Isosceles triangle, Golden ratio, Great Pyramid at Giza & Penrose Tiling

7/28/2019 Isosceles triangle, Golden ratio, Great Pyramid at Giza & Penrose Tiling

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ISOSCELESTRIANGLE,GoldenTriangle,FibonacciSeries,GreatPyramidof

GizaandTilingsDrNKSrinivasanIntroductionThismathtutorialisdesignedtotakeyouthroughall

thesetopicsandshowhowsomeexcellentmath

relationshipsareusedinreallife.Iwillshowyouwhy

theGreatPyramidofGizaiaslantedatanangleof

nearly52degrees[51.83degreestobeexact.]Youwill

alsolearnabitoftrigonometry,ifyouhavenot

learntoruseditmuchsofar.Whatisanisoscelestriangle?Torecallyourmiddleschoolgeometry,"Isosceles

triangle"isatriangleinwhichtwosidesareofequal

length.IfweconstructatriangleABCwithAB=AC,andthe

baseBCnotequaltoABorAC,wehaveanisosceles

triangle.


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WewillcallthevertexAastheapexandBCasthe

base.Iwillaskyoutodrawormakeconstructionaswego

alonginapaper.Keepapaper,rule,protractorsndan

eraserready!IsoscelesRightTriangleWewillbeginwithaverysimpleisoscelestriangle.Drawasquareofside,say3inches;callitsquare

ABCD.Drawoneofthediagonals,sayAC.Nowyouhavemadetwo'congruent'righttriangleswith

twosidesequal.Thesidesare,ofcourse3inches,with

90degreesandthehypotenuseisthediagonalwhich

willbe:AC=2x3[SquarerootIwillwriteasfollows:sqrt(2)in

future.]SomePropertiesofIsoscelestriangles.Youcancalltheseproperties'theorems'asyour

geometrytextbookwillstate.Ijustcallthem


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'properties'.1Thebaseanglesofanisoscelestriangleareequal.IfyouhavedrawnatrianglewithAastheapexandAB=

AC,andBCasthebase,thenangleABC=angleACB.Iftheapexangleis36degrees,thebaseanglesare:(1/2)[180-36]=72degrees.2DrawthealtitudeorheightofthetriangleADfrom

theapexAtothebase.YouwillnoticethatDisthe

midpointofthebaseBCorBD=DC.[SinceADisalsoperpendiculartoBC,wesaythatADis

the'perpendicularbisector'ofbaseBC.]Nowyouhavedividedthetriangleintotwocongruent

righttriangles,ABDandACD---withthesamelegsand

hypotenusesABandAC.Youwillalsofindthatyouhavebisectedtheapex

angle:angleBAD=angleCAD.

Thesetwopropertiesarethemainonesrequiredforyour

workahead.


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GoldenTriangleConstructatriangleasfollows:--DrawalineBCoflengthsay5inches.---Findthemidpointanderectaprependicularline.---Usingaprotractor,drawthelinesBAandCA

enclosinganangleof72degreeswiththebaselineBC.TheApexAisnowlocatedbyyou.----Measuretheapexangle.[Itshouldbecloseto36degrees...nosurprisehere

becausetheapexangle=180-72-72=36.Nowyouhaveconstructeda"Goldentriangle"with

36-72-72angles.Nowwecanfindsomeinterestingpropertiesofthis"G

Triangle".---MeasurethedistancesBAandCA.Theyareequal.FindtheratioofBA/BC.Youwillfindthatitisequal

to[nearly]1.618[or1.62]whichiscalledthe"Golden

Ratio"--usuallywrittenas"Phi".Notethat1/Phi=1/1.618=0.618.{ActuallyPhiisanirrationalnumberbutwetakeitas


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1.618forourcalculations...]GoldenRatio

Whatisthis"GoldenRatio"?

YoumighthavereadthattheGreekswere

fascinatedbythisratio.Theyhadagreat

senseofbeautyoraestheticsfornice

buildingsandsculpturesandtheratioof

widthtoheightofthebuildingstheywould

liketoconstructwiththisgoldenratio---

suchasParthenoninAthensthattheratio

ofheighttowidth=h/w=1.618.

Youmustknowthatthisfascinationfor

"goldenratio"wasfoundinearlier

civilizationstoo...theSumeriansandthe

Egyptiansdidusethisratioaswell.


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DrawalineABandlocateapointCbetween

AandBsuchthat

AC/BC=AB/AC.-----------------------X----------------.

ACB

ThisratiowouldbetheGoldenratioor

goldensection.IfwecallAC=aandBC=b,itfollows:

a/b=(a+b)/a=1+(b/a)

Ifwecalla/b=x,weget

x=1+(1/x)orx=(1+x)/x

Simplifyingweget:x2=1+x

orx2-x-1=0Weshallreturntothisequationlater.

Xisthe'GoldenRatio'.


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FibonacciSeries

FibonacciwasanItalianmathematician,

muchlaterin13thCentury,whostudiedthe

numberofrabbitsbornfromonegeneration

tothenext...thesenumbersforman

interestingsequenceorseries.

Youcanbuildupthesequenceveryeasily:

Startwith0and1;addtheprevioustwo

numbers:

0+1=1

1+1=2

2+1=3

3+2=5

5+3=8

8+5=13


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13+8=21

21+13=34

-----andsoon.Thenumbersgetbiggerandbiggerand

animalfarmersknowprettywellthatrabbits

breedvaryfast!

Letuswriteoutaformulaforthisseries: F(n+2)=F(n+1)+F(n)

Thismeansthatthen+2termisthesumof

theprevioustwoterms,namely(n+1)term

andnthterm.

Whilethesenumberskeepgrowing,theratiooftwo

consequentnumbersreachesalimit---Thisiscalled'convergence'ofaseries.LetuswritetheFibonacciseriesfirstforeaseof

calculations:0,1,2,3,5,8,13,21,34,55,89,144.......Taketheratiooftwoconsecutivenumbers:8/5=1.6


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13/8=1.625---------55/34=1.617689/55=1.6181818144/89=1.61797Youcanseenowthattheratio'converges'to1.618.ThisnumberistheGoldenRatio='Phi'=1.61803.....NatureseemstousethisFibosequenceinseveral

growthpatterns--suchaspetalsofflowers.Youmaylike

toreadabouttheminotheressays.GoldenRatioandAlgebraYoumaybefamiliarwith'solvingquadraticequations'

usingstrangeformulasandgetting'roots'ofequations.TaketheequationgivenearlierfortheGoldenRatio:x2-x-1=0Youcannotsolvethisequationbyfactorizing.Soweuse

thewell-known'QuadraticFormula':Ifax2+bx+c=0,thenthenx=[-b+/-Sqrt(b.b-4ac)]/2


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Intheequationgivenabove,a=1b=-1c=-1.Therootsare:x=[1+/-sqrt(5)]/2=1.6180=PhiTakingthepositiveroot,wegetphi=(1+5)/2.Notethatsqrt(5)isanirrationalnumber:findthis

numberfromyourcalculator!:sqrt(5)=2.236079.....=

2.236(nearly)Phi=3.236...../2=1.618033988......WegottheGoldenRatiobysolvingthisequation.Tounderstandthisequation,letusnotethatPhi2=1+PhiorPhi2-Phi=1

orPhi2

-Phi-1=0

ThisistheequationwestartedwithxreplacingPhi.[Youmayliketorememberthisimportantrelationabout

theGoldenRatio,Phi:1+Phi=Phi2Dividingbyphi:1/phi+1=phior1/phi=phi-1=0.618GoldenRatioandContinuedFractionThereisanelegantandsimplewayofgettingtheGolden


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ratio.Letuswritethecontinuedfractiononlywith

"1"s:R=1+1/1+1/1+1/....andsoon.Howtosolvethis?Thisfraction,repeatedorcontinued

toinfinity,appearsbizarreandintractable..Not

really.Itisextremelysimple:Notethattheleftsideexpressioncanalsobewritten

intermsof'R":R=1+1/RMultiplyingbyR,weget:R2=R+1whichisthesameastheequationwehaveforphi;So:R=Phi=1.618...GoldenRatioandGoldentriangleWestartedwiththeGoldentriangle--anisosceles

trianglewithangles36-72-72.Wehavetotieupthisto

Goldenratioandthenmoveontoconstructionof

'Pentagons'.Inthegoldentriangle,simplycalledG-T,wehavean


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apexangleof36degreesandtheheightADdividesthe

baseBCintoBDandCD.Lookupthetrigonometrictablesoryourcalculator.Sin(18deg)=0.3090IntheG-T,thehalfofapexangleis18degrees.So,sin(18)=BD/ABorSin18=CD/ACNotethatAB=ACandBD=CDcos(36)=0.8092cos(pi/5)=2cos(36)=1.618Cos(72)=0.309=BD/ABorBC/AB=0.618=1/phiNowyouseetheconnectionbetweengoldentriangleand

goldenratio.InaGoldentriangle,theratioofAB/BC=Phi.Goldentrianglerepeatedagain!Onceyoudrawagoldentriangle,youcanrepeatthat

easilyinsidethattriangle.

DrawagainagoldentriangleABCwithapexAandthe

baseBC.Thebaseanglesare72degrees.BisecttheangleABCsothattheanglebisectormeetsAC

atD.


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NowyouhavecreatedonemoreG-Tri:TriangleBDC,with

36attheapexBandangleBDC=72.Nowyouhavealsocreatedonemoretrianglewhichisan

isoscelestriangle:triangleABDwithanobtuseangle

of108andtwobaseanglesof36degrees.Thistrianglewewillcallflattriangle,becauseit

hasobtuseangleof108.Boththesetriangles,36-72-72triangleand108-36-36

triangle,wewilluseforconstructingPentagonsand

nicetiles.Thesetwotrianglesbecomethebuilding

blocksforourfurtherwork.Wewillcall36-72-72thick

triangleand108-36-36aflattriangleorthintriangle.YoucanbisecttheangleACBandletthebisectormeetBDatE.YouhaveformedonemoresmallerG-Tinside

triangleBCD.GoldenrectangleAsasimpleexercise,youcandrawa'goldenrectangle

usingtwoconsecutivenumbersintheFiboseries.


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A'GoldenRectangle'withsides34x55.Theratiois

55/34.[Youcandrawarectanglewith3.4inx5.5in.]

Itisclaimedthatmanyartistsusedsuchrectangles

withintheirpaintingsasinnerframes...Seearticleson

"LastSupper'byLeonardodaVinci.Pentagon,PentagramAregularpentagonisafigurewithfiveequalsidesand

aninternalangleof108.ApentagonarisesfromtheGoldentriangleandalsohas

goldenratioembeddedinit.Drawapentagonorfivepointedstarorcopyfromabook

orcopyfromthelogoofChrystlercarorfromthe

figuregivenhere.

A


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Pentagram

Markthefivecornersorverticesofthepentagonas

A,B,C,DandEinclock-wiserotation.DrawthediagonalsfromA,thatis,ACandAD.Measuretheratiodiagonal/side:AC/AB.Youwillfindthattheratioofside:diagonal=1:

1.618=1:phi.Youwillfindthattheinternalangleof108arisesfrom

thelittleisoscelestrianglewith108-36-36nearA.Pentagonisfoundinnatureinsomeflowersand

vegetables,butyoufindthefigurealsoincorporated


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inman-madebuildingssuchasthePentagon,the

DepartmentofDefensebuilding,inWashingtonDC.Youcandraweasilypentagramsforfun.Someusepentagonwithmysticmeanings----thosegroups

likeneo-pagansandfreemasons!Someusepentagonasa

logoforpeaceactivities.AGoldenrighttriangleJohannesKepler,thefamousastronomer,wasvery

interestedinGoldenRatiothathedevisedaright

triangleusingthisratio.Youknowalreadythat1+phi=(phi)2

UsingthePythagoriantheorem,wecanconstructaright

trianglewiththethreesidesintheratio:1:sqrt(phi):phi.Youcandrawatrianglewithlegs3inand(3x1.272=

)3.82inandfindthehypotenusetobe(3x1.618x1.618)

=7.85in.Thistriangleiscalleda"Keplertriangle".Nowwhatwouldbetheslopingangleofthisright

triangle,sayalpha?:


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tan(alpha)=sqrt(phi)/1=1.272.Lookupatrigtableoruseacalculatortofindthe

anglealpha:Itis51.8degreesorprettycloseto52

degrees.Keepthisnumberinmindwhenwediscussthe

pyramids.

Keplertriangle

GreatPyramidatGiza.ConsiderthepyramidatGiza---withasquarebaseand

slopingsides...Wehavearighttrianglethere...formed

bytheheight,halfthesideatthebaseandthesloping

linefromthetopofthepyramidtothebaseline.Thereisacontroversyhere.Somescholarsbelievethat

thistriangleisthesameastheKeplertriangleand

thattheEgyptiansknewabouttheGoldenratioandused

it.Theslopingangleisabout52degrees....


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Butthiscanbeexplainedbythesimpler3:4:5triple

forarighttriangleaswell....[Notethattheheightof

thepyramidhadbeenalteredduetothetopportion

beingbrokenormodifiedovertheyears;sotheheight

couldvaryabit.]Ifyoutakethelegsintheratioof

3:4,thentan(alpha)=4/3=1.33andtan53

=1.327...prettyclose! ThereisapossibilitythattheEgyptiansusedthe

Goldenratiotriangle(aKeplertriangle)for

constructingthepyramidorjustthesimpler3-4-5right


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triangle!LogspiralandGoldenSpiral[Youmayskipthissectionifthemathisfoundtoo

toughorunfamiliartoyou.]Alogarithmicspiralfollowsthepolarequation:r=aebwherethetaistheangleinpolarcoordiantes.andaandbareconstants. Asthetaincreasefrom0to2pi,rincreaseexponentially,

sweepingalargearea.Ifbischosensuchthatb=ln(phi)/90[fordegrees],we

getaGoldenspiral.[NotethattheArchimedesspiralissomewhatsimpler,

followingthepolarequation:r=k.]Byrepeatedconstructionofgoldentriangleswithinasingle

goldentriangle,wecangeneratethisspiral.Notethatineach

step,youarebisectinga72deganglecornerofthetriangle. InNature,onefindsmanybeautifulillustrationsoflog


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spirals---nautilus[mollusk]shells,spiralgalaxies,sun

flowerheads.theapproachofaninsecttowardsalightsource

andthenervesofcorneaintheeye. Anotherinstanceoflogspiralisthepathofacharged

particleinamagneticfiledperpendiculartoit---inacyclotron. HereIgivethefamiliarpictureofM51'Whirlpool'Spiral

Galaxy.

M51SpiralGalaxy--Alogspiral


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PenroseTilingGoldenrhombusesYoucanconstructarhombusbyplacingthebasesoftwo

isoscelestrianglestogether.Drawarhombusforthegoldentrianglewith36-72-72

angles.Youobtainarhombuswiththeangles

36-144-36-144.Drawanotherrhombuswiththeflattriangle--108-36-36

angles.Youobtainarhombuswiththeangles


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72-108-72-108.[Makeseveralsmallcut-outsofthesetriangleswhich

wewilluselater.]Thesearecalled'Goldenrhombuses'whichcanbeusedto

makenicepatterns.TessallationandPenrosetilingNowwecometothefascinatingtopicofformingtiling

patternusingpentagonlikefive-foldrotational

symmetry.Wecanusegoldentrianglesof36-72-72anglesandalso

flattriangleswith108-36-36angles.Formrhombusesas

explainedintheprevioussection.Byjuxtapositioning

oftheserhombusesyoucancovera2dimensionalspace

orformtiledfloororcarpets.Yougetverybeautiful

patterns.Studythefiguresofthesetilesandaftera

fewtrials,youwillgetthehangofplacingthe

rhombusesinaproperarrangement.

This5-foldsymmetrywasconsideredimpossibletill

RogerPenrose,aMathematicsProfessoratOxford

University,UKdevelopedsuch'tiles'in1974,called


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aperiodictiling.[IncidentallyRogerPenrosehad

collaboratedwithStephenHawkingontheoretical

physics/astrophysicstoo.]ProfPenrosepatentedthe

tilepatterninUK,USAandJapan.PleaselookupthesePenrosetilecolorpictures.[See"PenroseTiling"intheMathematicsPortalofMay

2009,selectedas'pictureofthemonth",in

Internet,shownabove.]Youcanconstuctnicequiltsorwallhangingswith

Penrosetiling.[Note1:GirihtilesinanIslamicmosquein

Isfahan,Iran,builtin1453,containsomeaperiodic

Penrosetilingpatterns.ThiswasdiscoveredbyPeterJ

LuofHarvardUniversity.Note2:Certain3-dimensionalcrystalsshowfivefold

symmetrylikePenrosetilesdoin2dimensionsandthese

arecalled'Quasicrystals"orFibonacci

crystals--originallydiscoveredbyDanShechtmanin1984

inanaluminumalloy.ThecompoundAlMn6presentin

thisalloyhadthisquasi-crystallinestructure.

ShechtmangotNobelprizeinChemistryin2011==forthe


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discoveryofquasicrystals.]Tryexperimentingwithgoldentriangles,rhombusesand

tilesand3Dobjectsandhavefun!HowtoconstructaPenrosetiling?Firstmakedrawingsofthetworhombuses,calledthick

andthinones,usingthetwotriangles,namely36-72-72

(thick)and108-36-36(thin).[orusing'dart'and'kite'

proto-tilessuggestedbyMartinGardner.]Step2:Makecutoutsoftheserhombususinga

cardboardsheet;youcanusethecerealboxesas

convenientcardboardsheets.Makeatleast30ofeach

kindofrhombuses.[Somewebsitesgivetemplatesfor

theserhombuses;youcanprintoutandcutthemwith

papercuttersorphotoshears.]Step3.Colourtherhombusdifferently:redcolorfor

thickoneandblueforthinoneorusecolorsheets.Step4:Trytoarrangethemwithpentagonlikefigure

atthecenterorsomeotherarrangement;Severalclever

methodshavebeendevised.Seesomeofthewebsitesfor

guidance.


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Step5Ifyouhavearrivedatapleasingdesign,youcan

takeaphotographandthentransferthepiecesto

anotherheaviercardboardorplasticorplywoodsheet

andpastethemwithglueforawallhangingorframing.Step6Youmayalsostudy"Pentagridmethod"which

wasdevelopedbyNicolaasdeBruijnandavailablein

websites.

ArtistsandtheGoldenratioDidLeonardoDaVinciusethegoldenratiowhile

painting"MonaLisa"or"TheLastSupper"?DidSeurat

usethisratioextensively?Somearthistoriansand

analyststhinkso.Explorefurthertoformyouropinion!

ReferencesMARIOLIVIO--Phi-thegoldenratio...MARTINGARDNER----PenrosetilestoTrapdoor

ciphers--themathematicalassociationofAmerica,,1997

Isosceles triangle, Golden ratio, Great Pyramid at Giza & Penrose Tiling

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