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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