22 Three-cushion?
17:23 Yes.
17:24 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.
17:44 [Music]
17:58 One, two, three.
18:00 [Music]
18:10 It takes an expert to make several shots in succession.
18:13 One, two, three, four, five, six.
18:20 Wow! That was not that good 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]
19:11 Hitting the ball on the right side will make it hug the rail.
19:14 These trick shots take a lot of practice.
19:18 Ha-ha! He missed it! Ha-ha-ha!
19:21 One, two,
19:23 [Music]
19:26 three.
19:29 What's the best way to do about that?
19:32 Oh, this game takes precise calculation.
19:35 He figures out each shot in his head.
19:38 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.
20:23 [Music]
20:27 Natural angle, two.
20:29 Cue position, one and a half, two, two and a half, three, three and a half.
20:34 Two from three and a half is one and a half.
20:37 So, shoot halfway between the first and second diamonds.
20:41 [Music]
20:44 That's better to it. The best way.
20:47 [Music]
20:51 Let's see now.
20:53 If I set it here, hit at about there, and, uh, no, 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 Uh, you know, divided into, uh, two.
21:09 I guess I should shoot about here.
21:11 No, no, Donald. There's no guesswork to mathematics.
21:15 It's quite simple. 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! Oh, boy! It's a switch!
21:44 If I set it here, uh, three and a half plus four, two.
21:47 Four and a half minus three, uh, two.
21:49 Somebody, uh, we have a way to find out my diamonds.
21:51 You're making it tough for yourself, Donald.
21:54 [Music]
22:05 So, here's our guess for my smuggled Swiss cash, Dillon.
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.
22:36 [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.
23:37 A lens is a section of a sphere.
23:39 All optical instruments are created through mathematics.
23:42 [Music]
23:47 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 We roll it and we have a...
24:10 The circle has been the basis for many of man's important inventions.
24:20 The mind can create the most amazing things.
24:24 If we spin the triangle, we have a...
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.
25:27 Number, please.
25:32 [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-do.
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, what's the matter with these doors?
26:48 Hey, these doors won't open.
26:51 They're locked.
26:53 Of course they are locked.
26:54 These are the doors of the future.
26:57 And the key is...
26:58 Mathematics.
27:00 Right, 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."