Geometry Applications: Polar Coordinates

Geometry Applications: Polar Coordinates

[Music]

[Music]

THE GUGGENHEIM MUSEUM IN NEW YORK CITY

IS FRANK LLOYD WRIGHT'S MASTERPCE.

WRIGHT WAS ONE OF THE GREATEST ARCHITECTS

OF THE 20TH CENTURY AND THE GUGGENHEIM WAS THE

CULMINATION OF A LIFETIME OF INNOVATIVE WORK.

DESIGNED OVER A 15 YEAR PERIOD STARTING IN 1943

AND FINALLY COMPLETED IN 1959,

SIX MONTHS AFTER THE DEATH OF THE ARCHITECT,

THE GUGGENHEIM SPEAKS TO THE ARTISTIC ACHIEVEMENT

IN THE LANGUAGE OF MATHEMATICS.

THE OUTER STRUCTURE OF THE GUGGENHEIM

HAS AN ALMOST CYLINDRICAL SHAPE,

BUT INSIDE IT IS A CONTINUOUS SPIRAL GALLERY

THAT EXTENDS FROM GROUND LEVEL ACROSS FIVE LEVELS.

THE SHAPE IS SIMILAR TO THAT OF A SPIRAL STAIRCASE.

BUT ONE THAT IS WIDER AT THE TOP.

WHEN SEEN FROM ABOVE, THIS SPIRAL HAS THE

SAME SHAPE AS A NAUTILUS SHELL.

IN FACT THIS SPIRAL SHAPE OCCURS OFTEN IN NATURE.

WHY DID FRANK LLOYD WRIGHT CHOOSE THIS SHAPE

FOR THE EXHIBITION GALLERY OF THE GUGGENHEIM?

FIRST THE SPIRAL DOES HAVE A PLEASING DESIGN

AND THE FACT THAT IT OCCURS NATURALLY

IS CONSISTENT WITH WRIGHT'S DESIRE TO HAVE

HIS ARCHITECTURE CONNECT TO NATURE AND BE ORGANIC.

BUT THE SHAPE THAT WRIGHT CHOSE

HAS A SPECIFIC MATHEMATICAL MEANING.

THIS SPIRAL IS KNOWN AS A LOGARITHMIC SPIRAL AND

CAN BE EASILY GRAPHED ON A POLAR COORDINATE SYSTEM.

POLAR COORDINATES INVOLVE TWO VALUES

USUALLY LABELED r AND THETA.

THESE COORDINATES ARE MEASURED

RELATIVE TO THE ORIGIN.

THE r VALUE IS THE DISTANCE TO THE ORIGIN AND CAN BE

CONSIDERED THE RADIUS OF AN IMAGINARY CIRCLE.

SO AN R VALUE CAN RANGE OVER THE ENTIRE

SWEEP OF THE CIRCLE.

THE THETA VALUE IS THE ANGLE THAT r MAKES

RELATIVE TO THE x AXIS.

HERE ARE SOME SAMPLE POLAR COORDINATES.

SIMILAR TO THE xy COORDINATES SYSTEM

YOU CAN GRAPH EQUATIONS WITH VARIABLES.

IN THE xy SYSTEM RECALL THAT THE

SIMPLEST FUNCTION GRAPH IS THAT OF y=x.

THIS IS A LINEAR GRAPH FOR INCREASING VALUES OF x.

THE EQUIVALENT GRAPH IN THE POLAR

COORDINATE SYSTEM IS r EQUALS THETA.

LET'S EXPLORE THIS GRAPH ON THE NSPIRE

AND SEE HOW ITS PROPERTIES ARE REFLECTED

IN THE ARCHITECTURE OF THE GUGGENHEIM MUSEUM.

TURN ON THE TI-NSPIRE.

CREATE A NEW DOCUMENT.

YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT.

CREATE A GRAPH WINDOW.

BY DEFAULT THE GRAPH WINDOW IS SET UP FOR AN

xy CARTESIAN COORDINATE GRAPH.

CHANGE THE GRAPH TYPE TO POLAR.

PRESS MENU AND UNDER GRAPH TYPE SELECT POLAR.

NOTICE THAT THE EQUIVALENT OF THE FUNCTION ENTRY LINE

IS DIFFERENT FOR THE POLAR GRAPH.

IN THIS ENTRY LINE, r IS A FUNCTION OF THETA

AND BY DEFAULT, THETA GOES FROM 0 TO 2Pi.

YOU WANT TO GRAPH r EQUALS THETA.

TO INPUT THE SYMBOL FOR THETA

PRESS THE LIBRARY BUTTON WHICH IS THE BUTTON

THAT LOOKS LIKE AN OPEN BOOK.

NEXT, PRESS 3 TO GO TO THE SYMBOLS TAB.

USE THE NAVIGATION ARROWS TO SELECT THETA.

PRESS ENTER.

THIS TAKES YOU BACK TO THE ENTRY LINE.

PRESS ENTER AGAIN.

THIS IS THE GRAPH OF A LOGARITHMIC SPIRAL.

IT DOESN'T QUITE MATCH THE SPIRAL FROM THE GUGGENHEIM

OR FOR THAT MATTER, THE KIND OF SWIRL YOU SEE

IN A NAUTILUS SHELL.

THIS IS BECAUSE THE VALUES OF THETA

GO FROM ZERO TO 2Pi.

IF WE INCREASE THE RANGE OF THETA VALUES

THIS SHOULD INCREASE THE NUMBER OF SWIRLS.

SO PRESS CONTROL AND G

TO BRING BACK THE POLAR EQUATION ENTRY LINE.

BY DEFAULT IT IS ON EQUATION r2.

SO PRESS THE UP ARROW TO GO TO THE FIRST EQUATION.

USE THE NAVIGATION ARROWS TO GO TO THE

UPPER RANGE OF THETA VALUES.

PRESS THE CLEAR KEY TO DELETE THE 6.28

WHICH IS WHAT CORRESPONDS TO 2Pi.

INPUT 4Pi.

PRESS 4 FOLLOWED BY THE LIBRARY KEY

WHICH SHOULD ALREADY BE IN THE SYMBOLS TAB.

IF NOT, PRESS 3.

USE THE NAVIGATION ARROWS TO SELECT THE SYMBOL Pi.

PRESS ENTER TO GO BACK TO THE POLAR EQUATION

ENTRY LINE.

PRESS ENTER AGAIN TO RE-GRAPH THE EQUATION.

NOTICE HOW THE SPIRAL IS LARGER

AND MAY EXTEND OUTSIDE THE SCREEN.

IF SO, PRESS MENU AND UNDER ZOOM PRESS ZOOM OUT.

MOVE THE POINTER TO THE CENTER OF THE SCREEN

AND PRESS ENTER ENOUGH TIMES

TO SHOW ALL THE GRAPH ONSCREEN.

THEN PRESS ESCAPE.

WHEN THETA RANGED FROM ZERO TO 2Pi

THE SPIRAL INTERSECTED THE x AXIS 3 TIMES.

WITH THE CHANGE TO 4Pi

THE SPIRAL INTERSECTS THE x AXIS 5 TIMES.

SO CLEARLY EVERY TWO Pi INCREMENT ADDS

TWO MORE INTERSECTIONS TO THE x AXIS.

NOW YOU CAN SEE HOW THE LOGARITHMIC SPIRAL

IS SIMILAR TO THE ONE FROM THE GUGGENHEIM.

IN FACT, IF YOU WANT TO MODEL THE SPIRALING GALLERY

CREATE A NEW GRAPH OF r EQUALS 0.8 THETA.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

SO YOU CAN SEE HOW A POLAR GRAPH OF r EQUALS THETA

IS ONE OF THE SIMPLEST POLAR GRAPHS.

BUT THE LOGARITHMIC SPIRAL HAS ITS ORIGINS

IN THE xy CARTESIAN WORLD.

THE SEQUENCE OF NUMBERS KNOWN AS

THE FIBONACCI SEQUENCE

IS AT THE HEART OF THE LOGARITHMIC SPIRAL.

THE FIBONACCI SEQUENCE IS SIMPLE TO GENERATE.

HERE ARE THE FIRST FEW TERMS IN THE SEQUENCE:

EACH TERM IN THE SEQUENCE IS THE SUM OF

THE PREVIOUS TWO TERMS.

THE SEQUENCE IS NAMED AFTER A RENAISSANCE

MATHEMATICIAN WHO WAS STUDYING THE POPULATION

GROWTH PATTERN OF PAIRS OF RABBITS.

START WITH ONE PAIR OF RABBITS.

LET'S LABEL THIS TERM IN OUR SEQUENCE f0.

SO WE GET f0=1.

AFTER ONE MONTH THIS PAIR OF RABBITS

GIVES BIRTH TO ANOTHER PAIR OF RABBITS.

SO THERE IS ONE PAIR OF NEW RABBITS

AND WE CAN WRITE THIS AS f1 EQUALS 1.

AFTER ANOTHER MONTH THE ORIGINAL PAIR OF RABBITS,

f0, HAS ANOTHER PAIR OF RABBITS

AND THE SECOND PAIR OF RABBITS, F1,

ALSO HAS ANOTHER PAIR OF RABBITS

WHICH GIVES US THIS TERM:

USING THIS SAME PATTERN WE CAN GENERATE

THE OTHER NUMERICAL TERMS IN THE FIBONACCI SEQUENCE.

BUT MOST IMPORTANT, WE CAN GENERATE A TERM

THAT APPLIES TO THE ENTIRE SEQUENCE:

THIS KIND OF SEQUENCE IS KNOWN AS

A RECURSIVE SEQUENCE.

THE RESULTS OF PREVIOUS TERMS

ARE FED BACK INTO THE NEXT TERM

AND YOU CAN SEE HOW THIS RECURSIVE PATTERN

ACCOUNTS FOR THE POPULATION GROWTH

IN THE ORIGINAL SITUATION WITH THE RABBITS.

BUT HOW DOES THIS PATTERN RELATE TO THE

LOGARITHMIC SPIRAL?

SUPPOSE WE START WITH A ONE BY ONE SQUARE

REPRESENTING f0.

WE THEN ADD SUBSEQUENT SQUARES

WHOSE DIMENSIONS MODEL THE FIBONACCI SEQUENCE.

IF YOU ARRANGE THESE SQUARES

IN A COUNTER CLOCKWISE MANNER

THEN YOU HAVE THE SHELL FOR CREATING

THE LOGARITHMIC SPIRAL AS SHOWN HERE.

THE FIBONACCI SEQUENCE IS A MODEL OF

SIMPLE RECURSIVE GROWTH...

THE KIND THAT OCCURS IN NATURE.

THE SIMPLEST WAY FOR AN OBJECT TO GROW IN SIZE

IS OUTWARD.

THUS, NEW GROWTH IS BUILT UPON WHAT CAME BEFORE

AND IT SPIRALS OUTWARD

IN A RECURSIVE NUMERICAL SEQUENCE

AND GRAPHICALLY IN THE FORM OF A LOGARITHMIC SPIRAL.

AND THERE'S MORE...

LOOK AT THE TERMS OF THE FIBONACCI SEQUENCE:

AND LOOK AT THIS RATIO:

IN OTHER WORDS, WHAT IS THE RATIO OF ONE TERM

IN THE SEQUENCE AND THE PREVIOUS TERM?

FOR THE FIBONACCI SEQUENCE

THE RATIO CHANGES FROM TERM TO TERM.

THIS IS DIFFERENT FROM WHAT'S CALLED

A GEOMETRIC SEQUENCE WHERE THE RATIO

OF ONE TERM AND THE NEXT IS CONSTANT.

FOR EXAMPLE, THIS IS A GEOMETRIC SEQUENCE:

IF YOU DIVIDE ANY PAIR OF CONSECUTIVE TERMS

THE RATIO IS THREE.

A GEOMETRIC SEQUENCE HAS A COMMON RATIO.

BUT WITH THE FIBONACCI SEQUENCE

THE RATIO VARIES BUT YOU'LL SEE THAT THE RATIO

APPROACHES A PARTICULAR NUMBER AS n INCREASES.

AS n APPROACHES INFINITY THE FIBONACCI RATIO

APPROACHES THE GOLDEN RATIO.

SYMBOLIZED BY THE GREEK LETTER PHI,

THE GOLDEN RATIO IS FOUND THROUGHOUT

MANY WORKS OF ART.

FROM THE RATIO OF THE SIDES OF THE PARTHENON

TO THE DIMENSIONS IN RENAISSANCE ART.

YOU CAN THINK OF THE GOLDEN RATIO

AS HUMANITY'S VERSION OF THE FIBONACCI SEQUENCE.

IT IS A PLEASING RATIO AND THE WAY

THAT ARTISTS THROUGHOUT THE AGES

HAVE HONORED THE ARTS AND OTHER ARTISTS.

AND THIS BRINGS US BACK TO THE GUGGENHEIM MUSEUM

AND FRANK LLOYD WRIGHT'S ARTISTIC ACHIEVEMENT.

A MUSEUM IS WHERE ARTISTIC ACHIEVEMENTS OF THE PAST

ARE COLLECTED AND CAN BE VIEWED AND ADMIRED.

THE COLLECTIVE WISDOM AND TALENT OF AGES PAST

ARE HOUSED IN THE MUSEUM.

AND WHAT BETTER WAY OF ENCAPSULATING THE NOTION

OF ARTISTIC GROWTH AND ACHIEVEMENT

THAN IN A MUSEUM THAT ITSELF

IS THE EMBODIMENT OF THAT GROWTH.

FOR NOT ONLY IS THE LOGARITHMIC SPIRAL

AT THE HEART OF THE GUGGENHEIM

A SYMBOL OF ORGANIC GROWTH

AS IT SPIRALS OUTWARD IN THE MANNER OF SHELLS,

FLOWERS AND GALAXIES,

IT POINTS TO THAT ARTISTIC RATIO

SOMETIMES REFERRED TO AS THE DIVINE RATIO.

THE OUTWARD, UPWARD SPIRAL OF THE GUGGENHEIM

IS FRANK LLOYD WRIGHT'S FINAL STATEMENT

ON HIS ARTISTIC ACHIEVEMENTS.

IT IS BOTH HEROIC AND HUMBLE AT THE SAME TIME.