Geometry Applications: Polygons

[Music]

[Music]

Title: Geometry Applications: Polygons

Title: Geometry Applications: Polygons

Title: Geometry Applications: Geometry Basics: Polygons

Title: Geometry Applications: Geometry Basics: Polygons

THE WORLD'S MOST FAMOUS OFFICE BUILDING,

THE PENTAGON, WAS ORIGINALLY

MEANT TO BE LOCATED ELSEWHERE.

DURING WORLD WAR II THIS BUILDING WAS TO BE

BUILT ACROSS FROM THE LINCOLN MEMORIAL

AND NEAR ARLINGTON MEMORIAL CEMETERY.

IN FACT, SOME FARMLAND HAD BEEN CHOSEN

AND IT HAPPENED TO BE IN THE SHAPE OF A PENTAGON.

NOT AN EVEN SIDED REGULAR PENTAGON,

BUT A PENTAGON NONETHELESS.

FOR SIMPLICITY THE ARCHITECTS DECIDED TO

MAKE THE OFFICE BUILDING IN THE SHAPE OF THE FARMLAND.

BUT PROBLEMS EMERGED WHEN IT BECAME APPARENT

THAT THE PROPOSED LOCATION OF THE NEW OFFICE BUILDING

WOULD INTERFERE WITH THE VIEW OF ARLINGTON CEMETERY

WHEN SEEN FROM THE WASHINGTON SIDE OF

THE POTOMAC RIVER.

A DECISION WAS QUICKLY MADE TO RELOCATE THE BUILDING

FARTHER DOWN THE RIVER.

THE DEMANDS OF WARTIME PREVENTED

AN ENTIRELY NEW BLUEPRINT FOR THE OFFICE BUILDING.

BUT THE ARCHITECTS DID MAKE ONE SIGNIFICANT CHANGE.

THE BUILDING WOULD BE A REGULAR PENTAGON

AND THUS THE PENTAGON, AS THE BUILDING

HAS SINCE COME TO BE KNOWN, CAME TO BE.

IN THEIR RUSH TO CREATE THE NEW OFFICE BUILDING,

THE DESIGNERS CAME UP WITH A BUILDING

WHOSE SHAPE OFFERS SOME CLEAR ADVANTAGES.

LET'S COMPARE A SQUARE

INSCRIBED IN A CIRCLE OF RADIUS R

TO A REGULAR PENTAGON INSCRIBED IN THE SAME CIRCLE.

THE INCREASE IN THE NUMBER OF SIDES FOR THE PENTAGON

RESULTS IN A SIZABLE INCREASE IN THE PERIMETER.

THIS IS IMPORTANT FOR A BUILDING THAT NEEDED

A HUGE NUMBER OF OFFICE SUITES.

FURTHERMORE, THE FIVE SIDES

OFFER MORE ANGLES OF APPROACH

FOR THE SURROUNDING ROADS AND HIGHWAYS.

THIS EXAMPLE ILLUSTRATES THE VERSATILITY OF POLYGONS.

WITH AN INCREASE IN THE NUMBER OF SIDES

COME OTHER PROPERTIES THAT MAKE POLYGONS

THE SOLUTION TO CERTAIN DESIGN PROBLEMS.

IN THIS PROGRAM YOU WILL EXPLORE SOME REAL WORLD

PROBLEMS THAT INVOLVE THE USE OF POLYGONS.

SPECIFICALLY, THIS PROGRAM WILL ADDRESS

THE FOLLOWING CONCEPTS:

THE FOLLOWING CONCEPTS:

Title: Geometry Applications: Regular Polygons

Title: Geometry Applications: Regular Polygons

Title: Geometry Applications: Regular Polygons

IT IS LATE IN THE DAY IN MARRAKESH,

THE CAPITAL CITY OF MOROCCO.

THE MERCHANTS HAVE CLOSED THEIR SHOPS,

THE TOWN SQUARE HAS LOST ITS BUSTLE.

THIS MEDIEVAL CITY IS THE JEWEL IN THE CROWN

OF THE MOORISH EMPIRE

WHICH INCLUDES PART OF MODERN-DAY SPAIN.

BUT IT IS ALSO LATE IN THE DAY FOR THIS EMPIRE.

THE YEAR IS AROUND 1100 AD.

WITHIN 300 YEARS EUROPE WILL EMERGE FROM THE DARK AGES

WITH THE BEGINNING OF THE RENAISSANCE...

AND THE MOORS WILL NO LONGER BE THE RULERS

OF THIS PART OF EUROPE.

THROUGHOUT THIS PERIOD OF CHANGE,

MARRAKESH RETAINS ITS IDENTITY AS A

CULTURAL CENTER OF ISLAMIC ART AND ARCHITECTURE.

ISLAMIC ART RELIES ON GEOMETRIC PATTERNS.

THESE INTRICATE PATTERNS ARE MEANT TO SUGGEST

THE SWIRLS OF PLANTS AND SHAPES OF ORGANISMS.

AND THESE PLEASING DESIGNS ARE OFTEN REFERRED TO AS

"ARABESQUES", A REFERENCE TO THEIR CONSTANT USE

THROUGHOUT THE ARAB WORLD.

BUT THESE INTRICATE DESIGNS ARE BASED ON

REGULAR POLYGONS.

A REGULAR POLYGON IS A CLOSED FIGURE

WHOSE SIDE LENGTHS AND INTERIOR ANGLE MEASURES

ARE CONGRUENT.

A SQUARE IS A REGULAR QUADRILATERAL.

ALL SIDES HAVE THE SAME LENGTH

AND ALL INTERIOR ANGLES ARE 90 DEGREES.

THE SUM OF THEIR INTERIOR ANGLES IS 360 DEGREES.

THIS IS A POLYGON AND THIS IS A REGULAR POLYGON.

THE PENTAGON ON THE LEFT HAS DIFFERENT SIDE LENGTHS

AND DIFFERENT INTERIOR ANGLE MEASURES.

AND YET THE SUM OF THE INTERIOR ANGLES

FOR BOTH PENTAGONS IS 540 DEGREES.

HERE ARE SOME OTHER REGULAR POLYGONS.

THESE INTRICATE DESIGNS ARE NOTEWORTHY

FOR THEIR PRECISION.

BUT HOW WERE MEDIEVAL ARTISANS ABLE TO ACHIEVE

SUCH PRECISION WITH THE AVAILABLE TOOLS OF THE TIME?

THESE TOOLS WOULD HAVE INCLUDED A COMPASS

AND STRAIGHTEDGE.

TRY THIS.

TAKE A PENCIL, RULER, PROTRACTOR AND COMPASS

AND TRY TO CONSTRUCT A REGULAR POLYGON.

CONSTRUCTING A SQUARE IS EASY,

BUT STARTING WITH A PENTAGON

THE TASK IS SIGNIFICANTLY MORE CHALLENGING.

HOW CAN YOU PRECISELY DRAW A REGULAR POLYGON?

HOW CAN YOU MAKE SURE THAT THE SIDE LENGTHS

ARE THE SAME LENGTH?

HOW CAN YOU MAKE SURE THAT THE ANGLE MEASURES

ARE THE SAME?

MOST IMPORTANT, HOW CAN YOU MAKE IDENTICAL COPIES

OF THE POLYGON SHAPE FOR THE NUMEROUS CERAMIC TILES

THAT WOULD BE USED TO COVER A BUILDING?

THE ISLAMIC WORLD DURING THE MIDDLE AGES

WAS AT ITS ARTISTIC AND INTELLECTUAL HEIGHTS.

ISLAMIC SCHOLARS READ AND INCORPORATED

THE WORKS OF THE ANCIENT GREEKS,

AND THIS OF COURSE INCLUDED THE WRITINGS OF EUCLID.

SO KEY CONCEPTS FROM GEOMETRY

BECAME PART OF THE ISLAMIC WORLD.

THIS IS ESPECIALLY HELPFUL BECAUSE EUCLID OFFERS

A CLEAR WAY TO CREATE REGULAR POLYGONS

USING ONLY A COMPASS AND STRAIGHTEDGE.

ISLAMIC ARTISANS USED THESE TECHNIQUES EXTENSIVELY.

THROUGHOUT THE ISLAMIC WORLD ARE MOSQUES

AND OTHER BUILDINGS WITH INTRICATE CERAMIC TILES

WITH COLORFUL ARABESQUES.

EUCLID SHOWED THAT WHEN IT COMES TO CONSTRUCTING

A REGULAR POLYGON, A COMPASS AND STRAIGHTEDGE

ARE HARDLY LIMITATIONS.

LET'S TAKE A LOOK AT HOW TO CONSTRUCT

A REGULAR HEXAGON.

THE METHOD OF CONSTRUCTING A REGULAR POLYGON

BEGINS WITH A CIRCLE.

AND A CIRCLE IS WHAT A COMPASS

IS IDEAL FOR CONSTRUCTING.

FOR THIS ACTIVITY YOU WILL NEED A COMPASS, RULER,

PENCIL AND PAPER.

PAUSE THE VIDEO AS NEEDED TO COMPLETE EACH STEP.

START BY CONSTRUCTING A CIRCLE

AND MAKE A NOTE OF ITS CENTER.

USE THE STRAIGHTEDGE TO DRAW A DIAMETER.

YOU NOW WANT TO CONSTRUCT

THE PERPENDICULAR BISECTOR OF THE DIAMETER.

YOU CAN DO THIS WITH THE COMPASS.

EXTEND THE COMPASS BEYOND THE LENGTH OF THE RADIUS.

PLACE ONE END OF THE COMPASS

ON THE CIRCLE THAT YOU JUST CONSTRUCTED

AT THE POINT WHERE THE DIAMETER

INTERSECTS THE CIRCLE.

CONSTRUCT PAIRS OF INTERSECTING ARCS

ABOVE AND BELOW THE CIRCLE AS SHOWN.

NEXT, USE THE STRAIGHTEDGE TO DRAW THE PERPENDICULAR

BISECTOR THROUGH THE INTERSECTING ARCS

IN THE CENTER OF THE CIRCLE.

RE-SIZE THE COMPASS TO THE SIZE OF THE CIRCLE'S RADIUS.

PLACE ONE END OF THE COMPASS

WHERE THE DIAMETER INTERSECTS THE CIRCLE.

CONSTRUCT ANOTHER CIRCLE

THAT INTERSECTS THE FIRST CIRCLE.

NOTICE THAT THERE ARE TWO POINTS OF INTERSECTION.

NOW TAKE THE STRAIGHTEDGE AND

CONSTRUCT THE FOLLOWING QUADRILATERAL.

SIMPLY CONNECT THE INTERSECTION POINTS

AS SHOWN.

NOTICE THAT ALL THESE SEGMENT LENGTHS

ARE CONGRUENT.

IN ALL CASES A SEGMENT IS THE RADIUS

OF ONE OF THE CIRCLES.

THIS MEANS THAT THIS QUADRILATERAL

IS A RHOMBUS, A TYPE OF PARALLELOGRAM

WITH ALL SIDES THE SAME LENGTH.

THE STEPS IN CREATING THE RHOMBUS CAN BE REPEATED

AGAIN AT THE OTHER END OF THE DIAMETER

OF THE FIRST CIRCLE.

SO PLACE THE COMPASS AT THE SECOND

INTERSECTION POINT AND CONSTRUCT A THIRD CIRCLE

WITH THE SAME RADIUS AS THE OTHER TWO CIRCLES.

NEXT, USE THE STRAIGHTEDGE

TO CONSTRUCT ANOTHER RHOMBUS

WHICH INTERSECTS THE FIRST RHOMBUS AT ONE POINT.

FINALLY, USE THE STRAIGHTEDGE AGAIN

TO CONSTRUCT THE TWO EQUILATERAL TRIANGLES

AS SHOWN.

BY HIGHLIGHTING THE EXTERIOR STRAIGHT SIDES

YOU CAN SEE A HEXAGON.

BUT THIS ISN'T JUST ANY HEXAGON.

IT'S A REGULAR HEXAGON WITH EACH SIDE

THE SAME LENGTH THE SIZE OF THE RADIUS.

THE INTERIOR ANGLES OF THE HEXAGON ARE CONGRUENT,

BUT WHAT IS THE ANGLE MEASURE?

NOTICE THAT YOU HAVE ALSO CONSTRUCTED

SIX EQUILATERAL TRIANGLES.

AN EQUILATERAL TRIANGLE IS ALSO A REGULAR POLYGON.

ALL SIDES ARE CONGRUENT AS ARE ALL ANGLE MEASURES.

SINCE THE SUM OF THE ANGLES OF A TRIANGLE

EQUALS 180 DEGREES, AND SINCE THE ANGLE MEASURES

OF AN EQUILATERAL TRIANGLE ARE CONGRUENT,

THIS GIVES RISE TO THIS EQUATION.

3X = 180 DEGREES, SO X = 60 DEGREES.

THIS MEANS THAT THE HEXAGON HAS THIS ARRAY

OF INTERIOR ANGLES.

THE TWELVE 60 DEGREE ANGLES EQUAL A SUM OF 720 DEGREES.

IN FACT, A SIMPLE WAY TO CALCULATE THE SUM OF THE

INTERIOR ANGLES OF ANY POLYGON RELIES ON THE FACT

THAT A POLYGON, WHETHER REGULAR OR NOT,

HAS TRIANGLES EMBEDDED IN IT.

START AT ONE OF THE VERTICES OF THE POLYGON

AND DRAW DIAGONALS FROM THAT VERTEX

TO THE OTHER VERTICES.

THIS CREATES AN ARRAY OF EMBEDDED TRIANGLES.

SINCE THE INTERIOR TRIANGLES OF EACH TRIANGLE

IS 180 DEGREES, THEN THE SUM OF THE INTERIOR ANGLES

OF THE POLYGON IS 180 X THE NUMBER OF TRIANGLES.

THIS IS SUMMARIZED WITH THE FOLLOWING FORMULA.

FOR ANY (N) SIDED POLYGON THE SUM OF THE

INTERIOR ANGLES IS (N-2) TIMES 180 DEGREES.

SO SIMPLY USING A COMPASS AND STRAIGHTEDGE,

AN ISLAMIC ARTISAN COULD CREATE PRECISE DIAGRAMS

OF REGULAR POLYGONS AND CREATE ELABORATE ARABESQUES.

THESE TECHNIQUES CAN ALSO BE USED ON THE TI-NSPIRE.

LET'S USE THE NSPIRE TO CREATE A REGULAR POLYGON.

TURN ON THE TI-NSPIRE.

CREATE A NEW DOCUMENT.

YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT.

CREATE A GEOMETRY WINDOW.

MAKE SURE THE VIEW SHOWS THE PLANE GEOMETRY VIEW.

START BY CONSTRUCTING A CIRCLE.

PRESS MENU AND UNDER SHAPES SELECT CIRCLE.

USE THE NAV PAD TO MOVE THE POINTER

TO THE MIDDLE OF THE SCREEN.

PRESS ENTER TO DEFINE THE RADIUS OF THE CIRCLE.

NEXT, USE THE NAV PAD TO MOVE THE POINTER

AWAY FROM THE CENTER.

NOTICE HOW THE CIRCLE TAKES SHAPE.

AFTER MOVING A QUARTER OF THE SCREEN AWAY,

PRESS ENTER TO DEFINE THE CIRCLE.

RECALL THAT WITH THE COMPASS AND STRAIGHTEDGE

DRAWING A DIAMETER WAS A STRAIGHTFORWARD PROCESS.

WITH THE NSPIRE THERE ARE SEVERAL WAYS OF DOING THIS.

THE MOST STRAIGHTFORWARD IS TO USE THE LINE TOOL.

PRESS MENU AND UNDER "POINTS & LINES" SELECT LINE.

MOVE THE POINTER TO THE CIRCLE ITSELF

UNTIL YOU SEE THE LABEL "POINT ON".

PRESS ENTER.

MOVE THE POINTER TO THE CENTER OF THE CIRCLE.

MAKE SURE THE CENTER POINT IS HIGHLIGHTED.

YOU'LL SEE THE LABEL "POINT" APPEAR ON SCREEN.

PRESS ENTER.

PRESS THE ESCAPE KEY AND MOVE THE POINTER

TO THE END OF THE LINE YOU'VE JUST CREATED.

PRESS AND HOLD THE CLICK KEY TO SELECT THE CIRCLE.

MOVE THE POINTER TO THE OTHER END OF THE CIRCLE

TO EXTEND THE LINE.

PRESS ENTER.

NEXT, CREATE A LINE

PERPENDICULAR TO THE DIAMETER.

PRESS MENU AND UNDER "CONSTRUCTION"

SELECT PERPENDICULAR.

MOVE THE POINTER ALONG THE DIAMETER

AND STOP WHEN THE CENTER IS HIGHLIGHTED.

PRESS ENTER TWICE TO CONSTRUCT THE

PERPENDICULAR DIAMETER.

YOU NOW HAVE TWO PERPENDICULAR DIAMETERS

THAT INTERSECT THE CIRCLE.

NOTICE THAT NOT ALL THE INTERSECTIONS

HAVE A POINT ASSOCIATED WITH THEM.

LET'S ADD POINTS AT ALL THESE INTERSECTIONS.

PRESS MENU AND UNDER "POINTS & LINES"

SELECT INTERSECTION POINTS.

SELECT A DIAMETER AND A CIRCLE.

MOVE THE POINTER ABOVE THE DIAMETER AND PRESS ENTER.

THEN MOVE THE POINTER ABOVE THE CIRCLE

AND PRESS ENTER AGAIN.

REPEAT WITH THE OTHER DIAMETER AND THE CIRCLE.

YOU NOW HAVE THE FRAMEWORK NEEDED FOR

CONSTRUCTING A REGULAR POLYGON ON THE NSPIRE.

YOU MAY WANT TO SAVE THIS FILE AS A TEMPLATE

FOR OTHER POLYGON CONSTRUCTION ACTIVITIES.

IN THE CASE OF A REGULAR PENTAGON,

THE FINISHED POLYGON INSCRIBED IN THE CIRCLE

LOOKS LIKE THIS.

TO CONSTRUCT THE FIRST SIDE OF THE PENTAGON

YOU WILL NEED TO FIND THE MIDPOINT OF THIS RADIUS.

YOU THEN CONSTRUCT A SEGMENT

FROM THE MIDPOINT TO THIS POINT ON THE CIRCLE.

NEXT, FIND THE ANGLE BISECTOR FOR THIS ANGLE.

FROM THE POINT WHERE THE ANGLE BISECTOR

INTERSECTS THE VERTICAL DIAMETER, CONSTRUCT A LINE

PARALLEL TO THE HORIZONTAL DIAMETER.

WHERE THIS PARALLEL LINE INTERSECTS THE CIRCLE

IS ALSO ONE OF THE VERTICES OF THE PENTAGON.

SIMPLY CONNECT THIS POINT

AND THE TOP POINT ON THE CIRCLE.

LET'S CONTINUE WITH THE NSPIRE

TO CONSTRUCT THE FIRST SIDE OF THE PENTAGON.

FIND THE MIDPOINT OF THE HORIZONTAL RADIUS

ON THE LEFT.

PRESS MENU AND UNDER CONSTRUCTION

SELECT MIDPOINT.

YOU NEED TO DEFINE THE TWO POINTS OF THE SEGMENT

FOR FINDING THE MIDPOINT, SO USE THE NAV PAD

TO MOVE THE POINTER ABOVE THE CENTER OF THE CIRCLE.

PRESS ENTER.

THEN MOVE THE POINTER TO THE LEFT

TO HIGHLIGHT THE POINT ON THE CIRCLE.

PRESS ENTER AGAIN.

YOU SHOULD NOW SEE THE MIDPOINT

OF THE HORIZONTAL RADIUS.

NOW CONSTRUCT A SEGMENT FROM THE MIDPOINT

TO THE UPPER POINT ON THE CIRCLE.

PRESS MENU AND UNDER "POINTS & LINES"

SELECT SEGMENT.

MOVE THE POINTER ABOVE THE MIDPOINT AND PRESS ENTER.

THEN MOVE THE POINTER ABOVE THE SECOND POINT

AND PRESS ENTER AGAIN.

NOTICE THE ANGLE FORMED WITH THE VERTEX

AT THE MIDPOINT.

NEXT, CONSTRUCT THE ANGLE BISECTOR.

PRESS MENU AND UNDER CONSTRUCTION

SELECT ANGLE BISECTOR.

WITH THE ANGLE BISECTOR TOOL YOU NEED TO DEFINE

THREE POINTS THAT INCLUDE THE ANGLE, SO USE THE NAV PAD

TO MOVE THE POINTER TO THE TOP POINT ON THE CIRCLE.

PRESS ENTER.

THEN MOVE THE POINTER TO THE VERTEX OF THE ANGLE

OVER THE MIDPOINT AND PRESS ENTER AGAIN.

FINALLY, MOVE THE POINTER OVER THE CENTER.

PRESS ENTER.

YOU WILL SEE THE ANGLE BISECTOR

INTERSECT THE VERTICAL RADIUS.

CREATE AN INTERSECTION POINT.

PRESS MENU AND UNDER "POINTS & LINES"

SELECT INTERSECTION POINT.

MOVE THE POINTER OVER THE ANGLE BISECTOR

AND PRESS ENTER.

THEN MOVE THE POINTER ABOVE THE VERTICAL RADIUS.

PRESS ENTER AGAIN.

FINALLY, CREATE A LINE PARALLEL TO THE

HORIZONTAL DIAMETER THAT CROSSES THE

INTERSECTION POINT YOU'VE JUST CREATED.

PRESS MENU AND UNDER CONSTRUCTION

SELECT PARALLEL.

MOVE THE POINTER ABOVE THE HORIZONTAL DIAMETER

AND PRESS ENTER.

THEN MOVE THE POINTER ABOVE THE INTERSECTION POINT

AND PRESS ENTER AGAIN.

YOU SHOULD SEE THE PARALLEL LINE.

WHERE THIS LINE INTERSECTS THE CIRCLE

IS WHERE THE PENTAGON INTERSECTS THE CIRCLE TOO.

USE THE SEGMENT TOOL TO CONNECT THESE POINTS

TO THE UPPER POINT ON THE CIRCLE.

YOU NOW HAVE TWO OF THE FIVE SIDES OF THE PENTAGON.

BUT MOST IMPORTANT, YOU HAVE A METHOD FOR

CREATING ALL SIDES OF THE PENTAGON.

YOU CAN APPLY THIS SAME METHOD

TO CREATE THE REMAINING SIDES.

BUT FIRST CLEAR THE SCREEN

OF THE LINES YOU JUST CREATED,

KEEPING THE TWO SIDES OF THE PENTAGON.

PRESS THE ESCAPE KEY.

HOVER OVER EACH LINE, SEGMENT AND POINT

YOU WANT TO HIDE.

PRESS CONTROL AND MENU AND SELECT THE HIDE OPTION.

DO THIS UNTIL WHAT YOU HAVE VISIBLE ARE THE CIRCLE,

THE CENTER AND THE TWO SIDES OF THE PENTAGON.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

NOW STARTING AT THE PENTAGON VERTEX ON THE

UPPER LEFT SIDE OF THE CIRCLE, CONSTRUCT A LINE

FROM THIS POINT TO THE CENTER OF THE CIRCLE.

PRESS MENU AND UNDER "POINTS & LINES"

SELECT LINE.

MAKE SURE THE LINE CROSSES THE TWO POINTS.

ALSO, EXTEND THE LINE SO THAT IT INTERSECTS

THE CIRCLE TWICE.

THEN ADD AN INTERSECTION POINT.

CONSTRUCT THE LINE

PERPENDICULAR TO THE DIAMETER.

PRESS MENU AND UNDER CONSTRUCTION

SELECT PERPENDICULAR.

MAKE SURE THE PERPENDICULAR LINE

CROSSES THE CENTER OF THE CIRCLE.

THEN ADD INTERSECTION POINTS

TO THIS PERPENDICULAR LINE.

FIND THE MIDPOINT OF THE LOWER RADIUS.

PRESS MENU AND UNDER CONSTRUCTION

SELECT MIDPOINT.

AFTERWARD, CONSTRUCT A LINE SEGMENT FROM THE

MIDPOINT TO THE LOWER POINT ON THE CIRCLE

THAT INTERSECTS THE PERPENDICULAR LINE.

NEXT, CONSTRUCT THE ANGLE BISECTOR

OF THE ANGLE FORMED FROM THE MIDPOINT.

ADD AN INTERSECTION POINT WHERE THE ANGLE BISECTOR

INTERSECTS THE RADIUS.

CREATE THE LINE PARALLEL TO THE LOWER RADIUS

THAT CROSSES THE INTERSECTION POINT

OF THE ANGLE BISECTOR.

WHERE THE PARALLEL LINE INTERSECTS THE CIRCLE

IS THE NEXT PENTAGON VERTEX.

USE THE SEGMENT TOOL TO CONSTRUCT THE THIRD SIDE.

COMPLETE THE NEXT SIDE OF THE PENTAGON

USING THE SAME METHOD.

BE SURE TO HIDE THE PREVIOUSLY CREATED

POINTS, LINES AND SEGMENTS,

KEEPING ONLY THE CIRCLE, CENTER, AND PENTAGON SIDES.

ONCE YOU HAVE CREATED THE FOURTH SIDE YOU WILL ALSO

HAVE CONSTRUCTED THE FIFTH SIDE BY SIMPLY CREATING

A SEGMENT CONNECTING THE REMAINING TWO SIDES.

WHEN YOU ARE DONE

YOUR SCREEN SHOULD LOOK LIKE THIS.

MANY OF THE INTRICATE POLYGON-BASED PATTERNS

IN ISLAMIC ART FORM WHAT ARE CALLED TESSELLATIONS.

A TESSELLATION IS FORMED FROM REPEATED USES

OF A SHAPE, OR SHAPES, TO CREATE A TILED SURFACE

WITH NO GAPS BETWEEN TILES.

A REGULAR TESSELLATION IS MADE UP OF REGULAR POLYGONS.

FOR EXAMPLE, REGULAR HEXAGONS

FORM THIS TYPE OF TESSELLATION.

HERE IS A TESSELLATION MADE UP OF OCTAGONS

AND QUADRILATERALS.

HERE IS AN ISLAMIC TILE PATTERN

THAT INCLUDES OVERLAPPING TESSELLATIONS.

HERE YOU CAN SEE RECTANGLES, TRAPEZOIDS,

PARALLELOGRAMS, TRIANGLES AND HEXAGONS.

FINALLY, THERE ARE ALSO SHAPES USED IN ISLAMIC ART

THAT RELY ON AN UNDERLYING POLYGON SHAPE

ALTHOUGH THE POLYGON ISN'T VISIBLE.

LET'S USE THE NSPIRE TO CONSTRUCT A

SIX-PETAL FLOWER SHAPE.

CREATE A NEW GEOMETRY WINDOW.

YOU COULD ADD A PAGE TO YOUR EXISTING DOCUMENT

WITH A NEW GEOMETRY WINDOW.

PRESS THE HOME KEY AND SELECT GEOMETRY.

MAKE SURE THAT THE PLANE GEOMETRY VIEW IS SELECTED.

CREATE A LINE SEGMENT.

PRESS MENU AND UNDER "POINTS & LINES"

SELECT SEGMENT.

CREATE A HORIZONTAL SEGMENT

ACROSS THE MIDDLE PART OF THE SCREEN.

NEXT, ACTIVATE THE CIRCLE TOOL.

CREATE A CIRCLE WHOSE CENTER IS IN THE MIDDLE

OF THE LINE SEGMENT.

THEN MOVE THE POINTER TO THE LEFT

TO THE POINT ON THE CIRCLE

WHERE IT INTERSECTS THE HORIZONTAL LINE.

MAKE THIS THE CENTER OF A NEW CIRCLE

BY PRESSING ENTER.

MOVE THE POINTER TO THE CENTER OF THE CIRCLE

AND PRESS ENTER AGAIN.

YOU NOW HAVE TWO OVERLAPPING CIRCLES.

THE CENTER OF EACH CIRCLE IS PART OF THE OTHER CIRCLE.

REPEAT THIS FOR THE OTHER POINT

WHERE THE FIRST CIRCLE INTERSECTS THE LINE.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

NOTICE THERE ARE FOUR POINTS OF INTERSECTION

AMONG THE THREE CIRCLES YOU CONSTRUCTED.

CONTINUING WITH THE CIRCLE TOOL,

LET EACH OF THOSE POINTS BE THE CENTER OF A NEW CIRCLE

THAT EXTENDS AS FAR AS THE CENTER OF THE FIRST CIRCLE.

YOU WILL BE CONSTRUCTING FOUR NEW CIRCLES

AS SHOWN HERE.

MAKE SURE NOT TO CREATE NEW POINTS, BUT SIMPLY

USE THE EXISTING POINTS TO CREATE THE NEW CIRCLES.

IF YOU MAKE A MISTAKE ALONG THE WAY, YOU CAN

ALWAYS PRESS CONTROL AND Z TO UNDO A STEP.

YOUR SCREEN SHOULD LOOK LIKE THIS.

NOTICE THE FLOWER PETAL SHAPE THAT EMERGES

FROM THESE INTERSECTING CIRCLES.

THIS IS A TYPICAL TEMPLATE THAT COULD BE USED

FOR CREATING AN ARABESQUE.

SO THE TECHNIQUES THAT EUCLID DEVISED

FOR CONSTRUCTING REGULAR POLYGONS

GAVE ISLAMIC ARTISANS A METHOD FOR CREATING

ENDLESS NUMBERS OF DESIGNS AND ALL WITH

PRECISE GEOMETRY THAT COULD BE REPLICATED.

THIS ALLOWED FOR THE MASS PRODUCTION OF CERAMIC TILES.

BY SIMPLY TAKING THE GEOMETRIC PATTERN,

AN ARTISAN COULD CREATE A STENCIL PATTERN

FOR COPYING THE PATTERN TO A CERAMIC TILE,

PAINTING IT WITH GLAZE, APPLYING HEAT TO THE TILE

TO GIVE IT THE GLOSS OF A FINISHED TILE,

AND USING THE TILES TO CREATE A DECORATIVE PATTERN.

AND AS MARRAKESH SHOWS,

THERE ARE INFINITE DESIGNS POSSIBLE.

THE PETRONAS TOWERS IN KUALA LUMPUR

ARE THE PRIDE OF MALAYSIA.

THESE GRAND STRUCTURES

ARE THE WORLD'S TALLEST TWIN TOWERS.

THESE 88-FLOOR TOWERS TOOK SEVEN YEARS TO BUILD

AND ARE A TESTAMENT TO MALAYSIA'S ECONOMIC GROWTH

AS WELL AS THE GROWING INFLUENCE

OF THE ASIAN ECONOMY IN THE GLOBAL MARKETPLACE.

THE TWO TOWERS ARE CONNECTED BY

A TWO-LEVEL BRIDGE MIDWAY UP EACH TOWER,

MAKING IT THE HIGHEST TWO LEVEL BRIDGE IN THE WORLD.

PROVIDING A PANORAMIC VIEW OF KUALA LUMPUR,

THE SKYBRIDGE IS ALSO AN EMERGENCY EVACUATION

FROM ONE TOWER TO ANOTHER...

AN ISSUE OF CONCERN IN THE POST-911 WORLD.

MALAYSIA IS A PREDOMINANTLY

MUSLIM COUNTRY AND THIS IS REFLECTED

IN THE ARCHITECTURE OF THE PETRONAS TOWERS.

WHEN VIEWED FROM ABOVE, THE TOWERS REVEAL

A FAMILIAR PATTERN FOUND IN ISLAMIC ART.

NOTE THE OVERLAPPING SQUARES, WHICH IS A COMMON

DESIGN FOUND THROUGHOUT THE ISLAMIC WORLD.

THE CIRCULAR ARCS WERE ADDED TO THE DESIGN

OF THE PETRONAS TOWERS AND WE'LL ANALYZE THESE SHAPES

IN MORE DETAIL LATER.

BUT FOR NOW LET'S FOCUS ON THE OVERLAPPING SQUARES.

AS YOU SAW IN THE PREVIOUS SECTION,

ISLAMIC ARTISANS USED A COMPASS AND STRAIGHTEDGE

TO CREATE ELABORATE PATTERNS THAT WERE PRECISE

BUT COULD ALSO BE REPLICATED QUITE EASILY.

LET'S USE THE TI-NSPIRE TO CONSTRUCT AND EXPLORE THIS

STAR PATTERN MADE UP OF THE TWO OVERLAPPING SQUARES.

TURN ON THE TI-NSPIRE.

CREATE A NEW DOCUMENT.

YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT.

CREATE A GEOMETRY WINDOW AND MAKE SURE

THAT THE VIEW IS THE PLANE GEOMETRY VIEW.

CREATE A HORIZONTAL LINE SEGMENT.

PRESS MENU AND UNDER "POINTS & LINES"

SELECT SEGMENT.

NOW CONSTRUCT A LINE

PERPENDICULAR TO THIS LINE SEGMENT.

PRESS MENU AND UNDER "CONSTRUCTION"

SELECT PERPENDICULAR.

MOVE THE POINTER ABOVE THE HORIZONTAL LINE

AND PRESS ENTER.

CONSTRUCT A CIRCLE WHOSE CENTER IS AT THE POINT

WHERE THE TWO LINES INTERSECT.

PRESS MENU AND UNDER SHAPES SELECT CIRCLE.

AFTER CREATING A CIRCLE, CONSTRUCT TWO MORE CIRCLES

WHOSE CENTERS ARE AT THE POINTS WHERE THE FIRST

CIRCLE AND THE HORIZONTAL LINE SEGMENT INTERSECT.

THESE CIRCLES SHOULD HAVE THE SAME RADIUS AS THE

FIRST CIRCLE AND INTERSECT THAT CIRCLE'S CENTER.

REPEAT THIS PROCESS FOR THE TWO POINTS

WHERE THE VERTICAL LINE INTERSECTS THE FIRST CIRCLE.

IN EACH CASE THE NEW CIRCLE WILL BE CENTERED

AT THE INTERSECTION POINT AND ALSO TOUCH THE CENTER

OF THE FIRST CIRCLE.

YOUR SCREEN SHOULD LOOK LIKE THIS.

MAKE A NOTE OF THESE FOUR HIGHLIGHTED

INTERSECTION POINTS.

USE THE SEGMENT TOOL TO CREATE TWO LINE SEGMENTS

THAT INTERSECT AT THE CENTER OF THE FIRST CIRCLE.

PRESS MENU AND UNDER "POINTS & LINES"

SELECT SEGMENT.

CONNECT THE PAIRS OF POINTS AS SHOWN.

YOU NOW HAVE ALL THE POINTS NECESSARY

TO CONSTRUCT THE OVERLAPPING SQUARES.

LET'S START WITH THE SQUARE

THAT HAS THE HORIZONTAL BASE.

NOTE THESE FOUR HIGHLIGHTED POINTS.

CONTINUING WITH THE SEGMENT TOOL,

CREATE THE SQUARE DEFINED BY THESE FOUR POINTS.

NEXT, USING THESE FOUR HIGHLIGHTED POINTS

CONSTRUCT THE OTHER SQUARE.

THIS SHAPE, AND VARIATIONS OF IT,

ARE FOUND IN MANY ARABESQUES.

THE OUTLINE OF THIS SHAPE IS KNOWN AS A

COMPOSITE FIGURE.

A COMPOSITE FIGURE IS A COMPLEX SHAPE

MADE UP OF SIMPLER GEOMETRIC SHAPES.

IN THE CASE OF THE ISLAMIC STAR THE PRIMARY SHAPES

INCLUDE THE SQUARE WITH THE HORIZONTAL BASE

AND FOUR ISOSCELES TRIANGLES.

THE TOTAL AREA OF THIS COMPOSITE FIGURE

IS THE SUM OF THE AREA OF THE LARGE SQUARE

AND THE COMBINED AREAS OF THE ISOSCELES TRIANGLES.

BUT HOW CAN WE CALCULATE THE AREA OF THE TRIANGLES?

FOR THAT, LET'S LOOK AT A VARIATION

OF THIS COMPOSITE FIGURE.

RATHER THAN LOOKING AT IT AS A COMBINATION

OF ONE SQUARE AND FOUR TRIANGLES,

NOTE THAT THERE IS ALSO A REGULAR HEXAGON.

LOOKED AT THIS WAY,

THE COMPOSITE FIGURE IS ONE HEXAGON

AND SIX CONGRUENT ISOSCELES TRIANGLES.

LET'S LOOK AT THIS TRIANGLE IN MORE DETAIL.

RECALL THAT FOR AN ISOSCELES TRIANGLE,

TWO SIDES AND TWO ANGLES ARE CONGRUENT.

BUT THIS ISOSCELES TRIANGLE IS A CORNER OF A SQUARE

SO THIS IS A SPECIAL TYPE OF ISOSCELES TRIANGLE...

NAMELY AN ISOSCELES RIGHT TRIANGLE.

THIS IS ALSO KNOWN AS THE 45-45-90 DEGREE

RIGHT TRIANGLE.

THESE RIGHT TRIANGLES HAVE SIDE LENGTHS THAT

FORM THE RATIO ONE, ONE, SQUARE ROOT OF TWO.

SUPPOSE THAT THE SIDE LENGTH OF THIS TRIANGLE IS S.

THEN THE LENGTH OF A HYPOTENUSE

IS SQUARE ROOT OF 2S.

THE AREA OF THIS ISOSCELES TRIANGLE IS ONE HALF BASE

TIMES HEIGHT, OR 1/2 S SQUARED.

GOING BACK TO THE COMPOSITE FIGURE, WE CAN SEE

THAT THE LARGE SQUARES HAVE THESE SIDE LENGTHS.

THE TOTAL AREA OF THE COMPOSITE FIGURE

IS THE SUM OF THE SQUARE AND FOUR TIMES THE AREA

OF ONE OF THE ISOSCELES TRIANGLES.

THE SOLUTION IS SHOWN HERE.

ANOTHER WAY TO CALCULATE THE AREA

IS TO USE THE PROPERTIES OF REGULAR POLYGONS.

FOR ANY REGULAR POLYGON THINK OF A CIRCLE

IN WHICH THE POLYGON IS INSCRIBED.

THE INTERIOR ANGLES ARE CONGRUENT AND THE MEASURE

OF EACH ANGLE IS FOUND USING THIS FORMULA

WHERE N IS THE NUMBER OF SIDES OF THE POLYGON.

THERE ARE N ISOSCELES TRIANGLES.

THE BASE OF EACH TRIANGLE IS ONE OF THE EDGES

OF THE POLYGON OF LENGTH S.

THE ALTITUDE OF EACH TRIANGLE, ALSO REFERRED TO

AS THE APATHEM, IS PERPENDICULAR TO SIDE S.

SO THE AREA OF EACH TRIANGLE IS ONE-HALF AS.

SINCE A IS ONE OF THE LEGS OF A RIGHT TRIANGLE,

THEN YOU CAN SOLVE FOR A IN TERMS OF N AND S.

USE THE PYTHAGOREAN THEOREM AND SOLVE FOR A.

WE GET A EQUALS THE SQUARE ROOT OF THE QUANTITY

R SQUARED MINUS S OVER TWO SQUARED

WHERE R IS THE RADIUS OF THE CIRCLE

AND S IS THE LENGTH OF THE POLYGON SIDE.

USING THIS EXPRESSION FOR A IN THE AREA FORMULA

WE GET THIS EQUATION AND THIS EQUATION CAN BE USED

TO FIND THE AREA OF ANY REGULAR POLYGON.

THIS FORMULA COULD BE USED TO FIND THE AREA

OF THE REGULAR HEXAGON SHOWN EARLIER.

THE DESIGN OF THE PETRONAS TOWERS

USES A VARIATION OF THE ISLAMIC SHAPE.

NOTICE THE ADDITIONAL CIRCULAR SECTIONS.

THESE WERE ADDED TO INCREASE THE AMOUNT OF

OFFICE SPACE AVAILABLE, SO THIS FORMS A NEW

COMPOSITE FIGURE SHOWN HERE.

THE QUESTION BECOMES, HOW MUCH MORE AREA HAS BEEN

ADDED TO THE AREA OF THE ORIGINAL COMPOSITE FIGURE?

LET'S ANALYZE THE FIGURES MORE CLOSELY.

A CLOSE UP SHOWS THAT THE CIRCULAR REGION

IS PART OF A CIRCLE WHOSE CENTER IS AT

THE CORNER OF TWO ADJACENT TRIANGLES.

THE RADIUS EXTENDS HALF THE LENGTH OF THE TRIANGLE SIDE.

THIS MEANS THAT THE CIRCLE HAS A RADIUS OF S OVER 2.

THE AREA OF THE FULL CIRCLE IS PI S SQUARED OVER 4.

BUT ONLY A PORTION OF THE CIRCULAR REGION IS USED.

WHAT PORTION? WE KNOW THAT THE INTERIOR ANGLE

OF THE ISOSCELES TRIANGLE IS 45 DEGREES.

THIS MEANS THAT THE CIRCLE

SWEEPS OUT A REGION OF 135 DEGREES.

SINCE A FULL CIRCLE SWEEPS OUT 360 DEGREES,

THE FRACTIONAL AMOUNT THAT THIS PORTION OF THE CIRCLE

SWEEPS OUT IS 135 OVER 360, OR THREE-EIGHTHS.

SO THE AREA OF ONE CIRCULAR SECTION IS THREE-EIGHTHS

OF PI S SQUARED OVER 4, OR THREE PI S SQUARED OVER 32.

AND NOW WE CAN CALCULATE THE TOTAL AREA OF THE

COMPOSITE FIGURE THAT MAKES UP ONE FLOOR IN EACH TOWER.

TAKE THE TOTAL AREA YOU FOUND FOR THE

OVERLAPPING SQUARE SECTION AND ADD IT

TO THE COMBINED AREAS OF THE CIRCULAR SECTIONS.

THE TOTAL AREA IS THIS EXPRESSION.

WHAT PERCENT INCREASE DOES THE ADDITION

OF THE CIRCULAR SECTIONS REPRESENT?

FOR THAT, LET'S USE THE PERCENT INCREASE FORMULA.

DIVIDE THE TOTAL AREA OF THE CIRCULAR SECTIONS

BY THE TOTAL AREA OF THE OVERLAPPING SQUARES.

YOU CAN CREATE A CALCULATOR WINDOW IN THE NSPIRE

TO CALCULATE THE PERCENT INCREASE.

IT TURNS OUT TO BE A ROUGHLY 17% INCREASE IN AREA.

THIS IS A SIZABLE INCREASE IN THE SQUARE FOOTAGE OF

THE AREA, YET THE ADDITION OF THE CIRCULAR SECTIONS

DOES NOT INTERFERE WITH THE OVERALL DESIGN OF THE TOWER.

AND FROM A DISTANCE THESE CIRCULAR SECTIONS

ADD A SMOOTHNESS TO THE LOOK OF THE TOWERS.

COMPOSITE FIGURES PROVIDE A WAY TO CREATE

A VARIETY OF NEW SHAPES WHILE STILL MAINTAINING

THE PROPERTIES OF THE ORIGINAL SHAPES.

AND THE PETRONAS TOWERS PROVIDE A

SOPHISTICATED USE OF SUCH COMPOSITE FIGURES.