-
Very strange.
-
Huh, that's an odd-looking creature
-
What kind of a crazy place is this?
-
Well, what do you know? Square roots!
-
Pi is equal to 3.141592653589747 etc. etc. etc.
-
Hello? (echo)
-
Hello, Donald.
-
That's me! Where am I?!
-
Mathmagic land.
-
Mathmagic land? Never heard of it.
-
It's the land of great adventure.
-
Well, who are you?
-
I'm a spirit, the true spirit of adventure.
-
That's for me! What's next?
-
A journey through the wonderland of mathematics.
-
Mathematics? That's for eggheads!
-
Eggheads? Now hold on, Donald.
-
You like music don't you?
-
Yeah.
-
Well, without eggheads, there would be no music.
-
Bah.
-
Come on, let's go to ancient Greece, to the time of Pythagoras, the master egghead of them all.
-
Pythagoras?
-
The father of mathematics and music.
-
Mathematics and music?
-
Ahh, you'll find mathematics in the darndest places.
-
Watch
-
First we'll need a string
-
Hey!
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Stretch it good and tight; pluck it!
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Now divide in half. Pluck again.
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You see? It's the same tone, one octave higher.
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Now divide the next section.
-
And the next.
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Pythagoras discovered the octave had a ratio of two to one.
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With simple fractions, he got this [major triad]
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And from this harmony in numbers, developed the musical scale of today. [major scale]
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By golly, you DO find mathematics in the darndest places.
-
You can imagine how excited Pythagoras was,
-
when he shared his findings with his pals and fraternity of eggheads, known as the Pythagoreans.
-
They used to be meet in secret to discuss their mathematical discoveries.
-
Only members were allowed to attend.
-
They had a secret emblem, the pentagram.
-
Let's see what the topic is for today.
-
[major scale]
-
[music]
-
What's going on?
-
Shh! It's a jam session.
-
Gimme something with a beat!
-
Shhh!
-
[percussion]
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So from these eggheads, the Pythagoreans,
-
with their mathematical formula
-
came the basis of our music of today.
-
[big band]
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Pythag, old boy, put her there
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Now I'll be a goshdarned egghead [laugh]
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It was our old friend Pythagoras who discovered that the pentagram was full of mathemagic.
-
The two shorter lines combined exactly equal the third
-
and this line shows the magic proportions of the famous golden section
-
The second and third lines exactly equal the fourth
-
Once again we have the golden section
-
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
-
The star contains the golden rectangle many times over
-
It's a most remarkable shape
-
It can mathematically reproduce itself indefinitely
-
All these rectangles have exactly the same proportions
-
This figure also contains a magic spiral
-
that repeats the proportions of the golden section into infinity
-
To the Greeks, the golden rectangle represented a mathematical law of beauty
-
We find it in their classical architecture
-
The Parthenon, perhaps one of the most famous of early Greek buildings,
-
contains many golden rectangles.
-
These same golden proportions are also found in their sculpture.
-
In the centuries that followed
-
the golden rectangle dominated the idea of beauty in architecture throughout the Western world.
-
The Cathedral of Notre Dame is an outstanding example.
-
The Renaissance painters knew this secret well.
-
Today, the golden rectangle is very much a part of our modern world.
-
Modern painters have rediscovered the magic of these proportions.
-
Indeed, this ideal proportion is to be found in life itself.
-
Boy, oh boy, oh boy!
-
This is mathematics? I like mathematical figures like that.
-
Ah, ah, ah, Donald.
-
Let me try it!
-
No, no.
-
Ideal proportion
-
Not quite
-
Uh uh. No, I'm afraid not.
-
Well, we can't all be mathematically perfect.
-
Oh yeah?
-
There, I knew I could do it.
-
Now that you're all pent up in a pentagon
-
let's see how nature uses the same mathematical form.
-
The petunia
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The star jasmine
-
The starfish
-
The wax flower
-
There are literally thousands of members in good standing
-
in nature's Pythagorean society of the star.
-
All nature's works have a mathematical logic
-
and her patterns are limitless.
-
The magic proportions of the golden section
-
are often found in the spirals of nature's designs.
-
The profusion of mathematical forms brings to mind the words of Pythagoras:
-
"Everything is arranged according to number and mathematical shape."
-
Yes, there is mathematics in music,
-
in art, in just about everything.
-
And as as the Greeks had guessed, the rules are always the same.
-
Well, Donald, did you enjoy your geometrical journey?
-
Gee, Mr. Spirit, there's a lot more to mathematics than two times two!
-
That's right, Donald
-
And you can find mathematics in games, too!
-
Games! Oh, boy!
-
Let's begin with a game that's played on squares.
-
Checkers?
-
No, chess.
-
Chess?!
-
A mathematical contest between two minds.
-
It's a game that has been enjoyed for centuries by kings and commoners.
-
In fact, Louis Carroll, a famous mathematician with a literary mind,
-
used chess as a setting for his classic tale, Through the Looking Glass.
-
Alice found herself face to face with a none-too-friendly group of chess pieces.
-
Good heavens, what's this?
-
Upon my soul, it appears to be a lost pawn!
-
I'm no pawn, I'm Donald Duck!
-
He says he's Donald Duck!
-
Preposterous!
-
Or, it could be an Alice.
-
Alice?!
-
No, no no. It's a lost pawn.
-
Lost pawn? Stop that pawn!
-
Ow, Mr. Spirit! Help, help, help!
-
Whew, that was close!
-
Now you can look at this game from a safer perspective.
-
Chess is a game of calculated strategy,
-
and since the board is geometrical,
-
the moves are mathematical.
-
Checkmate, and the game is over.
-
That's very interesting. What's next?
-
Practically all games are played on geometrical areas.
-
The baseball field is a diamond.
-
Oh boy!
-
And without mathematics, we couldn't even keep score.
-
Football is played on a rectangle divided by yard lines.
-
Basketball is a game of circles, spheres and rectangles.
-
Even hopscotch has its multiple squares.
-
What's next?
-
Tiddlywinks?
-
No, a mathematical game played on a field of two perfect squares
-
using three perfect spheres
-
and a lot of diamonds.
-
In other words, billiards.
-
Oh boy! That's for me!
-
You know the game, don't you Donald?
-
Of course, the cue ball has to hit the other two balls
-
like this!
-
Now let's see how an expert at three-cushion billiards uses his head.
-
Three-cushion?
-
Yes. The cue ball not only has to hit both the other balls,
-
but it must contact at least three cushions before it hits the final ball.
-
One, two, three
-
One, two, three
-
It takes an expert to make several shots in succession
-
One, two, three, four
-
five, six.
-
Wow! That was a lucky shot!
-
Luck? No. It's skill.
-
For this game, you have to know all the angles.
-
One, two, three, four, five, six, seven.
-
That's amazing! How does he do it?
-
First, there's technique.
-
He's striking the cue ball low, so it'll spin backwards.
-
Hitting the ball on the right side will make it hug the rail.
-
These trick shots take a lot of practice.
-
Hahaha! He missed that time!
-
One, two...
-
three.
-
What's so mathematical about that?
-
Oh, this game takes precise calculation.
-
He figures out each shot in his head.
-
He could play it like this, but it calls for quite a bit of luck.
-
There is a better choice.
-
For this, he uses the diamond markings on the rail as a mathematical guide.
-
First, he figures the natural angle for hitting the object balls.
-
And then he finds that his cue ball must bounce off the number three diamond.
-
Next, he gets ready for the shot and he needs a number for his cue position.
-
This calls for a different set of numbers.
-
Very confusing, isn't it?
-
Not when you get the hang of it.
-
You see, the cue position is four.
-
Now, a simple subtraction.
-
Three from four is one.
-
So, if he shoots for the first diamond, he should make it.
-
It's called "playing the diamond system".
-
Natural angle, 2.
-
Cue position, one and a half, two, two and a half, three,
-
three and a half.
-
Two from three and a half is one and a half.
-
So, shoot halfway between the first and second diamonds.
-
There's nothing to it! Let me try!
-
Let's see now.
-
If I shoot it here, and it bounces there, and uh, no there.
-
If I shoot it here...
-
Four and half minus three, three and a half plus four...
-
Add it to two...
-
And dividing it.. and...
-
I guess I should shoot about here.
-
No, no, Donald. There's no guesswork to mathematics.
-
It's quite simple.
-
Natural angle for the hit: two.
-
Cue position: three and a half.
-
How much is three and a half minus two?
-
Uhhh... one and a half!
-
Hey! It works! Oh boy!
-
It's a cinch!
-
If i hit it here, add three and a half plus four
-
Four and a half minus three... [???]
-
You're making it tough for yourself, Donald.
-
How do you like that for mathematics, Mr. Spirit?
-
Wonderful, Donald. And now you're ready for the most exciting game of all.
-
Oh, boy!
-
And the playing field for this game is in the mind.
-
Uh oh, look at the condition of your mind!
-
Antiquated ideas, bungling, false concepts, superstitions, confusion!
-
To think straight, we'll have to clean house.
-
There, that's more like it.
-
A nice clean sweep.
-
This game is played with circles and triangles.
-
Think of a perfect circle.
-
A perfect circle. Perfect. Circle.
-
Perfect. Ahhhhh.
-
Put a triangle inside and turn it.
-
Now spin the circle, and what have you got?
-
A ball!
-
Yes, a sphere.
-
The shape of things is first discovered in the mind.
-
Slice off the top and we have a...
-
A magnifying glass!
-
That's right.
-
A lens is a section of a sphere.
-
All optical instruments are created through mathematics.
-
You see, there's a lot more to mathematics than just numbers and equations.
-
Let's get back to our circle and triangle.
-
Roll it and we have a...
-
A... a wheel!
-
The circle has been the basis for many of man's important inventions.
-
The mind can create the most amazing things.
-
If we spin the triangle, we have a...
-
Cone!
-
Slice the cone.
-
giggle
-
The cone is full of useful mathematical shapes.
-
Slice it again. Slice it several times.
-
The orbits of all planets and satellites can be found in the cone.
-
No matter how you slice it, it's always mathematics.
-
A slice like this gives us the reflector of a search light.
-
A slice like this, the mirror of a giant telescope.
-
A line on a cone, and we have a drill.
-
And the spring.
-
Now you're ticking.
-
Number, please?
-
The mind is the birthplace for all of man's scientific achievements.
-
The mind knows no limits when used properly.
-
Think of a pentagram, Donald.
-
Now, put another inside.
-
A third. And a fourth.
-
No pencil is sharp enough to draw as fine as you can think
-
and no paper large enough to hold your imagination.
-
In fact, it is only in the mind that we can conceive infinity.
-
Mathematical thinking has opened the doors to the exciting adventures of science.
-
I'll be dog-darned!
-
I've never seen so many doors before.
-
Each discovery leads to many others.
-
An endless chain.
-
Hey! Hey! Whatsa matter with these doors?
-
Hey! These doors won't open! They're locked!
-
Of course they're locked.
-
These are the doors of the future,
-
and the key is...
-
Mathematics!
-
Right. Mathematics.
-
The boundless treasures of science are locked behind those doors.
-
In time, they will be opened by the curious and inquiring minds of future generations.
-
In the words of Galileo:
-
"Mathematics is the alphabet with which God has written the universe."
