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1
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Mighty strange.
2
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That's an odd-looking creature.
3
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What kind of a crazy place is this?
4
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Well, what do you know? Square roots!
5
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Pi is equal to 3.141592653589747
6
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etcetera, etcetera, etcetera.
7
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Huh?
8
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Hello!
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(ECHOING) Hello! Hello!
10
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SPIRIT: Hello, Donald.
11
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That's me. Where am I
12
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SPIRIT: Mathemagic Land.
13
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Mathmagic Land? Never heard of it.
14
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SPIRIT: It 's the land of great adventure.
15
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Well, who are you?
16
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SPIRIT: I' m a spirit.The True Spirit of Adventure.
17
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That's for me. What's next?
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SPIRIT: A journey throughthe wonderland of mathematics.
19
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Mathematics? That's for eggheads.
20
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SPIRIT: Eggheads?Now hold on, Donald.
21
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You like music, don't you?
22
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Yeah.
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SPIRIT: Well, without eggheads,there would be no music.
24
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(EX CLAIMS)
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SPIRIT: Come on,let's go to ancient Greece.
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To the time of Pythagoras,the master egghead of them all.
27
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Pythagoras?
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SPIRIT: The father of mathematicsand music.
29
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DONALD: Mathematics and music?
30
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SPIRIT: You'll find mathematicsin the darndest places.
31
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Watch.
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-SPIRIT: First, we'll need a string.
-Hey!
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SPIRIT: Stretch it good and tight,plunk it.
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(TWANGING)
35
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Now, divide in half.
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Plunk again.
37
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You see?It's the same tone, one octave higher.
38
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Now divide the next section,
39
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and the next.
40
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Pythagoras discoveredthe octave had a ratio of 2-to-1.
41
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With simple fractions, he got this.
42
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(MAJOR SCALE PLAYING)
43
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And from this harmony in numbersdeveloped the musical scale of today.
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By golly!
45
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You do find mathematics
in the darndest places.
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SPIRIT: You can imaginehow excited Pythagoras was
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when he shared his findingswith his pals, a fraternity of eggheads,
48
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known as the Pythagoreans.
49
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(WHISPERING) They used to meetin secret to discuss
50
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their mathematical discoveries.
51
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Only members were allowed to attend.
52
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They had a secret emblem,the pentagram.
53
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Let's see what the topic is for today.
54
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(PLAYING MAJOR SCALE)
55
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(PLAYING MAJOR SCALE)
56
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(PLAYING OUT OF TUNE)
57
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(PLAYING MAJOR SCALE)
58
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(PLAYING OUT OF TUNE)
59
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(PLAYING SOFT MUSIC)
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What's going on?
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SPIRIT: Shh, it's a jam session.
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DONALD: Give me
something with a beat.
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SPIRIT: Shh!
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(PLAYING UPBEAT DRUM MUSIC)
65
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(PLAYING UPBEAT MUSIC)
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SPIRIT: So, from these eggheads,the Pythagoreans,
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with their mathematical formula,came the basis of our music of today.
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(PLAYING JAZZ MUSIC)
69
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(PLAYING CLASSICAL MUSIC)
70
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Pythag, old boy, put her there.
71
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Well, I'll be a goshdarn egghead.
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SPIRIT: It was our old friendPythagoras who discovered
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that the pentagram wasfull of mathemagic.
74
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The two shorter lines combinedexactly equal the third,
75
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and this line showsthe magic proportions
76
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of the famous golden section.
77
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The second and third linesexactly equal the fourth.
78
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Once again,we have the golden section.
79
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But this is only the beginning.
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Hidden within the pentagram
81
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is a secret for creatinga golden rectangle,
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which the Greeks admiredfor its beautiful proportions
83
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and magic qualities.
84
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The star contains the golden rectanglemany times over.
85
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It's a most remarkable shape.
86
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It can mathematicallyreproduce itself indefinitely.
87
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All these rectangles haveexactly the same proportions.
88
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This figure also contains a magic spiral
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that repeats the proportionsof the golden section
90
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into infinity.
91
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To the Greeks,the golden rectangle represented
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a mathematical law of beauty.
93
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We find it in their classical architecture.
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The Parthenon,
95
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perhaps one of the most famousof early Greek buildings,
96
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contains many golden rectangles.
97
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These same golden proportionsare also found in their sculpture.
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In the centuries that followed,
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the golden rectangledominated the idea
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of beauty and architecturethroughout the Western world.
101
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The cathedral of Notre Dameis an outstanding example.
102
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The Renaissance paintersknew this secret well.
103
00:10:21,585 --> 00:10:26,613
Today, the golden rectangle isvery much a part of our modern world.
104
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Modern painters have rediscoveredthe magic of these proportions.
105
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Indeed, this ideal proportionis to be found in life itself.
106
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DONALD: Boy, oh, boy! Oh, boy!
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This is mathematics?
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l got mathematical figures like that.
109
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SPIRIT: Ah, ah, ah, Donald.
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Let me try it.
111
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SPIRIT: No, no.
112
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ldeal proportion.
113
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SPIRIT: Not quite.
114
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Uh-uh. No, I' m afraid not.
115
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Well, we can't allbe mathematically perfect.
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Oh, yeah?
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There! I knew I could do it.
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SPIRIT: Now that you'reall pent up in a pentagon,
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let's see how nature uses thissame mathematical form.
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The petunia,
121
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the Star Jasmine,
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the starfish,
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the waxflower.
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There are literally thousandsof members in good standing,
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in nature'sPythagorean Society of the Star.
126
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All nature's workshave a mathematical logic,
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00:12:01,418 --> 00:12:03,545
and her patterns are limitless.
128
00:12:27,644 --> 00:12:30,374
The magic proportionsof the golden section
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are often found inthe spirals of nature's designs.
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The profusion of mathematical forms
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brings to mind thewords of Pythagoras,
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"Everything is arranged according tonumber and mathematical shape."
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Yes, there is mathematics in music,in art, in just about everything.
134
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And as the Greeks had guessed,the rules are always the same.
135
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Well, Donald, did you enjoyyour geometrical journey?
136
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Gee, Mr. Spirit, there's a lot more
to mathematics than two times two.
137
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SPIRIT: That's right, Donald,
138
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and you can find mathematicsin games, too.
139
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Games? Oh, boy!
140
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SPIRIT: Let's begin with a gamethat's played on squares.
141
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-Checkers?
-No.
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-Chess.
-Chess?
143
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A mathematical contestbetween two minds.
144
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It's a game that has been enjoyedfor centuries by kings and commoners.
145
00:14:06,009 --> 00:14:07,704
In fact, Lewis Carroll,
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a famous mathematicianwith a literary mind,
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used chess as a settingfor his classic tale,
148
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Through the Looking Glass.
149
00:14:16,620 --> 00:14:19,521
Alice found herself face-to-face
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with a none too friendlygroup of chess pieces.
151
00:14:22,359 --> 00:14:24,953
Good heavens! What's this?
152
00:14:25,528 --> 00:14:29,294
Upon my soul,
it appears to be a lost pawn.
153
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I'm no pawn. I' m Donald Duck.
154
00:14:32,269 --> 00:14:34,601
He says he's Donald Duck.
155
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Preposterous!
156
00:14:36,640 --> 00:14:38,471
Or, it could be an Alice.
157
00:14:38,541 --> 00:14:39,565
Alice?
158
00:14:39,643 --> 00:14:43,204
No, no, no. It 's a lost pawn.
159
00:14:43,280 --> 00:14:44,747
Lost pawn?
160
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Stop that pawn!
161
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Help! Mr. Spirit! Help! Help! Help!
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00:15:09,706 --> 00:15:10,695
(GROANS)
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Whew! That was close.
164
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SPIRIT: Now you can lookat this game from a safer perspective.
165
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Chess is a game of calculated strategy
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and since the board is geometricalthe moves are mathematical
167
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Checkmate and the game is over.
168
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That's very interesting. What's next?
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SPIRIT: Practically, all gamesare played on geometrical areas.
170
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The baseball field is a diamond.
171
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Oh, boy!
172
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SPIRIT: And without mathematics,we couldn't even keep score.
173
00:16:16,306 --> 00:16:19,935
Football is played on a rectangledivided by yard lines.
174
00:16:22,412 --> 00:16:25,973
Basketball is a game of circles,spheres and rectangles.
175
00:16:31,121 --> 00:16:33,817
Even hopscotch hasits multiple squares.
176
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What's next, tiddlywinks?
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SPIRIT: No.
178
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A mathematical game playedon a field of two perfect squares,
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using three perfect spheresand a lot of diamonds.
180
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In other words, billiards.
181
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Oh, boy! That's for me!
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SPIRIT: You know the game,don't you, Donald ?
183
00:17:04,988 --> 00:17:09,584
Of course, that cue ball has
to hit the other two balls.
184
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Like this!
185
00:17:16,499 --> 00:17:18,831
SPIRIT: Now, let's seehow an expert at three-cushion billiards
186
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uses his head.
187
00:17:22,705 --> 00:17:24,070
DONALD: Three cushion?
188
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SPIRIT: Yes, the cue ballnot only has to hit both the other balls,
189
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but it must contact at leastthree cushions before it hits
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the final ball.
191
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One, two, three.
192
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One, two, three.
193
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It takes an expert to makeseveral shots in succession.
194
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One, two,
195
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three, four,
196
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five, six.
197
00:18:21,297 --> 00:18:24,425
Wow! That was a lucky shot.
198
00:18:24,934 --> 00:18:28,028
SPIRIT: Luck? No, it's skill.
199
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For this game,you have to know all the angles.
200
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DONALD: One, two, three, four, five,
201
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six, seven.
202
00:18:57,934 --> 00:19:01,165
That's amazing! How does he do it?
203
00:19:01,237 --> 00:19:03,034
SPIRIT: First, there's technique.
204
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He's striking the cue ball low
205
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so it'll spin backwards.
206
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Hitting the ball on the right sidewill make it hug the rail
207
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These trick shots take a lot of practice.
208
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(LAUGHING) He missed it!
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SPIRIT: One, two,
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three.
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DONALD: What's so mathematicalabout that?
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SPIRIT: Oh, this game takesprecise calculation.
213
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He figures out each shot in his head.
214
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He could play it like this,but it calls for quite a bit of luck.
215
00:19:42,979 --> 00:19:44,742
There is a better choice.
216
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For this he usesthe diamond markings on the rail
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as a mathematical guide.
218
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First, he figures the natural anglefor hitting the object balls,
219
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and then he finds that his cue ball must
220
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bounce off the number 3 diamond.
221
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Next, he gets ready for the shot,
222
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and he needs a numberfor his cue position.
223
00:20:02,265 --> 00:20:04,995
This calls for a different set of numbers.
224
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DONALD: Very confusing, isn't it?
225
00:20:08,237 --> 00:20:10,068
SPIRIT: Not whenyou get the hang of it.
226
00:20:10,139 --> 00:20:13,006
You see, the cue position is 4.
227
00:20:13,076 --> 00:20:16,876
Now, a simple subtraction.3 from 4 is 1.
228
00:20:16,946 --> 00:20:20,382
So, if he shoots for the first diamond,he should make it.
229
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It's called playing the Diamond System.
230
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Natural angle, 2.
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Cue position, 1.5, 2, 2.5, 3, 3.5.
232
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2 from 3.5 is 1.5.
233
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So, shoot halfway betweenthe first and second diamonds.
234
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There's nothing to it. Let me try.
235
00:20:51,681 --> 00:20:53,410
Let's see now.
236
00:20:53,516 --> 00:20:56,781
lf I hit it here and it bounced there...
237
00:20:56,853 --> 00:21:00,345
And, no, here. lf I hit it here...
238
00:21:00,823 --> 00:21:05,624
4.5 minus 3... 3.5 plus 4 added to 2.
239
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Divided into...
240
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I guess I should shoot about here.
241
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SPIRIT: No, no, Donald.There's no guess work to mathematics.
242
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It's quite simple.Natural angle for the hit, 2.
243
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Cue position, 3.5.
244
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How much is 3.5 minus 2?
245
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DONALD: Uh, 1.5.
246
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Hey, it works! Oh, boy!
247
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It's a cinch!
248
00:21:44,367 --> 00:21:46,392
If l hit it here...
249
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3.5 plus 4 to 4.5 minus 3...
250
00:21:50,973 --> 00:21:52,235
Divided by that...
251
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SPIRIT: You're making ittough for yourself, Donald.
252
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How do you like that
for mathematics, Mr. Spirit?
253
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SPIRIT: Wonderful, Donald.
254
00:22:11,127 --> 00:22:14,585
And now you're readyfor the most exciting game of all.
255
00:22:14,931 --> 00:22:16,455
Oh, boy!
256
00:22:16,999 --> 00:22:20,162
SPIRIT: And the playing fieldfor this game is in the mind.
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Uh-oh!Look at the condition of your mind.
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Antiquated ideas,bungling, false concepts,
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superstitions, confusion.
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To think straight,we'll have to clean house.
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There! That's more like it.A nice, clean sweep.
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This game is playedwith circles and triangles.
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Think of a perfect circle.
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A perfect circle.
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Perfect circle.
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Perfect. Ah!
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Put a triangle inside and turn it.
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Now, spin the circleand what have you got?
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DONALD: A ball!
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SPIRIT: Yes, a sphere.
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The shape of thingsis first discovered in the mind.
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Slice off the top and we have a...
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DONALD: A magnifying glass!
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SPIRIT: That's right.Alens is a section of a sphere.
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All optical instrumentsare created through mathematics.
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You see,there's a lot more to mathematics
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than just numbers and equations.
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Let's get back to our circle and triangle.
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Roll it and we have a...
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DONALD: A wheel
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SPIRIT: The circle has been the basisfor many of man's important inventions.
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The mind can createthe most amazing things.
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lf we spin the triangle, we have a...
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-DONALD: Cone!
-Slice the cone.
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(DONALD CHUCKLING)
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SPIRIT: The cone is fullof useful mathematical shapes.
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Slice it again. Slice it several times.
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The orbits of all planets and satellitescan be found in the cone.
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No matter how you slice it,it's always mathematics.
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A slice like this gives usthe reflector of a search light.
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A slice like this,the mirror of a giant telescope.
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Aline on a cone and we have a drill,
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and a spring.
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Now you're ticking.
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(TELEPHONE DIALING)
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OPERA TOR: Number, please.
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(TRUMPET MUSIC PLAYING)
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(ENERGETIC MUSIC PLAYING)
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SPIRIT: The mind is the birthplacefor all of man's scientific achievements.
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The mind knows no limitswhen used properly.
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Think of a pentagram, Donald.
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Now, put another inside,a third and a fourth.
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No pencil is sharp enoughto draw as fine as you can think,
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and no paper large enoughto hold your imagination.
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In fact, it is only in the mindthat we can conceive infinity.
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Mathematical thinkinghas opened the doors
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to the exciting adventures of science.
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I'll be doggone!
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I've never seen so many doors before.
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SPIRIT: Each discovery leadsto many others.
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An endless chain.
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Hey! Hey!
What's the matter with these doors?
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Hey! These doors won't open.
They're locked!
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SPIRIT: Of course, they're locked.These are the doors of the future,
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and the key is...
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Mathematics!
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SPIRIT: Right. Mathematics.
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The boundless treasures of scienceare locked behind those doors.
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In time, they will be openedby the curious and inquiring minds
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of future generations.
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In the words of Galileo,
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"Mathematics is the alphabet
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"with which God haswritten the universe. "26874
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