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[SPOOKY MUSIC PLAYS]
MIGHTY STRANGE.
HA! THAT'S AN ODD-LOOKING CREATURE.
WHAT KIND OF A CRAZY PLACE IS THIS?
Donald: WELL, WHAT DO YOU KNOW!
SQUARE ROOTS!
PI IS EQUAL TO 3.141592653589747...
ET CETERA, ET CETERA, ET CETERA.
HUH?
HELLO!
[ECHOING] HELLO!
Spirit: 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.
AWW!
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?
AH, YOU'LL FIND MATHEMATICS IN THE DARNEDEST PLACES.
WATCH.
FIRST WE'LL NEED A STRING.
HEY!
GOOD AND TIGHT.
PLUNK IT.
[LOW NOTE PLAYS]
ALF.
PLUNK AGAIN.
[HIGHER NOTE PLAYS]
YOU SEE?
IT'S THE SAME TONE, ONE OCTAVE HIGHER.
NOW DIVIDE THE NEXT SECTION.
[HIGHER NOTE PLAYS]
AND THE NEXT.
[HIGHER NOTE PLAYS]
PYTHAGORAS DISCOVERED
THE OCTAVE HAD A RATIO OF 2 TO 1.
WITH SIMPLE FRACTIONS HE GOT THIS...
♪ ♪
♪ ♪
AND FROM THIS HARMONY IN NUMBERS
DEVELOPED THE MUSICAL SCALE OF TODAY.
[PLAYS SCALES]
[BOING]
BY GOLLY,
YOU DO FIND MATHEMATICS IN THE DARNEDEST PLACES.
YOU CAN IMAGINE HOW EXCITED PYTHAGORAS WAS
WHEN HE SHARED HIS FINDINGS WITH HIS PALS--
A FRATERNITY OF EGGHEADS KNOWN AS THE PYTHAGOREANS.
Spirit, whispering: THEY USED TO MEET IN SECRET
TO DISCUSS THEIR MATHEMATICAL DISCOVERIES.
THEY HAD A SECRET EMBLEM:
THE PENTAGRAM.
LET'S SEE WHAT THE TOPIC IS FOR TODAY.
[PLAYING SCALES]
[PLAYING SCALES]
[HITS SOUR NOTE]
[HITS SOUR NOTE]
[REPEATS SAME NOTE]
[PLAYING SONG]
WHAT'S GOING ON?
Shh! It's a jam session.
Donald: GIVE ME SOMETHING WITH A BEAT.
Shh!
[DONALD PLAYING BONGOS]
[SOUR NOTE]
[PLAYING FASTER]
[BONGOS CONTINUE]
[PLAYING UPBEAT TUNE]
SO FROM THESE EGGHEADS-- THE PYTHAGOREANS--
WITH THEIR MATHEMATICAL FORMULA,
CAME THE BASIS OF OUR MUSIC OF TODAY.
[PLAYING JAZZ]
[PLAYING CLASSICAL MUSIC]
[PLAYING SCALES]
[PLAYING SCALES]
PYTHAG, OLD BOY, PUT IT THERE.
WELL, I'LL BE A GOSH-DARN EGGHEAD.
[LAUGHS]
IT WAS OUR OLD FRIEND PYTHAGORAS WHO DISCOVERED
THAT THE PENTAGRAM WAS FULL OF MATHMAGIC.
THE 2 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
LD.
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.
Donald: BOY, OH, BOY, OH, BOY.
THIS IS MATHEMATICS?
I LIKE MATHEMATICAL FIGURES LIKE THAT.
UH-UH-UH, 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?
UNH! UHH!
THERE, I KNEW I COULD DO IT.
NOW THAT YOU'RE ALL PENT UP IN A PENTAGON,
LET'S SEE HOW NATURE USES THIS 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 THE GREEKS HAD GUESSED,
THE RULES ARE ALWAYS THE SAME.
WELL, DONALD,
DID YOU ENJOY YOUR GEOMETRICAL JOURNEY?
THERE'S A LOT MORE TO MATHEMATICS
THAN 2 TIMES 2.
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 2 MINDS.
IT'S A GAME THAT HAS BEEN ENJOYED FOR CENTURIES
BY KINGS AND COMMONERS.
IN FACT, LEWIS 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!
HELP, MR. SPIRIT!
HELP, HELP, HELP!
[HONK, DING]
UHH! UHH!
UHH! UHH!
UH-UH-UHH!
OHH! 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
L AREAS.
D
OH, BOY!
AND WITHOUT MATHEMATICS,
WE COULDN'T EVEN KEEP SCORE.
UNH!
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 2 PERFECT SQUARES
USING 3 PERFECT SPHERES
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 2 BALLS...
LIKE THIS.
NOW LET'S SEE HOW AN EXPERT AT 3-CUSHION BILLIARDS
USES HIS HEAD.
Donald: 3-CUSHION?
Spirit: YES.
THE CUE BALL NOT ONLY HAS TO HIT
THER BALLS,
BUT IT MUST CONTACT AT LEAST 3 CUSHIONS
BEFORE IT HITS THE FINAL BALL.
1, 2, 3.
1, 2, 3.
IT TAKES AN EXPERT TO MAKE SEVERAL SHOTS IN SUCCESSION.
1, 2, 3, 4...
5, 6.
WOW! THAT WAS A LUCKY SHOT.
LUCK? NO.
L.
FOR THIS GAME, YOU HAVE TO KNOW ALL THE ANGLES.
Donald: 1, 2, 3...
4, 5...
6, 7!
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
RAIL.
THESE TRICK SHOTS TAKE A LOT OF PRACTICE.
HA HA! HE MISSED IT! HA HA HA HA!
1, 2...
3.
Donald: WHAT'S SO MATHEMATICAL ABOUT THAT?
Spirit: 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.