Geometry Applications: Angles and Planes

[Music]

[Music]

Title: Geometry Applications: Angles and Planes

Title: Geometry Applications: Angles and Planes

Title: Geometry Basics

Title: Geometry Basics

THE ARECIBO OBSERVATORY IN PUERTO RICO

IS ONE OF THE LARGEST RADIO TELESCOPES ON EARTH

AND HAS BEEN USED TO MAKE SOME IMPORTANT DISCOVERIES

IN OUR SOLAR SYSTEM.

THE SOLAR SYSTEM IS MADE UP OF THE SUN,

THE PLANETS AND THEIR MOONS

AND OTHER CELESTIAL OBJECTS IN THE VICINITY OF THE SUN.

AS THE EARTH ROTATES AROUND THE SUN

ITS MOTION THROUGH SPACE IS ALONG A PLANE

THAT INTERSECTS THE SUN.

THIS PLANE IS CALLED THE ECLIPTIC.

THE EARTH SPINS ABOUT ITS OWN AXES

AND THE AXIS OF ROTATION IS AT A 23 DEGREE ANGLE

RELATIVE TO THE ECLIPTIC.

ACCORDING TO KEPLER'S LAWS, THE EARTH'S ORBIT IS AN

ELLIPSE WITH THE SUN AT ONE OF THE FOCI OF THE ELLIPSE.

SUPPOSE IT TAKES TIME (T1) FOR THE EARTH TO GO

FROM THIS POINT IN ITS ORBIT TO THIS POINT.

LET'S DEFINE AREA 1 AS THE PORTION OF THE ECLIPTIC

THAT INCLUDES PART OF THE ORBIT

AND THE CORRESPONDING AREA OF THE ELLIPSE.

LET'S DO THE SAME FOR TIME (T2) AND AREA 2

AT A DIFFERENT PART OF THE ORBIT.

ACCORDING TO KEPLER, IF T1 AND T2 ARE EQUAL

TO EACH OTHER, THEN SO ARE AREA 1 AND AREA 2.

AS YOU CAN SEE, MANY REAL-WORLD PROBLEMS

INVOLVE A KNOWLEDGE OF BASIC GEOMETRIC FIGURES.

IN THIS CASE ANGLES AND PLANES.

NOT ONLY THAT, IT'S IMPORTANT TO KNOW THE

PROPERTIES OF ANGLES AND PLANES IN ORDER TO USE YOUR

KNOWLEDGE OF GEOMETRY TO SOLVE REAL WORLD PROBLEMS.

IN THIS PROGRAM YOU WILL EXPLORE

THE FOLLOWING CONCEPTS:

HIMEJI CASTLE HOLDS A SPECIAL PLACE

IN JAPAN'S HISTORY.

BUILT IN THE MIDDLE AGES, IT HAS SURVIVED

CIVIL WARS AND WORLD WARS.

NOT BAD FOR A WOODEN STRUCTURE.

BUT IT IS THE TIMELESS GEOMETRY OF THIS CASTLE

THAT MAKES IT MOST IMPRESSIVE.

IT IS A CASE STUDY IN THE CREATIVE USE OF ANGLES.

HIMEJI CASTLE SERVED AS A FORTRESS

WHERE SAMURAI ARMIES COULD DEFEND THEMSELVES

FROM AN ATTACKING ARMY.

SO HIMEJI CASTLE WAS BUILT TO MAKE IT DIFFICULT

FOR AN ARMY TO INVADE AND EASIER FOR THOSE

INSIDE THE CASTLE TO DEFEND THEMSELVES.

LET'S LOOK AT HOW ANGLES PLAY A KEY ROLE

IN HIMEJI CASTLE'S DEFENSES.

LET'S FIRST START WITH ITS ALTITUDE.

LIKE MANY CASTLES AND FORTRESSES, HIMEJI IS A

TALL STRUCTURE AND MADE ALL THE MORE TALLER

BECAUSE IT IS SITUATED ON A HILL.

THE TOTAL HEIGHT OF HIMEJI CASTLE IS 92 METERS

OR ABOUT THE LENGTH OF A FOOTBALL FIELD.

THE ADVANTAGE OF A TALL STRUCTURE IS THAT

IT INCREASES THE LINE OF SIGHT TO THE HORIZON.

THIS MAKES IT MUCH EASIER TO SEE AN INVADING ARMY

FROM FARTHER AWAY,

ALLOWING THOSE IN THE CASTLE TO PREPARE DEFENSES.

THIS IS HOW LINE OF SIGHT WORKS.

IMAGINE A STRUCTURE SITTING ON THE EARTH'S SURFACE

WHICH IS DRAWN AS A CIRCLE.

DRAW A LINE FROM THE TOP OF THE STRUCTURE

TO A POINT ON THE CIRCLE.

A LINE THAT INTERSECTS A CIRCLE AT ONE POINT

IS TANGENT TO THE CIRCLE.

NOW, IMAGINE A TALLER STRUCTURE.

WE DRAW ANOTHER LINE FROM THE TOP OF THIS STRUCTURE

TO A POINT TANGENT TO THE CIRCLE.

COMPARING THE FIRST LINE OF SITE DIAGRAM TO THE SECOND

YOU'LL SEE THAT AS THE STRUCTURE'S HEIGHT INCREASES

THE MORE OF THE EARTH'S SURFACE IT CAN SEE.

THE LINE OF SIGHT IMPROVES FOR TALLER BUILDINGS.

LET'S EXPLORE THE ANGLES GENERATED FROM LINE OF SIGHT

ON THE TI-NSPIRE.

TURN ON THE TI-NSPIRE.

CREATE A NEW DOCUMENT.

YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT.

CREATE A GRAPHS AND GEOMETRY WINDOW.

CREATE A CIRCLE.

PRESS MENU AND UNDER SHAPES SELECT CIRCLE.

MOVE THE POINTER TO THE MIDDLE OF THE SCREEN AND

PRESS ENTER TO DEFINE THE CENTER OF THE CIRCLE.

NEXT, MOVE THE POINTER AWAY FROM THE CENTER

TO DEFINE THE SIZE OF THE RADIUS.

PRESS ENTER AGAIN.

CONSTRUCT THE TANGENT TO THE CIRCLE.

PRESS MENU AND UNDER "POINTS AND LINES"

SELECT TANGENT.

MOVE THE POINTER TO THE CIRCLE NEAR THE POINT

THAT WOULD CORRESPOND TO 1 O'CLOCK

IF THE CIRCLE WERE A CLOCK FACE.

PRESS ENTER.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

IF YOU NEED TO MOVE THE POINT, SIMPLY HIGHLIGHT IT

AND USE THE NAV PAD TO MOVE THE POINT ALONG THE CIRCLE.

YOU NOW HAVE A LINE TANGENT TO THE CIRCLE.

YOU'LL NOW CONSTRUCT A LINE SEGMENT FROM THE

CENTER OF THE CIRCLE TO A POINT ON THE TANGENT LINE.

PRESS MENU AND UNDER "POINTS AND LINES"

SELECT SEGMENT.

MOVE THE POINTER TO THE CENTER OF THE CIRCLE.

THE POINTER SHOULD CHANGE TO A HAND.

PRESS ENTER.

MOVE THE POINTER STRAIGHT UP

UNTIL IT INTERSECTS THE TANGENT LINE.

YOU SHOULD SEE A TEXT MESSAGE NEXT TO THE POINTER

SAYING "POINT ON".

PRESS ENTER.

REPEAT THIS TO CREATE A SEGMENT

FROM THE CENTER OF THE CIRCLE TO THE POINT

WHERE THE TANGENT LINE INTERSECTS THE CIRCLE.

YOU SHOULD NOW HAVE A TRIANGLE

WITH VERTICES AT THE CENTER OF THE CIRCLE,

ON THE CIRCLE ITSELF, AND ON THE TANGENT LINE.

THIS IS A MODEL OF THE LINE OF SIGHT.

THE PORTION OF THE SEGMENT ABOVE THE CIRCLE

REPRESENTS HIMEJI CASTLE.

THE LINE OF SIGHT IS ONE OF THE SIDES OF THE TRIANGLE.

LET'S NOW EXPLORE THE ANGLE MEASURES INVOLVED.

USE THE ANGLE MEASURE TOOL.

PRESS MENU, AND UNDER MEASUREMENT SELECT ANGLE.

WITH THE ANGLE MEASURE TOOL YOU NEED TO DEFINE

THE THREE POINTS THAT MAKE UP THE TRIANGLE.

YOU WILL BE USING THE VERTICES OF THE TRIANGLE.

TAKE TURNS MARKING THE THREE POINTS

THAT MAKE UP EACH OF THE ANGLES

OF THE TRIANGLE YOU HAVE CONSTRUCTED.

PRESS ENTER EACH TIME YOU HAVE HIGHLIGHTED

ONE OF THE POINTS.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

WHEN YOU ARE DONE PRESS THE ESCAPE KEY.

FIRST NOTICE THAT THE ANGLE FORMED BY

THE TANGENT AND THE RADIUS IS 90 DEGREES.

EVEN IF YOU MANIPULATE THE VERTEX

THE ANGLE REMAINS 90 DEGREES.

AND IN FACT THIS IS A PROPERTY OF ALL

TANGENTS TO CIRCLES: THEY ARE PERPENDICULAR

TO THE RADIUS AT THE POINT OF INTERSECTION.

THIS MEANS THAT THE LINE OF SIGHT ANGLE IS AN

ACUTE ANGLE OR AN ANGLE LESS THAN 90 DEGREES.

LET'S CHANGE THE HEIGHT OF A TOWER

TO SEE HOW THE LINE OF SIGHT CHANGES.

MOVE THE POINTER TO THE POINT ON THE TANGENT LINE.

SELECT THE POINT BY PRESSING AND HOLDING THE CLICK KEY.

MOVE THE POINT UPWARD.

NOTICE HOW THE HEIGHT INCREASES.

BECAUSE THE POINT IS ON THE TANGENT LINE,

THE POINT WILL MOVE UP AND TO THE LEFT.

GO TO THE POINT ON THE CIRCLE AND MOVE IT CLOCKWISE

UNTIL THE LINE REPRESENTING HIMEJI CASTLE IS VERTICAL.

THIS MOVEMENT OF THE POINT OF THE CIRCLE

REPRESENTS THE INCREASE IN THE LINE OF SIGHT.

NOTICE ALSO THAT AS THE HEIGHT OF THE STRUCTURE

INCREASES, THE ACUTE ANGLE DECREASES IN SIZE.

HIMEJI CASTLE RISES HIGH IN THE AIR SO THAT IT HAS

A LONG LINE OF SIGHT TO THE HORIZON SO THAT

IT CAN IDENTIFY ATTACKERS FROM A LONG DISTANCE.

THE LINE OF SIGHT FORMS A LINE TANGENT TO THE

ARC OF THE EARTH'S CIRCULAR SHAPE.

WHAT IS THE LENGTH OF THIS LINE OF SIGHT

AND HOW CAN WE USE IT?

BECAUSE THE TANGENT AND THE RADIUS FORM A RIGHT ANGLE

THE TRIANGLE THAT YOU CREATED EARLIER

IS A RIGHT TRIANGLE.

SO THE LINE OF SIGHT DISTANCE

IS ONE OF THE LEGS OF A RIGHT TRIANGLE.

ALSO, THE RADIUS OF THE CIRCLE IS ACTUALLY

THE RADIUS OF THE EARTH.

WE CAN CREATE AN EQUATION TO SOLVE.

LET'S USE VARIABLE H FROM THE HEIGHT OF THE CASTLE.

R FOR THE RADIUS OF THE EARTH

AND L AS THE LINE OF SIGHT DISTANCE.

USING THE PYTHAGOREAN THEOREM

WE GET THE EQUATION L SQUARED PLUS R SQUARED

EQUALS THE QUANTITY H + R SQUARED.

SOLVING FOR L WE GET THE EXPRESSION

THE SQUARE ROOT OF THE QUANTITY H SQUARED +2 HR.

WE KNOW THAT H, THE HEIGHT OF

THE HIMEJI CASTLE IS 92 METERS.

THE RADIUS OF THE EARTH IS ROUGHLY

6.4X10 TO THE 6TH METERS.

PLUGGING THESE VALUES INTO THE FORMULA FOR FINDING L

WE'D GET THAT THE LINE OF SIGHT DISTANCE

IS ABOUT 34,316 METERS, OR ABOUT 21 MILES.

GIVEN THAT AN INVADING ARMY TRAVELS AT A FAIRLY

SLOW SPEED, THE LOOKOUTS ON HIMEJI CASTLE

WITH THEIR COMMANDING VIEW OF THE COUNTRYSIDE

COULD PREPARE FOR IMMINENT INVASION WITH HOURS TO SPARE.

AND WHAT WOULD BE IN STORE FOR THE INVADERS

WOULD BE MORE HOSTILE GEOMETRY.

THE GROUNDS SURROUNDING THE CASTLE ARE VAST

AND THERE ARE PLENTY OF OBSTACLES

ON THE WAY TO THE MAIN TOWER.

FIRST, LIKE MANY CASTLES, HIMEJI HAS A MOAT

SURROUNDING THE CASTLE GROUNDS.

SECOND, THERE IS A HIGH WALL BEHIND THE MOAT

ENCLOSING THE CASTLE GROUNDS.

THERE IS A BRIDGE NEAR THE MAIN ENTRANCE

AND AN INVADING FORCE WOULD MOVE IN THIS DIRECTION.

BUT ONCE INSIDE AND PAST THE MAIN GATE

THE GROUNDS BECOME A HUGE MAZE WITH VARIOUS PATHWAYS

LEADING TO DEAD ENDS.

THE LONG WINDING PATH THROUGH THIS MAZE

IS ACTUALLY A SPIRAL PATH TO THE MAIN TOWER.

MOVING ALONG A SPIRAL PATH MEANS THAT YOU

TURN 360 DEGREES OR ONE FULL CIRCUIT

FROM YOUR INITIAL POSITION.

YOU CAN SEE THIS ON THE NSPIRE.

ACTIVATE THE SEGMENT TOOL.

CONSTRUCT AN ANGLE.

FIRST CREATE ONE SIDE OF THE ANGLE,

PRESSING ENTER FOR EACH END POINT.

THEN PRESS ENTER ONCE AGAIN OVER ONE OF THE ENDPOINTS.

CREATE A SECOND SEGMENT

IN THE PROCESS CREATING THE ANGLE.

NOW MEASURE THE ANGLE.

PRESS MENU AND UNDER MEASUREMENT SELECT ANGLE.

AFTER YOU HAVE MEASURED THE ANGLE

PRESS ESCAPE AND MOVE THE POINTER

TO ONE OF THE ENDPOINTS AS SHOWN.

HOLD THE CLICK KEY UNTIL YOU SELECT THE POINT.

MOVE THE POINT SO THAT THE ANGLE MEASURE IS 0 DEGREES.

NEXT, MOVE THE POINT UNTIL THE MEASURE IS 180 DEGREES.

WHAT DO YOU NOTICE?

IT'S A STRAIGHT LINE.

SO IF YOU WERE TO CONTINUE ROTATING THE POINT

SO THAT IT IS BACK TO WHERE YOU STARTED

THIS WOULD RESULT IN ANOTHER 180 DEGREES

FOR A TOTAL OF 360 DEGREES.

AN INVADING ARMY WOULD NOT REALIZE THAT IT IS GOING

IN CIRCLES SINCE THE SPIRAL PATH IS GRADUALLY CIRCULAR.

IN ADDITION, ON ITS WAY THROUGH THE MAZE

THE PATH GRADUALLY BECOMES STEEPER.

THAT PATH HAS A SLOPE OF ABOUT TWO DEGREES.

THIS IS AN ACUTE ANGLE SINCE IT IS LESS THAN 90 DEGREES.

TRY TO MODEL A TWO DEGREE ANGLE ON THE NSPIRE.

AS YOU CAN SEE, SUCH A SLOPE IS GRADUAL ENOUGH

THAT AN ARMY WOULD NEED TO TRAVEL FARTHER AND

EXERT ITSELF MORE TO REACH THE BASE OF THE TOWER.

FINALLY UPON REACHING THE BASE OF THE TOWER

THE ARMY WOULD BE STARING AT THIS IMPOSING STRUCTURE.

SEEN FROM THE SIDE, THE MAIN TOWER

MAKES AN OBTUSE ANGLE WITH THE GROUND

OR AN ANGLE GREATER THAN 90 DEGREES.

ALSO NOTICE THAT THE LINE OF SIGHT FROM GROUND LEVEL

PREVENTS AN ATTACKER FROM GETTING A VIEW OF

THE TOP OF THE TOWER GIVING THE CASTLE'S DEFENDERS

STILL ONE MORE ELEMENT OF SURPRISE

IN ATTACKING THE ARMY AT THE BASE OF THE TOWER.

HIMEJI CASTLE'S DEFENSES OWE A GREAT DEAL TO THE WAY

IN WHICH ANGLES ARE USED, FROM THE SLOPING PATHS

WHERE INVADERS ARE SLOWED DOWN,

TO THE BLIND CORNERS OF ITS BYZANTINE MAZE

WHERE DEFENDERS COULD AMBUSH THE INVADERS,

TO THE INSURMOUNTABLE WALLS OF THE MAIN TOWER.

THE PROPERTIES OF ANGLES ARE USED FOR

DEFENSIVE PURPOSES AND YET HIMEJI REMAINS

AN ELEGANT PIECE OF ARCHITECTURE THAT ADDS

MAJESTY AND SERENITY TO ITS SURROUNDINGS.

IN THE CANADIAN ROCKIES, AN AMAZING DISCOVERY

WAS MADE NEARLY A CENTURY AGO.

HIGH ON A MOUNTAIN NEAR WHAT IS NOW KNOWN AS

YOHO NATIONAL PARK IS A COLLECTION OF FOSSILS,

SOME OF WHICH EXIST NOWHERE ELSE ON EARTH.

KNOWN AS THE BURGESS SHALE,

THESE FOSSILS ARE OVER 500 MILLION YEARS OLD.

THE ORGANISMS FROM WHICH THESE FOSSILS CAME FROM

LIVED DURING THE CAMBRIAN PERIOD,

LONG BEFORE THE AGE OF DINOSAURS.

THE BURGESS SHALE FOSSILS REPRESENT THE MOST DIVERSE

SET OF ORGANISMS EVER FOUND, SOME LOOKING LIKE

ALIEN CREATURES FROM ANOTHER WORLD.

THESE FOSSILS SPEAK TO A WORLD

SO DIFFERENT FROM OURS.

A FOSSIL IS AN IMPRINT ON A ROCK OF AN ANIMAL OR PLANT

AND THIS IMPRINT SURVIVES LONG AFTER THE ORGANISM

HAS DIED AND DECAYED.

MOST FOSSILS SUGGEST WHAT AN ANIMAL OR PLANT

MAY HAVE LOOKED LIKE.

IN THE CASE OF THE BURGESS SHALE FOSSILS,

MUCH MORE WAS PRESERVED.

SOFT TISSUE AND EVEN SOME OF THE INTERNAL ORGANS

OF THESE STRANGE CAMBRIAN CREATURES

SURVIVED UNDER THE FAVORABLE CONDITIONS

OF THE BURGESS SHALE.

SHALE IS A KIND OF SEDIMENTARY ROCK

FORMED FROM LAYERS OF SAND, DIRT AND OTHER DEPOSITS.

ALTHOUGH THE BURGESS SHALE FOSSILS ARE FOUND ON

MOUNTAINTOPS, THE FOSSILS ORIGINATED IN THE OCEAN.

MOST OF THE BURGESS SHALE FOSSILS

ARE OF MARINE ORGANISMS.

TO GET A BETTER UNDERSTANDING

OF HOW FOSSILS ARE FORMED

LET'S FOLLOW THE FORMATION OF ONE.

THIS IS A TRILOBITE.

IT WAS A MARINE CREATURE

COMMON DURING THE CAMBRIAN PERIOD.

AND THERE ARE MANY TRILOBITES FOUND

AMONG THE BURGESS SHALE FOSSILS.

TRILOBITES ARE BELIEVED TO BE DISTANT RELATIVES

OF HORSESHOE CRABS.

WHEN A TRILOBITE DIED

ITS BODY WOULD SINK INTO THE SOFT WET SAND.

A SEAFLOOR IS A PLACE WHERE LAYERS

OF SEDIMENT ACCUMULATE.

OVER TIME LAYERS AND LAYERS OF SEDIMENT

WOULD ACCUMULATE OVER THE TRILOBITE.

SEDIMENTARY ROCKS FORM WHEN LAYERS OF DIRT

AND SEDIMENT PRESS INTO EACH OTHER.

THE PACKED IN LAYERS OF SEDIMENT

WILL EVENTUALLY BECOME AS HARD AS ROCK.

WHEN AN ORGANISM IS TRAPPED IN

ONE OF THESE LAYERS IT WILL BECOME FLATTENED

AND INCORPORATED INTO THE ROCK'S STRUCTURE.

THIS IS WHAT HAPPENS WITH THE TRILOBITE.

BURGESS SHALE ROCKS ARE LAYERED AND FOSSIL IMPRINTS

ARE FOUND THROUGHOUT MANY OF THESE ROCKS.

LOOK AT THIS LAYER OF SEDIMENTARY ROCKS.

GEOMETRICALLY, THESE SEDIMENTARY LAYERS

ARE AN EXAMPLE OF PARALLEL PLANES.

WHEN PLANES ARE PARALLEL THEY NEVER INTERCEPT

NO MATTER HOW FAR THE PLANES EXTEND.

WHEN SEEN FROM THE SIDE THE TWO-DIMENSIONAL PLANES

BECOME TWO ONE-DIMENSIONAL PARALLEL LINES.

JUST AS YOU CAN THINK OF A LINE AS AN INFINITE

SET OF COLLINEAR POINTS, YOU CAN THINK OF A PLANE

AS AN INFINITE SET OF PARALLEL LINES.

SUPPOSE THERE ARE TWO PARALLEL PLANES, P1 AND P2.

NOW SUPPOSE POINT A IS ON P1.

BECAUSE THE TWO PLANES ARE PARALLEL,

THEN POINT A CANNOT ALSO BE ON PLANE P2.

THE REASON FOR THIS IS SIMPLE.

IF POINT A WAS ON P1 AND P2, THAT MEANS THAT THE PLANES

INTERSECT AT THIS POINT, BUT BY DEFINITION THE PLANES ARE

PARALLEL SO THEY DON'T HAVE ANY POINTS OF INTERSECTION.

THIS IS A USEFUL CONCEPT WHEN IT COMES TO

STUDYING FOSSILS. HERE'S WHY:

AS YOU SAW, SEDIMENTARY ROCKS ARE MADE UP OF LAYERS

AND LAYERS OF SEDIMENT THAT OVER TIME BECOME SOLID ROCK.

THIS MEANS THAT LAYERS THAT ARE FARTHER DOWN

REPRESENT AN OLDER PERIOD OF TIME

THAN LAYERS CLOSER TO THE EARTH'S SURFACE.

FURTHERMORE, IF TWO DIFFERENT FOSSILS

ARE FOUND IN THE SAME LAYER,

THEN THE TWO ORGANISMS LIVED AT THE SAME TIME.

THESE ARE TWO IMPORTANT CONCEPTS IN FOSSIL ANALYSIS.

LET'S LOOK AT THEM IN MORE DETAIL.

IN THE FIRST CASE, LOOK AT THESE LAYERS

OF SEDIMENTARY ROCKS WHICH WE CAN THINK OF

AS A SERIES OF PARALLEL PLANES.

SUPPOSE THAT IN ONE LAYER OF ROCKS

A TRILOBITE FOSSIL IS FOUND, AND IN ANOTHER

HIGHER LAYER A FISH FOSSIL IS FOUND.

WHAT CAN YOU LOGICALLY DEDUCE

FROM THIS ARRANGEMENT OF FOSSILS?

SINCE THE TWO FOSSILS ARE ON

DIFFERENT PARALLEL PLANES

THEN THEY WERE FORMED AT DIFFERENT PERIODS OF TIME.

WE CAN THEREFORE CONCLUDE THAT THE

TRILOBITE FOSSIL WAS FORMED BEFORE THE FISH FOSSIL.

BUT HOW CAN WE CONCLUDE THAT, AS A SPECIES,

TRILOBITES EXISTED LONG BEFORE FISH DID?

ONCE AGAIN WE CAN RELY ON

THE PROPERTIES OF PARALLEL PLANES.

PALEONTOLOGISTS USE WHAT IS REFERRED TO AS AN

INDEX FOSSIL TO DETERMINE THE AGE OF THE ROCK LAYERS.

THIS IS HOW INDEX FOSSILS WORK:

AN INDEX FOSSIL IS USED TO IDENTIFY

WHICH GEOLOGIC PERIOD A FOSSIL COMES FROM.

THESE FOSSILS ARE FROM ORGANISMS THAT LIVED

DURING A PARTICULAR GEOLOGIC PERIOD,

WENT EXTINCT RELATIVELY QUICKLY IN GEOLOGIC TIME,

AND ARE THEREFORE A GOOD INDICATOR

OF THE AGE OF THE ROCKS AND OTHER FOSSILS

IN A PARTICULAR SEDIMENTARY LAYER.

AS A RESULT OF INDEX FOSSILS

PALEONTOLOGISTS HAVE DEVELOPED A GEOLOGIC

TIME SCALE THAT LOOKS LIKE A VERTICAL NUMBER LINE

AND ACCOUNTS FOR DIFFERENT PERIODS IN THE EARTH'S HISTORY

SPANNING HUNDREDS OF MILLIONS OF YEARS.

FOR EXAMPLE, TRILOBITES ARE AN INDEX FOSSIL

FOR THE CAMBRIAN PERIOD.

THIS SPIRAL SHELL FOSSIL IS AN INDEX FOSSIL

FOR THE JURASSIC PERIOD.

AND THIS FOSSIL SHELL IS AN INDEX FOSSIL

FOR THE TERTIARY PERIOD.

THE FULL SET OF INDEX FOSSILS

AND GEOLOGIC PERIODS IS MUCH MORE EXTENSIVE.

SO LET'S SAY THAT A PALEONTOLOGIST

DISCOVERS A NEW PLANT FOSSIL.

WHAT CAN SHE CONCLUDE ABOUT THE AGE OF THE FOSSIL?

IF THE FOSSIL LAYER ALSO HAD A TRILOBITE FOSSIL

THEN THE PALEONTOLOGIST CAN CONCLUDE

THAT THE PLANT IS FROM THE CAMBRIAN PERIOD.

ON THE OTHER HAND, IF THE FOSSIL LAYER

HAD AN INDEX FOSSIL FROM THE JURASSIC PERIOD,

THEN THE PALEONTOLOGIST CAN CONCLUDE

THAT THE PLANT IS FROM THAT PERIOD.

INDEX FOSSILS ARE LIKE POINTS ON PARALLEL PLANES.

IF POINTS A AND B ARE COPLANAR

THEN B IS NOT ON PLANE P1 AND P3.

SINCE THE TRILOBITE IS AN INDEX FOSSIL

FOR THE CAMBRIAN PERIOD, THEN SUCH FOSSILS

WILL NOT BE FOUND IN OTHER FOSSIL LAYERS.

AND ALL OF THIS IS BASED ON THE FACT THAT

THE SEDIMENTARY LAYERS ARE SIMILAR TO PARALLEL PLANES.

AND THESE PARALLEL PLANES ARE MAPPED

TO A VERTICAL COORDINATE SYSTEM.

THIS IS A THREE-DIMENSIONAL COORDINATE SYSTEM.

THE THREE AXES ARE LABELED X, Y AND Z.

THE VERTICAL AXIS IS THE Y AXIS.

AND IF WE DESIGNATE THAT AS THE NUMBER LINE

FOR THE GEOLOGIC TIME SCALE, THAT MEANS THAT

THE HORIZONTAL PLANE THAT CROSSES THE ORIGIN

REPRESENTS THE EARTH'S SURFACE.

AS A RESULT, THE PLANES REPRESENTING

THE SEDIMENTARY LAYERS ARE BELOW THIS PLANE.

THE PROPERTY OF PARALLEL PLANES INSURES

THAT IF AN INDEX FOSSIL IS FOUND IN ONE PLANE

IT WON'T BE FOUND ON ANOTHER PLANE.

TAKE A LOOK AT THESE TWO FOSSILS.

THE FOSSIL ON THE LEFT IS A TRILOBITE

FROM THE CAMBRIAN PERIOD AND THE ONE ON THE RIGHT

IS A SHELL FROM THE JURASSIC PERIOD.

SINCE THE TRILOBITE FOSSIL IS OLDER,

THEN A PALEONTOLOGIST WOULD EXPECT TO ALWAYS FIND

SUCH FOSSIL LAYERS BENEATH THE NEWER FOSSIL LAYERS.

AND YET SOMETIMES PALEONTOLOGISTS DO IN FACT

FIND THE OLDER FOSSIL LAYER ABOVE THE NEWER LAYER.

HOW CAN THIS BE?

DOESN'T THE PROPERTY OF PARALLELISM MEAN THAT

THE OLDER FOSSIL WILL ALWAYS BE BELOW THE NEWER ONE?

GEOMETRICALLY THE ANSWER IS YES.

SO WHAT CAUSES THE SITUATION WHERE

THE OLDER FOSSIL IS ABOVE THE NEWER ONE?

LOGICALLY, THE ANSWER IS THAT

THE PLANES ARE NO LONGER PARALLEL.

FOR EXAMPLE, LOOK AT THESE TWO PLANES.

THESE PLANES INTERSECT WHICH IS WHAT HAPPENS

WHEN TWO PLANES ARE NO LONGER PARALLEL.

WHEN TWO PLANES INTERSECT THEN IT IS POSSIBLE

FOR ONE PLANE TO BE ABOVE THE OTHER PLANE OR BELOW IT,

DEPENDING ON WHERE ON THE PLANE A POINT IS.

FOR EXAMPLE, POINTS A AND B ARE ON PLANE P1.

POINT A IS BELOW PLANE P2 BUT POINT B IS ABOVE P2.

WE HAVE JUST SEEN HOW SEDIMENTARY LAYERS

ARE LIKE PARALLEL PLANES,

SO HOW IS IT POSSIBLE FOR THEM TO INTERSECT?

IT TURNS OUT THAT SEDIMENTARY LAYERS

ARE PART OF A SYSTEM THAT ON OCCASION

CAUSES THESE LAYERS TO PUSH AGAINST EACH OTHER.

IN THIS ILLUSTRATION OF THE EARTH'S INTERIOR,

YOU CAN SEE THAT THERE ARE NUMEROUS

LAYERS INSIDE THE EARTH.

THE OUTERMOST LAYER IS THE EARTH'S CRUST.

THE CRUST LIES ON A HOT MOLTEN LAYER

CALLED THE MANTLE.

BECAUSE OF THE MANTLE THE EARTH'S CRUST

IS LIKE A BROKEN EGG SHELL CALLED TECTONIC PLATES.

THESE PLATES PUSH AGAINST EACH OTHER.

SINCE THE FOSSIL LAYERS ARE PART OF THE EARTH'S CRUST

THEN THESE LAYERS WILL PRESS INTO EACH OTHER.

WHEN THIS HAPPENS, SEDIMENTARY LAYERS

WILL SOMETIMES ACT LIKE INTERSECTING PLANES.

FOR EXAMPLE, WHEN TWO TECTONIC PLATES

PRESS AGAINST EACH OTHER,

ONE PLATE BUCKLES UNDER THE PRESSURE OF THE OTHER

IN A MANNER SIMILAR TO PLANES INTERSECTING.

IN FACT, THE COLLISION OF TECTONIC PLATES IS

WHAT CAUSES EARTHQUAKES, VOLCANOES AND MOUNTAINS.

NOTICE HOW THE SEDIMENTARY LAYERS ACROSS

THE TWO TECTONIC PLATES ARE NO LONGER PARALLEL.

FURTHERMORE, WITH SOME LAYERS THE OLDER FOSSILS

COULD BE FOUND ABOVE THE NEWER ONES.

WHEN TWO PLANES INTERSECT, THEY INTERSECT AT A LINE.

IN OTHER WORDS A FLAT, TWO-DIMENSIONAL SHAPE

WILL INTERSECT ANOTHER FLAT TWO-DIMENSIONAL SHAPE

ALONG A ONE-DIMENSIONAL LINE.

YOU CAN SEE A CLEAR EXAMPLE OF THIS

BY LOOKING AT THE SAN ANDREAS FAULT LOCATED AT

THE BOUNDARY OF THE PACIFIC AND NORTH AMERICAN PLATES.

THE SAN ANDREAS FAULT IS A ROUGHLY LINEAR PATH

THAT EXTENDS NEARLY 1,000 MILES

AND CROSSES THROUGH THE EASTERN PART OF CALIFORNIA.

THE SAN ANDREAS FAULT IS AN EXAMPLE OF A FAULT LINE

WHERE PARTS OF THE EARTH'S CRUST

PRESS AGAINST EACH OTHER.

EARTHQUAKES OFTEN OCCUR NEAR FAULT LINES

AND THE SAN ANDREAS FAULT IS RESPONSIBLE

FOR MANY OF CALIFORNIA'S EARTHQUAKES.

GEOMETRICALLY, THE FAULT LINE IS THE BYPRODUCT OF TWO

TECTONIC PLATES THOUGHT OF AS TWO PLANES INTERSECTING.

FAULT LINES ARE NEVER TRULY STRAIGHT.

BUT IF YOU LOOK AT THE INTERSECTION OF

TWO DIMENSIONAL SURFACES THAT AREN'T FLAT

YOU CAN STILL SEE THE INTERSECTION AS LINEAR.

GEOMETRICALLY A FOSSIL CAN BE

DESCRIBED AS A CLOSED FIGURE ON A PLANE.

IF YOU WERE TO DRAW A CLOSED FIGURE

IT WOULD INVOLVE PLACING YOUR PENCIL

ON A SHEET OF PAPER, DRAWING A SHAPE

AND RETURNING THE PENCIL POINT TO WHERE YOU STARTED

WITHOUT ONCE LIFTING THE PENCIL

FROM THE SHEET OF PAPER.

HERE ARE SOME EXAMPLES OF CLOSED FIGURES.

A CLOSED FIGURE ON A PLANE TAKES UP AN AMOUNT OF AREA.

YOU CAN ALSO MEASURE THE LENGTH AND WIDTH OF THE

FIGURE TO EITHER CALCULATE OR ESTIMATE THE AREA.

FOR SIMPLE CLOSED FIGURES LIKE SQUARES, RECTANGLES

AND CIRCLES, AREA FORMULAS ARE USED

TO CALCULATE AN EXACT VALUE OF THE AREA.

FOR MORE COMPLEX FIGURES YOU CAN ESTIMATE THE AREA

BY BREAKING THE FIGURE DOWN TO SIMPLER SHAPES.

THESE CONCEPTS ARE IMPORTANT

TO PALEONTOLOGISTS ANALYZING FOSSILS.

SO FAR THE EXAMPLES OF FOSSILS THAT YOU'VE SEEN

ARE OF ENTIRE ORGANISMS, LIKE TRILOBITES AND SHELLS.

THESE ARE SMALL CREATURES THAT

EASILY FIT INTO SEDIMENTARY LAYERS.

THE SAME GOES FOR PLANT LEAVES.

BUT MANY ORGANISMS LIKE DINOSAURS

DON'T LEAVE BEHIND INTACT FOSSILS.

FOR MOST DINOSAURS PALEONTOLOGISTS HAVE

USUALLY FOUND BONES AND PARTS OF SKELETONS

BUT NOT THE ENTIRE DINOSAUR.

AND YET PALEONTOLOGISTS HAVE A POWERFUL GEOMETRIC TOOL

TO USE WITH A CERTAIN TYPE OF DINOSAUR FOSSIL.

AS DINOSAURS ROAMED THE EARTH THEY LEFT BEHIND

FOOTPRINTS SOME OF WHICH SURVIVED MILLIONS OF YEARS.

A DINOSAUR'S SIZE WAS TOO LARGE

TO FIT INTO A SEDIMENTARY LAYER

BUT ITS FOOTPRINT WAS A DIFFERENT MATTER.

A FOOTPRINT EASILY FITS ON A TWO-DIMENSIONAL SURFACE.

IN FACT, THE LARGER AND HEAVIER THE DINOSAUR

THE DEEPER THE IMPRINT AND THE MORE LIKELY

THE FOOTPRINT COULD SURVIVE

THE SEDIMENTARY ROCK FORMATION PROCESS.

BY MEASURING THE SIZE OF A DINOSAUR'S FOOTPRINT

A PALEONTOLOGIST CAN ESTIMATE THE SIZE

OF THE DINOSAUR THAT MADE THE FOOTPRINT.

LET'S USE THE TI-NSPIRE TO MODEL THIS PROCESS.

TURN ON THE TI-NSPIRE.

CREATE A NEW DOCUMENT.

YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT.

CREATE A GRAPHS AND GEOMETRY WINDOW.

SUPPOSE A PALEONTOLOGIST

DISCOVERS THESE FOSSIL PRINTS.

USE THE SEGMENT TOOL TO REPRODUCE THE SHAPE.

PRESS MENU, AND UNDER "POINTS AND LINES"

SELECT SEGMENT.

CREATE A CLOSED FIGURE SIMILAR TO THE ONE SHOWN.

PRESS ENTER TO DEFINE THE FIRST ENDPOINT OF THE SEGMENT

THEN USE THE NAV PAD TO MOVE THE POINTER.

PRESS ENTER AGAIN TO DEFINE THE SECOND ENDPOINT.

USE THAT SAME POINT AS THE STARTING POINT

FOR THE NEXT SEGMENT.

PRESS ENTER TO START THE NEXT SEGMENT.

CONTINUE UNTIL YOU HAVE DRAWN THE CLOSED FIGURE.

MEASURE THE LENGTH OF THE FOOTPRINT.

PRESS MENU AND UNDER MEASUREMENT

SELECT LENGTH.

MOVE THE POINTER TO ONE ENDPOINT OF THE FOOTPRINT.

PRESS ENTER.

MOVE THE POINTER UP TO THE FARTHEST ENDPOINT

TO MEASURE THE LENGTH.

PRESS ENTER AGAIN.

NOTICE THAT THE MEASUREMENT APPEARS NEXT TO THE POINTER.

MOVE THE POINTER TO A CLEAR PART OF THE SCREEN

AND PRESS ENTER TO PLACE THE MEASUREMENT ON SCREEN.

BY DEFAULT, THE MEASUREMENT SCALE IS IN CENTIMETERS.

SO THE VALUE SHOWN FOR THE MEASUREMENT

WILL BE CENTIMETERS.

USE THE NAV PAD TO MOVE THE POINTER

TO THE UPPER RIGHT PART OF THE SCREEN.

CHANGE THE SCALE FROM CENTIMETERS TO FEET.

HOVER OVER THE SCALE.

PRESS MENU AND UNDER ACTIONS CHOOSE SELECT

AND PRESS ENTER.

NOW EDIT THE TEXT FIELD AND CHANGE CENTIMETERS TO FEET.

PRESS ENTER.

PALEONTOLOGISTS USE A FORMULA

FOR ESTIMATING THE HEIGHT OF THE DINOSAUR.

BASICALLY THEY MULTIPLY THE FOOTPRINT LENGTH

AND MULTIPLY IT BY FOUR.

CREATE A FORMULA THAT CAPTURES THE LENGTH OF THE

FOOTPRINT AND CALCULATES THE DINOSAUR HEIGHT.

PRESS MENU AND UNDER ACTIONS

SELECT TEXT.

MOVE THE POINTER TO A CLEAR PART OF THE SCREEN

AND PRESS ENTER.

INPUT THE FORMULA LENGTH TIMES 4 AND PRESS ENTER.

LINK THIS FORMULA TO THE LENGTH OF THE FOOTPRINT.

PRESS MENU AND UNDER ACTIONS SELECT CALCULATE.

MOVE THE POINTER ABOVE THE FORMULA YOU'VE JUST CREATED

AND PRESS ENTER.

THEN MOVE THE POINTER OVER THE VALUE

OF THE FOOTPRINT'S LENGTH AND PRESS ENTER.

YOU'LL SEE THE CALCULATED HEIGHT NEXT TO THE POINTER.

MOVE THE POINTER NEXT TO THE FORMULA AND PRESS ENTER

TO PLACE THE CALCULATED VALUE NEXT TO THE FORMULA.

MODIFY THE LENGTH OF THE FOOTPRINT TO SEE THE FORMULA

RECALCULATE THE HEIGHT OF THE DINOSAUR.

THE BURGESS SHALE FOSSILS REVEAL A

GREAT DEAL OF INFORMATION ABOUT LIFE ON EARTH.

PALEONTOLOGISTS USE A NUMBER OF MATHEMATICAL TOOLS

TO LEARN MORE ABOUT PREHISTORIC CREATURES.

THIS INCLUDES USING SOME KEY CONCEPTS FROM GEOMETRY.