0:00:59.166,0:01:01.491
Very strange.

0:01:13.230,0:01:16.783
Huh, that's an odd-looking creature

0:01:27.953,0:01:31.720
What kind of a crazy place is this?

0:01:45.520,0:01:49.141
Well, what do you know? Square roots!

0:01:51.264,0:01:59.776
Pi is equal to 3.141592653589747 etc. etc. etc.

0:02:01.545,0:02:04.706
Hello? (echo)

0:02:06.103,0:02:07.891
Hello, Donald.

0:02:08.061,0:02:11.402
That's me! Where am I?!

0:02:11.402,0:02:13.838
Mathmagic land.

0:02:13.838,0:02:17.411
Mathmagic land? Never heard of it.

0:02:17.642,0:02:20.879
It's the land of great adventure.

0:02:20.879,0:02:23.048
Well, who are you?

0:02:23.048,0:02:27.075
I'm a spirit, the true spirit of adventure.

0:02:27.075,0:02:29.788
That's for me! What's next?

0:02:29.788,0:02:32.974
A journey through the wonderland of mathematics.

0:02:32.974,0:02:36.561
Mathematics? That's for eggheads!

0:02:36.561,0:02:39.141
Eggheads? Now hold on, Donald.

0:02:39.141,0:02:41.654
You like music don't you?

0:02:41.792,0:02:42.726
Yeah.

0:02:42.757,0:02:46.171
Well, without eggheads, there would be no music.

0:02:46.263,0:02:47.129
Bah.

0:02:47.668,0:02:55.009
Come on, let's go to ancient Greece, to the time of Pythagoras, the master egghead of them all.

0:02:55.009,0:02:56.225
Pythagoras?

0:02:56.225,0:02:58.513
The father of mathematics and music.

0:02:58.513,0:03:00.435
Mathematics and music?

0:03:00.435,0:03:04.622
Ahh, you'll find mathematics in the darndest places.

0:03:05.268,0:03:06.445
Watch

0:03:07.198,0:03:08.747
First we'll need a string

0:03:08.747,0:03:09.701
Hey!

0:03:10.055,0:03:13.142
Stretch it good and tight; pluck it!

0:03:14.049,0:03:17.176
Now divide in half. Pluck again.

0:03:18.007,0:03:21.952
You see? It's the same tone, one octave higher.

0:03:21.952,0:03:24.507
Now divide the next section.

0:03:24.568,0:03:26.639
And the next.

0:03:26.639,0:03:31.690
Pythagoras discovered the octave had a ratio of two to one.

0:03:31.690,0:03:39.475
With simple fractions, he got this [major triad]

0:03:39.475,0:03:47.668
And from this harmony in numbers, developed the musical scale of today. [major scale]

0:03:49.299,0:03:54.789
By golly, you DO find mathematics in the darndest places.

0:03:55.251,0:03:57.737
You can imagine how excited Pythagoras was,

0:03:57.737,0:04:03.384
when he shared his findings with his pals and fraternity of eggheads, known as the Pythagoreans.

0:04:03.877,0:04:08.571
They used to be meet in secret to discuss their mathematical discoveries.

0:04:08.571,0:04:11.340
Only members were allowed to attend.

0:04:11.340,0:04:15.502
They had a secret emblem, the pentagram.

0:04:18.887,0:04:22.129
Let's see what the topic is for today.

0:04:22.467,0:04:26.229
[major scale]

0:04:45.662,0:04:48.231
[music]

0:04:48.985,0:04:50.981
What's going on?

0:04:50.981,0:04:53.396
Shh! It's a jam session.

0:04:55.350,0:04:57.498
Gimme something with a beat!

0:04:57.498,0:04:58.600
Shhh!

0:04:59.308,0:05:02.277
[percussion]

0:05:31.033,0:05:34.041
So from these eggheads, the Pythagoreans,

0:05:34.041,0:05:36.102
with their mathematical formula

0:05:36.102,0:05:38.756
came the basis of our music of today.

0:05:38.756,0:05:42.179
[big band]

0:07:06.487,0:07:09.377
Pythag, old boy, put her there

0:07:12.838,0:07:17.707
Now I'll be a goshdarned egghead [laugh]

0:07:17.707,0:07:24.252
It was our old friend Pythagoras who discovered that the pentagram was full of mathemagic.

0:07:26.867,0:07:31.703
The two shorter lines combined exactly equal the third

0:07:31.703,0:07:36.288
and this line shows the magic proportions of the famous golden section

0:07:37.027,0:07:40.558
The second and third lines exactly equal the fourth

0:07:40.558,0:07:43.799
Once again we have the golden section

0:07:44.829,0:07:47.224
But this is only the beginning

0:07:47.224,0:07:49.239
Hidden within the pentagram

0:07:49.239,0:07:52.345
is a secret for creating a golden rectangle

0:07:52.345,0:07:58.056
which the Greeks admired for its beautiful proportions and magic qualities

0:07:58.056,0:08:02.268
The star contains the golden rectangle many times over

0:08:29.283,0:08:31.675
It's a most remarkable shape

0:08:31.675,0:08:35.269
It can mathematically reproduce itself indefinitely

0:08:40.161,0:08:43.964
All these rectangles have exactly the same proportions

0:08:51.272,0:08:54.317
This figure also contains a magic spiral

0:08:54.317,0:08:59.256
that repeats the proportions of the golden section into infinity

0:09:00.410,0:09:05.556
To the Greeks, the golden rectangle represented a mathematical law of beauty

0:09:05.556,0:09:08.986
We find it in their classical architecture

0:09:08.986,0:09:13.203
The Parthenon, perhaps one of the most famous of early Greek buildings,

0:09:13.203,0:09:15.708
contains many golden rectangles.

0:09:38.477,0:09:42.332
These same golden proportions are also found in their sculpture.

0:09:59.101,0:10:00.885
In the centuries that followed

0:10:00.885,0:10:06.665
the golden rectangle dominated the idea of beauty in architecture throughout the Western world.

0:10:06.665,0:10:10.718
The Cathedral of Notre Dame is an outstanding example.

0:10:12.579,0:10:15.622
The Renaissance painters knew this secret well.

0:10:21.299,0:10:25.960
Today, the golden rectangle is very much a part of our modern world.

0:10:30.544,0:10:34.457
Modern painters have rediscovered the magic of these proportions.

0:10:38.226,0:10:42.179
Indeed, this ideal proportion is to be found in life itself.

0:10:42.379,0:10:44.829
Boy, oh boy, oh boy!

0:10:45.198,0:10:50.429
This is mathematics? I like mathematical figures like that.

0:10:50.429,0:10:52.428
Ah, ah, ah, Donald.

0:10:52.428,0:10:53.747
Let me try it!

0:10:53.747,0:10:54.661
No, no.

0:10:54.661,0:10:56.355
Ideal proportion

0:10:56.878,0:10:58.182
Not quite

0:10:59.520,0:11:01.992
Uh uh. No, I'm afraid not.

0:11:04.654,0:11:07.310
Well, we can't all be mathematically perfect.

0:11:07.310,0:11:08.460
Oh yeah?

0:11:11.229,0:11:13.830
There, I knew I could do it.

0:11:14.015,0:11:16.468
Now that you're all pent up in a pentagon

0:11:16.468,0:11:20.422
let's see how nature uses the same mathematical form.

0:11:20.683,0:11:21.906
The petunia

0:11:25.121,0:11:26.637
The star jasmine

0:11:31.099,0:11:32.508
The starfish

0:11:37.123,0:11:38.529
The wax flower

0:11:43.760,0:11:46.812
There are literally thousands of members in good standing

0:11:46.812,0:11:50.407
in nature's Pythagorean society of the star.

0:11:57.868,0:12:00.915
All nature's works have a mathematical logic

0:12:00.915,0:12:02.938
and her patterns are limitless.

0:12:27.338,0:12:30.004
The magic proportions of the golden section

0:12:30.004,0:12:33.411
are often found in the spirals of nature's designs.

0:12:49.150,0:12:54.090
The profusion of mathematical forms brings to mind the words of Pythagoras:

0:12:54.090,0:12:58.878
"Everything is arranged according to number and mathematical shape."

0:12:59.555,0:13:02.355
Yes, there is mathematics in music,

0:13:02.355,0:13:05.027
in art, in just about everything.

0:13:05.166,0:13:09.483
And as as the Greeks had guessed, the rules are always the same.

0:13:36.683,0:13:40.004
Well, Donald, did you enjoy your geometrical journey?

0:13:40.004,0:13:45.218
Gee, Mr. Spirit, there's a lot more to mathematics than two times two!

0:13:45.218,0:13:46.439
That's right, Donald

0:13:46.439,0:13:49.109
And you can find mathematics in games, too!

0:13:49.109,0:13:51.152
Games! Oh, boy!

0:13:51.629,0:13:54.671
Let's begin with a game that's played on squares.

0:13:54.671,0:13:55.782
Checkers?

0:13:55.782,0:13:57.058
No, chess.

0:13:57.058,0:13:58.223
Chess?!

0:13:58.223,0:14:01.337
A mathematical contest between two minds.

0:14:01.352,0:14:05.805
It's a game that has been enjoyed for centuries by kings and commoners.

0:14:05.805,0:14:10.269
In fact, Louis Carroll, a famous mathematician with a literary mind,

0:14:10.269,0:14:15.785
used chess as a setting for his classic tale, Through the Looking Glass.

0:14:16.201,0:14:22.071
Alice found herself face to face with a none-too-friendly group of chess pieces.

0:14:22.071,0:14:24.938
Good heavens, what's this?

0:14:25.123,0:14:28.896
Upon my soul, it appears to be a lost pawn!

0:14:29.050,0:14:31.873
I'm no pawn, I'm Donald Duck!

0:14:31.920,0:14:34.227
He says he's Donald Duck!

0:14:34.381,0:14:36.162
Preposterous!

0:14:36.270,0:14:38.266
Or, it could be an Alice.

0:14:38.266,0:14:39.291
Alice?!

0:14:39.291,0:14:42.127
No, no no. It's a lost pawn.

0:14:42.666,0:14:46.578
Lost pawn? Stop that pawn!

0:14:46.578,0:14:50.643
Ow, Mr. Spirit! Help, help, help!

0:15:10.736,0:15:13.032
Whew, that was close!

0:15:13.032,0:15:16.341
Now you can look at this game from a safer perspective.

0:15:22.264,0:15:25.048
Chess is a game of calculated strategy,

0:15:25.048,0:15:27.552
and since the board is geometrical,

0:15:27.552,0:15:29.737
the moves are mathematical.

0:15:53.152,0:15:55.836
Checkmate, and the game is over.

0:15:55.836,0:15:59.574
That's very interesting. What's next?

0:15:59.574,0:16:03.321
Practically all games are played on geometrical areas.

0:16:03.321,0:16:05.506
The baseball field is a diamond.

0:16:05.506,0:16:06.703
Oh boy!

0:16:12.795,0:16:16.008
And without mathematics, we couldn't even keep score.

0:16:16.008,0:16:20.077
Football is played on a rectangle divided by yard lines.

0:16:22.030,0:16:26.606
Basketball is a game of circles, spheres and rectangles.

0:16:30.729,0:16:33.353
Even hopscotch has its multiple squares.

0:16:44.153,0:16:45.432
What's next?

0:16:45.971,0:16:47.134
Tiddlywinks?

0:16:48.027,0:16:53.291
No, a mathematical game played on a field of two perfect squares

0:16:53.291,0:16:55.683
using three perfect spheres

0:16:55.683,0:16:57.629
and a lot of diamonds.

0:16:57.629,0:16:59.716
In other words, billiards.

0:16:59.716,0:17:02.609
Oh boy! That's for me!

0:17:02.609,0:17:04.726
You know the game, don't you Donald?

0:17:04.726,0:17:09.641
Of course, the cue ball has to hit the other two balls

0:17:09.641,0:17:10.646
like this!

0:17:16.061,0:17:20.142
Now let's see how an expert at three-cushion billiards uses his head.

0:17:22.280,0:17:23.807
Three-cushion?

0:17:23.807,0:17:27.664
Yes. The cue ball not only has to hit both the other balls,

0:17:27.664,0:17:31.703
but it must contact at least three cushions before it hits the final ball.

0:17:42.734,0:17:44.545
One, two, three

0:17:58.314,0:18:00.180
One, two, three

0:18:10.042,0:18:13.572
It takes an expert to make several shots in succession

0:18:13.572,0:18:16.501
One, two, three, four

0:18:18.301,0:18:19.558
five, six.

0:18:21.697,0:18:24.493
Wow! That was a lucky shot!

0:18:24.908,0:18:27.209
Luck? No. It's skill.

0:18:27.870,0:18:30.637
For this game, you have to know all the angles.

0:18:52.067,0:18:56.844
One, two, three, four, five, six, seven.

0:18:57.606,0:19:00.843
That's amazing! How does he do it?

0:19:00.843,0:19:02.912
First, there's technique.

0:19:02.912,0:19:06.663
He's striking the cue ball low, so it'll spin backwards.

0:19:11.017,0:19:14.860
Hitting the ball on the right side will make it hug the rail.

0:19:14.860,0:19:17.466
These trick shots take a lot of practice.

0:19:18.712,0:19:21.225
Hahaha! He missed that time!

0:19:21.225,0:19:22.545
One, two...

0:19:26.130,0:19:26.996
three.

0:19:29.596,0:19:32.155
What's so mathematical about that?

0:19:32.155,0:19:35.037
Oh, this game takes precise calculation.

0:19:35.037,0:19:37.594
He figures out each shot in his head.

0:19:38.671,0:19:42.700
He could play it like this, but it calls for quite a bit of luck.

0:19:42.700,0:19:44.538
There is a better choice.

0:19:44.538,0:19:49.360
For this, he uses the diamond markings on the rail as a mathematical guide.

0:19:49.360,0:19:53.355
First, he figures the natural angle for hitting the object balls.

0:19:53.355,0:19:57.674
And then he finds that his cue ball must bounce off the number three diamond.

0:19:57.674,0:20:01.797
Next, he gets ready for the shot and he needs a number for his cue position.

0:20:01.797,0:20:04.936
This calls for a different set of numbers.

0:20:05.198,0:20:07.459
Very confusing, isn't it?

0:20:07.844,0:20:09.889
Not when you get the hang of it.

0:20:09.889,0:20:12.734
You see, the cue position is four.

0:20:12.734,0:20:14.789
Now, a simple subtraction.

0:20:14.789,0:20:16.578
Three from four is one.

0:20:16.578,0:20:19.903
So, if he shoots for the first diamond, he should make it.

0:20:19.903,0:20:22.459
It's called "playing the diamond system".

0:20:27.536,0:20:29.440
Natural angle, 2.

0:20:29.440,0:20:32.953
Cue position, one and a half, two, two and a half, three,

0:20:33.230,0:20:34.413
three and a half.

0:20:34.413,0:20:37.012
Two from three and a half is one and a half.

0:20:37.012,0:20:41.333
So, shoot halfway between the first and second diamonds.

0:20:44.795,0:20:47.212
There's nothing to it! Let me try!

0:20:51.335,0:20:52.921
Let's see now.

0:20:53.167,0:20:58.485
If I shoot it here, and it bounces there, and uh, no there.

0:20:58.808,0:21:00.519
If I shoot it here...

0:21:00.519,0:21:04.169
Four and half minus three, three and a half plus four...

0:21:04.169,0:21:05.865
Add it to two...

0:21:07.711,0:21:09.821
And dividing it.. and...

0:21:09.821,0:21:11.713
I guess I should shoot about here.

0:21:11.713,0:21:15.570
No, no, Donald. There's no guesswork to mathematics.

0:21:15.570,0:21:17.062
It's quite simple.

0:21:17.062,0:21:20.167
Natural angle for the hit: two.

0:21:20.167,0:21:22.857
Cue position: three and a half.

0:21:22.857,0:21:25.696
How much is three and a half minus two?

0:21:25.696,0:21:27.991
Uhhh... one and a half!

0:21:38.714,0:21:41.852
Hey! It works! Oh boy!

0:21:41.852,0:21:43.594
It's a cinch!

0:21:44.610,0:21:48.097
If i hit it here, add three and a half plus four

0:21:48.097,0:21:50.803
Four and a half minus three... [???]

0:21:51.942,0:21:54.655
You're making it tough for yourself, Donald.

0:22:05.808,0:22:09.444
How do you like that for mathematics, Mr. Spirit?

0:22:09.444,0:22:14.461
Wonderful, Donald. And now you're ready for the most exciting game of all.

0:22:14.461,0:22:16.080
Oh, boy!

0:22:16.695,0:22:19.874
And the playing field for this game is in the mind.

0:22:20.982,0:22:24.665
Uh oh, look at the condition of your mind!

0:22:24.665,0:22:33.437
Antiquated ideas, bungling, false concepts, superstitions, confusion!

0:22:33.437,0:22:36.091
To think straight, we'll have to clean house.

0:22:48.091,0:22:50.461
There, that's more like it.

0:22:50.461,0:22:53.044
A nice clean sweep.

0:22:53.044,0:22:56.334
This game is played with circles and triangles.

0:22:56.334,0:22:58.588
Think of a perfect circle.

0:23:02.465,0:23:07.917
A perfect circle. Perfect. Circle.

0:23:08.748,0:23:11.695
Perfect. Ahhhhh.

0:23:11.695,0:23:15.057
Put a triangle inside and turn it.

0:23:15.057,0:23:18.895
Now spin the circle, and what have you got?

0:23:19.956,0:23:21.156
A ball!

0:23:21.713,0:23:23.576
Yes, a sphere.

0:23:23.576,0:23:27.819
The shape of things is first discovered in the mind.

0:23:27.819,0:23:30.325
Slice off the top and we have a...

0:23:33.309,0:23:35.745
A magnifying glass!

0:23:35.745,0:23:37.131
That's right.

0:23:37.131,0:23:39.901
A lens is a section of a sphere.

0:23:39.901,0:23:43.383
All optical instruments are created through mathematics.

0:23:46.783,0:23:51.399
You see, there's a lot more to mathematics than just numbers and equations.

0:23:52.429,0:23:55.246
Let's get back to our circle and triangle.

0:23:59.000,0:24:01.298
Roll it and we have a...

0:24:01.298,0:24:02.568
A... a wheel!

0:24:10.337,0:24:14.443
The circle has been the basis for many of man's important inventions.

0:24:20.197,0:24:23.552
The mind can create the most amazing things.

0:24:24.537,0:24:27.475
If we spin the triangle, we have a...

0:24:27.475,0:24:28.174
Cone!

0:24:28.174,0:24:29.531
Slice the cone.

0:24:29.531,0:24:30.466
<i>giggle</i>

0:24:30.466,0:24:33.938
The cone is full of useful mathematical shapes.

0:24:35.815,0:24:38.955
Slice it again. Slice it several times.

0:24:42.047,0:24:46.738
The orbits of all planets and satellites can be found in the cone.

0:24:46.738,0:24:50.555
No matter how you slice it, it's always mathematics.

0:24:50.555,0:24:54.710
A slice like this gives us the reflector of a search light.

0:24:55.541,0:24:59.356
A slice like this, the mirror of a giant telescope.

0:25:01.480,0:25:04.680
A line on a cone, and we have a drill.

0:25:08.449,0:25:10.157
And the spring.

0:25:14.342,0:25:15.447
Now you're ticking.

0:25:27.063,0:25:28.412
Number, please?

0:25:40.643,0:25:45.338
The mind is the birthplace for all of man's scientific achievements.

0:26:00.754,0:26:04.763
The mind knows no limits when used properly.

0:26:04.763,0:26:06.855
Think of a pentagram, Donald.

0:26:09.532,0:26:11.526
Now, put another inside.

0:26:11.526,0:26:13.981
A third. And a fourth.

0:26:14.535,0:26:18.069
No pencil is sharp enough to draw as fine as you can think

0:26:18.069,0:26:22.093
and no paper large enough to hold your imagination.

0:26:22.093,0:26:26.566
In fact, it is only in the mind that we can conceive infinity.

0:26:28.259,0:26:33.040
Mathematical thinking has opened the doors to the exciting adventures of science.

0:26:35.086,0:26:37.265
I'll be dog-darned!

0:26:37.265,0:26:40.030
I've never seen so many doors before.

0:26:40.030,0:26:43.023
Each discovery leads to many others.

0:26:43.023,0:26:44.804
An endless chain.

0:26:45.127,0:26:48.897
Hey! Hey! Whatsa matter with these doors?

0:26:48.897,0:26:52.775
Hey! These doors won't open! They're locked!

0:26:52.775,0:26:54.724
Of course they're locked.

0:26:54.724,0:26:56.852
These are the doors of the future,

0:26:56.852,0:26:58.568
and the key is...

0:26:58.568,0:27:00.220
Mathematics!

0:27:00.328,0:27:02.844
Right. Mathematics.

0:27:02.844,0:27:07.517
The boundless treasures of science are locked behind those doors.

0:27:07.517,0:27:13.698
In time, they will be opened by the curious and inquiring minds of future generations.

0:27:15.467,0:27:17.628
In the words of Galileo:

0:27:17.628,0:27:25.813
"Mathematics is the alphabet with which God has written the universe."