Geometry Applications: Coordinate Geometry

Geometry Applications: Coordinate Geometry

[Music]

[Music]

Title: Geometry Applications: Coordinate Geometry

Title: Geometry Applications: Coordinate Geometry

Title: Lattitude and Longitude

Title: Lattitude and Longitude

IN 1884, THE CITY OF GREENWICH ENGLAND

BECAME THE LOCATION OF THE PRIME MERIDIAN.

BY AGREEMENT AMONG A GROUP OF COUNTRIES,

GREENWICH WAS GIVEN LONGITUDE 0 DEGREES.

EVEN TODAY'S GPS SATELLITES USE GREENWICH

AS THE PRIME MERIDIAN IN THEIR DETERMINATIONS

OF LONGITUDE AND LATITUDE.

THE DESIGNATION OF GREENWICH AS THE LONGITUDINAL CENTER

OF THE WORLD ENDED A CENTURIES LONG ENDEAVOR

TO DEVELOP AN ACCURATE SYSTEM OF NAVIGATING SHIPS.

INVENTIONS THAT WE TAKE FOR GRANTED TODAY

WERE CRUCIAL IN THE DEVELOPMENT OF AN ACCURATE

SYSTEM OF LONGITUDE AND LATITUDE AND YET LATITUDE

WAS ALWAYS AVAILABLE AS AN ACCURATE MEASURE.

WHY IS LATITUDE SO MUCH EASIER TO MEASURE?

LETS TAKE A LOOK.

LONGITUDE AND LATITUDE ARE USED

AS A COORDINATE SYSTEM ON A SPHERE.

THE SPHERE REPRESENTS THE EARTH.

AS THE EARTH ROTATES, ITS AXIS OF ROTATION

CREATES A NATURAL ALIGNMENT

WITH STARS COLLINEAR WITH THE AXIS.

THIS IS HOW THE NORTH STAR CAME TO BE USED

AND NAMED AS A MEANS OF MEASURING LATITUDE.

LATITUDE IS A SET OF CONCENTRIC CIRCLES

ALONG THE SPHERE THAT ARE PARALLEL TO EACH OTHER.

DEPENDING ON YOUR POSITION ON THE EARTH'S SURFACE,

THE ANGLE TO THE NORTH STAR CHANGES.

LATITUDE IS AN ANGLE MEASURE

AND LATITUDE IS MEASURED RELATIVE TO THE EQUATOR

WHICH CORRESPONDS TO 0 DEGREES.

THE MAXIMUM VALUE FOR LATITUDE AT EACH POLE

IS 90 DEGREES.

SUBDIVISIONS OF LATITUDE ARE MINUTES AND SECONDS.

60 MINUTES EQUALS ONE DEGREE

AND 60 SECONDS EQUALS ONE MINUTE.

YOU WILL SOMETIMES SEE LATITUDE MEASURES

EXPRESSED IN DEGREES, MINUTES AND SECONDS

DEPENDING ON HOW PRECISE THE LOCATION.

LATITUDE CAN BE MEASURED ON LAND AND ON THE SEA

USING A NUMBER OF MEASUREMENT DEVICES

COMMON TO THE TIME.

SO LATITUDE BECAME AN ACCURATE WAY

OF MEASURING THE LOCATION OF A SHIP

NORTH OR SOUTH OF THE EQUATOR.

BUT THE CONVENIENCE OF LATITUDE MEASUREMENTS

COMES AT THE EXPENSE OF LONGITUDE

OR EAST-WEST MEASUREMENTS.

BECAUSE THE ROTATION OF THE EARTH IS ALONG

THE EAST-WEST DIRECTION THERE IS NO FIXED POINT

COMPARABLE TO THE NORTH STAR.

YET SAILORS NEEDED TO KNOW BOTH LONGITUDE

AND LATITUDE TO NAVIGATE THE SEAS.

THERE ARE NO LANDMARKS ON THE OCEAN.

AND ALTHOUGH MEASURING LATITUDE WAS POSSIBLE,

WITHOUT A LONGITUDE MEASUREMENT

A SAILOR WOULD BE UNSURE WHETHER THE SHIP

WAS MOVING IN AN EAST-WEST DIRECTION.

SHIPS LOST AT SEA WERE A COMMON OCCURRENCE

BEFORE A WAY OF MEASURING LONGITUDE WAS FOUND.

THE SOLUTION CAME WITH THE DEVELOPMENT

OF ACCURATE CLOCKS.

HERE'S WHY.

THE LONGITUDE COORDINATE SYSTEM

IS MADE UP OF A SERIES OF GREAT CIRCLES

THAT INTERSECT THE NORTH AND SOUTH POLES.

THESE GREAT CIRCLES ARE KNOWN AS MERIDIANS.

BECAUSE THE ROTATION OF THE EARTH'S AXIS IS A

360 DEGREE TURN, THEN LONGITUDE IS MEASURED

FROM 0 TO 180 DEGREES TO THE EAST OR WEST

OF THE MERIDIAN THAT CORRESPONDS TO 0 DEGREES,

ALSO KNOWN AS THE PRIME MERIDIAN.

GREENWICH ENGLAND IS ALONG THE PRIME MERIDIAN.

AS WITH LATITUDE, LONGITUDE CAN ALSO BE

EXPRESSED IN DEGREES, MINUTES AND SECONDS.

THE EARTH ROTATES 360 DEGREES IN A 24-HOUR PERIOD.

IF YOU DIVIDE 360 BY 24

YOU GET 15 DEGREES PER HOUR.

EVERY 15 DEGREE CHANGE IN YOUR LONGITUDE

MEANS THAT YOU ARE ONE HOUR AHEAD

OR BEHIND YOUR PREVIOUS LOCATION.

THIS IS HOW A LONGITUDE MEASUREMENT WORKS:

LET'S SAY A SHIP IS LEAVING GREENWICH ENGLAND.

THE SHIP'S CLOCKS ARE SET TO GREENWICH MEAN TIME.

AT NOON GREENWICH MEAN TIME

THE SUN IS DIRECTLY OVERHEAD.

BUT AFTER SEVERAL DAYS OF SAILING

YOU FIND THAT THE SUN IS OVERHEAD TO YOUR LOCATION

WHEN IT IS 2:00 PM IN GREENWICH.

THIS MEANS THAT YOU ARE TWO HOURS BEHIND GREENWICH

AND THEREFORE 30 DEGREES WEST OF GREENWICH.

AS YOU CAN SEE, LONGITUDE IS REALLY A TIME MEASUREMENT

AND HAVING ACCURATE CLOCKS IS ESSENTIAL.

PRIOR TO THE DEVELOPMENT OF

ACCURATE MARINE CHRONOMETERS

TIME MEASUREMENTS RELIED ON

PENDULUM CLOCKS AND SUNDIALS

BUT THESE TOOLS DON'T WORK AS WELL ON A SHIP.

THE MOVEMENT OF THE WATER AND THE SHIP

INTRODUCES A LOT OF INACCURACY

TO THE TIME MEASUREMENTS

AND GIVEN THAT THE MARGIN OF ERROR WAS ONE HOUR

IT WOULD HAVE BEEN EASY TO BE OFF BY MANY DEGREES

AND MANY MILES.

THE DEVELOPMENT OF ACCURATE CLOCKS

ALLOWED FOR THE ACCURATE MEASUREMENT OF LONGITUDE

AND CREATED A REVOLUTION IN SAILING AND NAVIGATION.

THE BRITISH WERE INSTRUMENTAL

IN THE DEVELOPMENT OF THE TECHNOLOGY

OF THESE INSTRUMENTS.

IT SHOULD NOT BE SURPRISING THAT, AS A RESULT,

THE BRITISH NAVY RULED THE SEAS FOR CENTURIES.

LONGITUDE AND LATITUDE MAKE UP A THREE-DIMENSIONAL

COORDINATE SYSTEM THAT IS VERY DIFFERENT

FROM THE CARTESIAN COORDINATE SYSTEM.

THE CARTESIAN SYSTEM IS TWO-DIMENSIONAL

AND RECTANGULAR WHILE THE LONGITUDE-LATITUDE

SYSTEM IS SPHERICAL. AND YET IT IS NECESSARY

TO USE INFORMATION FROM ONE SYSTEM IN ANOTHER.

WHILE LONGITUDE AND LATITUDE COORDINATES

ARE EXTREMELY USEFUL FOR SAILING AND NAVIGATION,

FINDING THE DISTANCE BETWEEN TWO POINTS

IS MUCH MORE SUITABLE TO THE RECTANGULAR SYSTEM

WHERE YOU CAN USE THE DISTANCE FORMULA.

BUT HOW DO YOU TRANSLATE THREE-DIMENSIONAL DATA

INTO TWO-DIMENSIONAL COORDINATES?

ONE WAY IS TO USE A TWO-DIMENSIONAL

PROJECTION OF A THREE-DIMENSIONAL OBJECT.

A SIMPLE WAY TO THINK OF THIS IS TO PICTURE

A PERSON'S SHADOW ON A SUNNY DAY.

THE PERSON IS THREE-DIMENSIONAL

BUT THEIR SHADOW IS TWO-DIMENSIONAL.

WITH THIS IN MIND TAKE THE THREE-DIMENSIONAL SPHERE

AND PROJECT IT TO A TWO-DIMENSIONAL

RECTANGULAR GRID...

THE RESULT IS WHAT IS KNOWN AS A MERCATOR PROJECTION.

THERE IS ACTUALLY AN INTERMEDIATE STEP

WHERE THE SPHERE IS PROJECTED TO A CYLINDER

AND THE CYLINDER IS FLATTENED TO A RECTANGLE.

THE CYLINDER IS A MUCH EASIER THREE-DIMENSIONAL

FIGURE TO FLATTEN INTO A TWO-DIMENSIONAL FIGURE

THAN A SPHERE.

THE EASE OF VIEWING A TWO-DIMENSIONAL

WORLD MAP COMES AT A PRICE.

THE PARTS OF THE MAP CLOSER TO THE POLLS ARE DISTORTED

COMPARED TO THE PARTS OF THE MAP NEAR THE EQUATOR.

FOR EXAMPLE, NOTICE HOW GREENLAND

IS NEARLY THE SAME SIZE AS THE CONTINENT OF AFRICA

WHEN IN FACT AFRICA IS MUCH LARGER.

TO SEE AN EXAMPLE OF CONVERTING FROM

ONE COORDINATE SYSTEM TO ANOTHER

LET'S USE THE TI-NSPIRE.

TURN ON THE TI-NSPIRE.

CREATE A NEW DOCUMENT.

YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT.

CREATE A GRAPH WINDOW.

PRESS CONTROL G TO HIDE THE FUNCTION ENTRY LINE.

YOU'LL BE CONSTRUCTING A CIRCLE

AND LOOKING AT ITS COORDINATES.

PRESS MENU AND UNDER SHAPES SELECT CIRCLE.

MOVE THE POINTER ABOVE THE ORIGIN AND PRESS ENTER.

PRESS AND HOLD THE RIGHT ARROW

TO MOVE THE POINTER ALONG THE X AXIS.

STOP AT X EQUALS FIVE AND PRESS ENTER.

YOU HAVE CONSTRUCTED A CIRCLE OF RADIUS FIVE.

NOW PLACE A POINT ON THE CIRCLE IN QUADRANT ONE.

PRESS MENU AND UNDER POINTS AND LINES SELECT POINT.

MOVE THE POINTER TO A POINT ON THE CIRCLE.

YOU SHOULD SEE THE ONSCREEN MESSAGE "POINT".

PRESS ENTER.

TO SEE THE CARTESIAN COORDINATES OF THIS POINT

PRESS MENU AND UNDER ACTIONS

SELECT COORDINATES AND EQUATIONS.

MAKE SURE THE POINTER IS ABOVE THE POINT.

PRESS ENTER TWICE AND THEN PRESS ESCAPE.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

YOU NOW HAVE THE xy RECTANGULAR COORDINATES

FOR THIS POINT ON THE CIRCLE.

AS YOU MOVE THE POINT THE COORDINATES CHANGE

TO OTHER RECTANGULAR COORDINATES.

ANOTHER TYPE OF COORDINATES

THAT YOU COULD USE ARE POLAR COORDINATES.

WITH POLAR COORDINATES YOU ARE INTERESTED IN

THE DISTANCE FROM THE ORIGIN

AND THE ANGLE RELATIVE TO THE X AXIS.

A POLAR COORDINATE IS SIMILAR TO

A LATITUDE COORDINATE.

BEFORE MEASURING THE POLAR COORDINATES

CONSTRUCT A RADIUS FOR THE CIRCLE

THROUGH THE POINT YOU CREATED.

PRESS MENU AND UNDER POINTS AND LINES SELECT SEGMENT.

MOVE THE POINTER ABOVE THE ORIGIN AND PRESS ENTER.

NEXT, MOVE THE POINTER ABOVE THE POINT

ON THE CIRCLE AND PRESS ENTER AGAIN.

YOU SHOULD NOW HAVE A RADIUS.

WE CAN NOW CREATE A CORRESPONDING PAIR OF

POLAR COORDINATES FOR THE POINT ON THE CIRCLE.

USE THE MEASUREMENT TOOL.

PRESS MENU AND UNDER MEASUREMENT SELECT LENGTH.

MOVE THE POINTER ALONG THE RADIUS

YOU JUST CONSTRUCTED AND PRESS ENTER.

YOU WILL SEE A RADIUS MEASURE.

NOW MOVE THE POINTER TO THE LOWER RIGHTHAND

PART OF THE SCREEN AND PRESS ENTER AGAIN

TO PLACE THE RADIUS MEASUREMENT ONSCREEN.

NEXT, MEASURE THE ANGLE THE RADIUS MAKES

WITH THE X AXIS.

MAKE SURE THAT THE ANGLE MEASURE DISPLAYED

IS IN ANGLES NOT RADIANS.

GO TO THE DOCUMENTS SETTINGS WINDOW

AND CHANGE FROM RADIANS TO ANGLES

UNDER THE GRAPHING ANGLE OPTION.

NOW MEASURE THE ANGLE.

PRESS MENU AND UNDER MEASUREMENT SELECT ANGLE.

MOVE THE POINTER ABOVE THE POINT ON THE CIRCLE

AND PRESS ENTER.

MOVE THE POINTER TO THE ORIGIN

AND PRESS ENTER AGAIN.

FINALLY, MOVE THE POINTER ABOVE POINT 5,0 ON THE X AXIS

AND PRESS ENTER ONCE MORE.

YOU SHOULD NOW SEE AN ANGLE MEASURE APPEAR.

PRESS ESCAPE.

MOVE THE POINTER ABOVE THE ANGLE MEASURE.

PRESS AND HOLD THE CLICK BUTTON

TO SELECT THE ANGLE MEASURE.

MOVE IT SO THAT IT IS TO THE RIGHT OF THE RADIUS MEASURE

SO THAT YOU GET WHAT LOOKS LIKE TWO COORDINATES

WHICH ARE THE POLAR COORDINATES.

YOU CAN ALSO MOVE THE xy COORDINATES

ABOVE THE POLAR COORDINATES

TO MAKE FOR AN EASY COMPARISON OF THE VALUES.

AS YOU MOVE THE POINT ON THE CIRCLE

YOU CAN SEE HOW BOTH SETS OF COORDINATES UPDATE.

DIFFERENT SITUATIONS MAKE EACH SET OF COORDINATES

THE BETTER CHOICE.

IF YOU NEED TO KNOW THE PRECISE LOCATION

OF THE POINT ALONG THE x AND y AXES THEN THE

RECTANGULAR COORDINATES ARE THE ONES TO USE.

IF, AS IN THE CASE OF A RADAR SCREEN,

YOU NEED TO TRACK THE DISTANCE AND ANGLE,

THEN POLAR COORDINATES ARE THE BETTER CHOICE.

AS YOU SAW WITH THE MERCATOR PROJECTION

THERE IS A RELATIONSHIP BETWEEN

THE TWO SETS OF COORDINATES.

IN THE CASE OF THE POLAR COORDINATES

TO CHANGE FROM CARTESIAN TO POLAR COORDINATES

WE USE THESE EQUATIONS.

ON SEPTEMBER 4, 1622 THE SPANISH GALLEON

NUESTRA SENORA DE ATOCHA SET SAIL FROM HAVANA, CUBA

BOUND FOR SPAIN.

IT WAS LOADED WITH TONS OF GOLD, SILVER, EMERALDS

AND ENOUGH TREASURE THAT WOULD ALLOW THE

SPANISH EMPIRE TO FUND ITS WARS IN EUROPE.

BUT THE ATOCHA HAD BEEN DELAYED IN ITS DEPARTURE

BY SIX WEEKS.

THIS PLACED THE DEPARTURE RIGHT IN THE MIDST OF THE

ANNUAL HURRICANE SEASON THAT TORMENTS THIS REGION.

YET ANY FURTHER DELAYS WERE OUT OF THE QUESTION.

SO THE ATOCHA DEPARTED

ACCOMPANIED BY 27 OTHER SHIPS.

AS THE SPANISH FEARED, THE FLOTILLA OF SHIPS

ENCOUNTERED A HURRICANE TWO DAYS INTO THE VOYAGE.

THE ATOCHA WAS PUSHED TOWARD THE REEFS

ALONG WHAT IS NOW THE FLORIDA KEYS.

SUSTAINING DAMAGE TO ITS HULL, COMBINED WITH THE

HEAVY WEIGHT OF ITS CARGO, THE ATOCHA SANK

OFF THE COAST OF THE DRY TORTUGAS ISLANDS.

ALL BUT A HANDFUL OF PEOPLE SURVIVED.

THE SPANISH SENT SHIPS TO TRY TO SALVAGE THE TREASURE

BUT THE TECHNOLOGY OF THE TIME MADE THIS DIFFICULT

AND THE SPANISH HAD LIMITED SUCCESS.

SO THE SPANISH GALLEON ATOCHA JOINED OTHER SHIPS

THAT OVER THE CENTURIES SANK ALONG THE WATERS

OFF THE FLORIDA KEYS.

IN 1969 A DETERMINED TREASURE HUNTER

BEGAN A QUEST THAT WOULD END 15 YEARS LATER

WITH THE RECOVERY OF MOST OF THE REMAINS OF THE SHIP.

DURING THAT TIME, BITS AND PIECES OF THE TREASURE

WERE FOUND OVER A LARGE AREA OF THE SEA.

WHEN A WOODEN SHIP SINKS, THE WATER CAUSES THE WOOD

TO, OVER TIME, DECOMPOSE AND BREAK APART.

THE CONTINUAL MOVEMENT OF THE SEA

CAUSES THE REMAINS OF THE SHIP TO DISPERSE.

THE CONTINUAL DEPOSITS OF SAND AND SILT

WILL BURY THE TREASURE, LITERALLY ADDING

AN ADDITIONAL LAYER OF DIFFICULTY TO FINDING IT.

TREASURE HUNTERS RELY ON TECHNOLOGICAL TOOLS:

METAL DETECTORS, SUCTION DEVICES FOR REMOVING

LAYERS OF SAND, SONAR DEVICES AND GPS UNITS.

OF THE VARIOUS DEVICES IT IS THE GPS THAT IS BY FAR

THE MOST CRUCIAL PART OF THE HUNT.

AND GPS RELIES ON A COORDINATE SYSTEM.

AS YOU HAVE SEEN, LONGITUDE AND LATITUDE

ARE A COORDINATE SYSTEM IDEAL FOR NAVIGATION.

IN THE AREA OF TREASURE HUNTING,

THE DECIMAL VERSIONS OF LONGITUDE AND LATITUDE

IN COMBINATION WITH A CARTESIAN SYSTEM ARE USED.

THE LOCATION OF THE WRECK OF THE ATOCHA IS SHOWN HERE.

24 DEGREES, 32.673 MINUTES NORTH,

82 DEGREES, 20.806 MINUTES WEST.

RECALL THAT NORTH REFERS TO NORTH OF THE EQUATOR

AND WEST REFERS TO WEST OF THE PRIME MERIDIAN

IN GREENWICH, ENGLAND.

TRY THIS.

IF YOU HAVE ACCESS TO A COMPUTER WITH AN INTERNET

CONNECTION, GO TO GOOGLE MAPS OR ANOTHER WEBSITE

THAT WILL ACCEPT LONGITUDE AND LATITUDE COORDINATES.

INPUT THIS LONGITUDE AND LATITUDE AT THE ENTRY LINE

AND PRESS ENTER.

SELECT THE SATELLITE VIEW.

YOU CAN SEE WHERE THE SHIP'S DEBRIS WAS FIRST FOUND.

MAKE A NOTE OF THE MAP SCALE.

EVEN A SMALL REGION AROUND THE SITE

IS IN THE NEIGHBORHOOD OF 20 MILES

AND A 20 MILE BY 20 MILE AREA IS 400 SQUARE MILES.

YOU CAN BEGIN TO SEE WHY IT TOOK SO LONG

TO FIND THE WRECKAGE.

LET'S USE THE TI-NSPIRE TO CREATE A WAY

TO CONVERT FROM LONGITUDE AND LATITUDE

COORDINATES TO x,y COORDINATES.

TURN ON THE TI-NSPIRE.

CREATE A NEW DOCUMENT.

YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT.

CREATE A SPREADSHEET WINDOW.

MOVE THE CURSOR TO THE VERY TOP OF COLUMN A.

INPUT THE LABEL "DEGREES" AND PRESS TAB

TO GO TO THE TOP OF THE NEXT COLUMN.

INPUT THE LABEL "MINUTES/SECONDS."

PRESS ENTER.

MOVE THE CURSOR TO CELL A1.

START WITH WHAT WILL BE THE x COORDINATE.

LONGITUDE IS THE HORIZONTAL MEASUREMENT SO USE THAT.

SINCE THE LOCATION OF THE ATOCHA IS WEST OF

THE PRIME MERIDIAN, USE A NEGATIVE VALUE.

INPUT -82 IN CELL A1

AND PRESS THE TAB KEY TO GO TO CELL B1.

IN THIS COLUMN INPUT THE MINUTES AND SECONDS

OF LONGITUDE.

INPUT -20.806 AND PRESS ENTER.

MOVE THE CURSOR TO CELL A2.

HERE YOU WILL INPUT THE DEGREES OF LATITUDE.

SINCE THE LOCATION OF THE ATOCHA

IS NORTH OF THE EQUATOR USE A POSITIVE VALUE.

INPUT 24 AND PRESS THE TAB KEY TO GO TO CELL B2.

INPUT 32.673 AND PRESS ENTER.

NOW GO TO THE COLUMN HEADING

AT THE VERY TOP OF COLUMN C.

INPUT THE LABEL HEADING "DECIMAL" AND PRESS ENTER.

THE CURSOR IS NOW AT THE FORMULA LINE.

ANY FORMULAS YOU INPUT WILL AFFECT

ALL THE ENTRIES IN COLUMN C.

LET'S INPUT A FORMULA THAT TAKES THE DATA

FROM COLUMNS A AND B

AND CONVERTS IT INTO A DECIMAL COORDINATE.

SO AT THE FORMULA LINE, INPUT THIS EXPRESSION:

A+B DIVIDED BY 60.

PRESS ENTER.

YOU WILL NOW SEE THE CORRESPONDING LONGITUDE

AND LATITUDE COORDINATES WRITTEN AS DECIMALS.

LET'S GRAPH THIS COORDINATE AND HAVE IT

REPRESENT WHERE THE ATOCHA ORIGINALLY SANK.

WE WILL THEN GRAPH SOME ADDITIONAL POINTS

THAT REPRESENT LOCATIONS WHERE

SOME OF THE SHIP'S TREASURE WAS FOUND.

PRESS THE HOME KEY TO CREATE A NEW WORKSHEET.

SELECT A GRAPH WINDOW.

BY DEFAULT THERE IS A FUNCTION ENTRY LINE

WHICH, FOR THIS ACTIVITY, YOU WON'T NEED.

SO PRESS CONTROL AND G TO HIDE THE ENTRY LINE.

NEXT CHANGE THE WINDOW SETTINGS TO ONLY

FOCUS ON THE AREA NEAR WHERE THE ATOCHA SANK.

PRESS MENU AND UNDER WINDOW/ZOOM

CHANGE THE SETTINGS TO THIS:

PRESS THE TAB KEY TO MOVE FROM ONE DATA FIELD

TO ANOTHER.

WHEN YOU HAVE CHANGED THE WINDOW SETTINGS

TAB YOUR WAY TO THE "OK" BUTTON.

PRESS ENTER.

NOW PLACE A POINT IN THE CENTER OF THE SCREEN.

PRESS MENU AND UNDER POINTS AND LINES SELECT POINT.

MOVE THE POINTER TO THE MIDDLE OF THE SCREEN

AND PRESS ENTER.

NOW DISPLAY THE COORDINATES.

MOVE THE POINTER ABOVE THE POINT

AND PRESS CONTROL AND MENU.

SELECT THE COORDINATES AND EQUATIONS OPTION.

YOU SHOULD NOW SEE THE COORDINATES OF THE POINT.

IF YOU NEED TO MOVE THE COORDINATES

MOVE THE POINTER ABOVE THE COORDINATES

AND PRESS AND HOLD THE CLICK KEY

TO SELECT THE COORDINATES.

USE THE NAVIGATION ARROWS TO MOVE THE COORDINATES

TO A NEW LOCATION AND THEN PRESS ENTER.

NOW CHANGE THE COORDINATES

TO THE ONES IN THE SPREADSHEET.

MOVE THE POINTER ABOVE THE TEXT OF

THE X COORDINATE AND PRESS ENTER.

THE TEXT FIELD IS NOW EDITABLE.

PRESS THE CLEAR KEY TO REMOVE THE CURRENT

X COORDINATE AND REPLACE IT WITH THE DECIMAL

LONGITUDE VALUE FROM THE SPREADSHEET.

YOU CAN TOGGLE BACK TO THE SPREADSHEET

BY PRESSING CONTROL AND THE LEFT ARROW,

COPYING THE CONTENTS OF CELL C1

AND PASTING IT INTO THE COORDINATE FIELD.

THE CONTROL LEFT OR RIGHT ARROW

TAKES YOU IN AND OUT OF THE DIFFERENT WINDOWS.

REPEAT WITH THE y COORDINATE.

MOVE THE POINTER ABOVE THE y COORDINATE

AND PRESS ENTER.

THEN INPUT OR PASTE IN THE DECIMAL LATITUDE VALUE.

YOUR SCREEN SHOULD LOOK LIKE THIS.

SINCE WE WILL BE GRAPHING OTHER COORDINATES

MAKE THIS POINT DISTINCTIVE

BY CHANGING ITS ATTRIBUTES.

MOVE THE POINTER ABOVE THE POINT

AND PRESS CONTROL AND MENU.

SELECT THE ATTRIBUTES OPTION.

USE THE LEFT OR RIGHT ARROWS

TO CHANGE THE APPEARANCE OF THE POINT.

THE X ICON MAY WORK BEST.

PRESS ENTER AFTER YOU HAVE SELECTED THE NEW ICON.

THE POINT ON SCREEN REPRESENTS WHERE

THE ATOCHA SANK BUT THE TREASURE WAS LOCATED

IN DOZENS OF LOCATIONS AROUND THIS POINT.

THIS TABLE SHOWS A HANDFUL OF POINTS

WHERE TREASURE WAS FOUND.

THE DATA POINTS ARE EXPRESSED IN LONGITUDE

AND LATITUDE COORDINATES.

USE THE SPREADSHEET TO CONVERT THESE

TO DECIMAL COORDINATES.

THEN GRAPH THESE POINTS.

FIRST PLACE POINTS ON THE GRAPH WINDOW.

DISPLAY THE COORDINATES AND THEN CHANGE THE VALUES

OF THE COORDINATES TO THE ONES FROM THE SPREADSHEET.

PAUSE THE VIDEO TO INPUT, CONVERT

AND GRAPH THE DATA POINTS.

WHEN YOU'RE DONE, TRY TO GET YOUR SCREEN

TO LOOK LIKE THIS.

YOU'LL SEE THAT THE TREASURE IS DISTRIBUTED

OVER A REGION AROUND THE POINT WHERE THE SHIP SANK.

AS EXPLAINED EARLIER THE DISPERSION OF THE WRECKAGE

HAS TO DO WITH THE MOVEMENT OF WATER

AND THE DECOMPOSITION OF THE WOOD ON THE SHIP.

HOW WIDE A REGION OF DISPERSAL IS THIS?

WE CAN USE THE SEGMENT TOOL TO CREATE A LINE

SEGMENT BETWEEN TWO POINTS AND THEN USE THE

MEASUREMENT TOOL TO MEASURE THE DISTANCE.

PRESS MENU AND UNDER POINTS AND LINES SELECT SEGMENT.

MOVE THE POINTER TO THE SITE OF THE

SINKING OF THE ATOCHA.

PRESS ENTER.

THEN MOVE THE POINTER TO ONE OF THE POINTS

WHERE WRECKAGE WAS FOUND.

PRESS ENTER AGAIN.

NOW MEASURE THIS SEGMENT.

PRESS MENU AND UNDER MEASUREMENT SELECT LENGTH.

MOVE THE POINTER ABOVE THE SEGMENT YOU JUST CREATED.

PRESS ENTER ONCE TO RECORD THE MEASUREMENT.

MOVE THE POINTER TO THE SIDE OF THE SEGMENT

AND PRESS ENTER AGAIN

TO PLACE THE MEASUREMENT ON SCREEN.

YOUR MEASUREMENT SHOULD BE AROUND 1/10 OF A DEGREE.

ASSUME THAT THIS REPRESENTS ABOUT

HALF OF THE FULL DISPERSAL OF THE WRECKAGE.

SO THE RADIUS OF A CIRCLE THAT WOULD ENCOMPASS THE

FULL AREA WOULD HAVE A RADIUS OF 2/10 OF A DEGREE.

HOW LONG A DISTANCE IS THIS?

NEAR THE EQUATOR WHICH APPLIES TO THE FLORIDA KEYS

ONE DEGREE OF LATITUDE AND LONGITUDE

CORRESPONDS TO 60 MILES.

SO 2/10 OF A DEGREE CORRESPONDS TO 12 MILES.

A CIRCLE OF RADIUS 12 MILES HAS AN AREA

OF ROUGHLY 452 SQUARE MILES.

SO YOU CAN SEE THAT EVEN THOUGH THE LOCATION

OF A WRECKAGE MAY BE KNOWN,

THE FORCES OF DISPERSAL CAN MAKE FINDING

THE WRECKAGE MORE AND MORE DIFFICULT

AS MORE AND MORE TIME GOES ON.

THE GUGGENHEIM MUSEUM IN NEW YORK CITY

IS FRANK LLOYD WRIGHT'S MASTERPIECE.

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.