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!
Stretch it good and tight; pluck it!
Now divide in half. Pluck again.
You see? It's the same tone, one octave higher.
Now divide the next section.
And the next.
Pythagoras discovered the octave had a ratio of two to one.
With simple fractions, he got this [major triad]
And from this harmony in numbers, developed the musical scale of today. [major scale]
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]
So from these eggheads, the Pythagoreans,
with their mathematical formula
came the basis of our music of today.
[big band]
Pythag, old boy, put her there
Now I'll be a goshdarned egghead [laugh]
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
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
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."