Cr Nbc Lecture Eng

7/27/2019 Cr Nbc Lecture Eng

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HIDDENSYMMESTRIES


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Platonsfivesolids

ChapterXIII,thelastofEuclidsElements,focusesondescribingEuclidsfivepolyhera.

Polyhedraarethree-dimensionalsolidslimitedbyplanes,the

samewaypoligonsareshapeslimitedonaplanebylines.A

polyhedronhasvertices,linesandfaceswhileapolygononly

hasverticesandlines.Aregularpolygonisonethatis

equiangular(allanglesareequalinmeasure)andequilateral

(allsideshavethesamelength).Aregularpolyhedronisoneofidenticalvertices,linesandfaces.Aregularpolygonis

completelysymmetriconitssurface;similarlyaregular

polyhedroniswhollysymmetricinspace.

Suprisingfact:onaplane,aregularpolygonmayhave

numerouslinesandverticeswhileinspace,thereareonlyfive

regularpolyhedra.

Acubeandapyramidarerelativelyfamiliar,whereastheother

solidsthathavetwelvefaces(dodecahedron),twentyfaces

(icosahedron)andeightfaces(octohedron)arelessso.An

octohedroncanbeconstructedbybasingitsvertexonthe

centroidofacubeseightfaces.


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Thereisarelationbetweenadodecahedronandan

icosahedron.Adodecahedronhasexactly20verticesandan

icosahedrondo12.Fromthecentroidsofadodecahedrons

faceswecanconstructanicosahedronandviceversa.

Platoncaredmuchaboutthemetaphysicalmeaningofthefive

regularpolyhedra.Hethoughttheycorrespondedtofour

elementsinnature:fire,water,airandearth.Apyramidlookedlikefire,itshouldbefire.Acubewasearthbecausetheearth

wasthoughttobesquare.Whyanoctohedronwasair,we

dontknow,nordoweknowwhyanicosahedronwouldbe

water.Adodecahedroncorrespondedtocosmos,forwhich

Platondidnotdefineclearlyeither.

AfterwardsAristotebroughtupether,theenvironmentfor

thefirstfourelements,asthe5thelementinplaceofPlatonscosmos.AlthoughthedefinitionofAristotesetherwas

clearerthenthatofPlatonscosmos,theirdisciples

encounteredmuchdifficultiesinunderstandingwhatit

actuallywas.Itwasconjecturedlaterthatetherwasafieldthat

allowedelectromagneticradiation.


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Andnow,awhowantstobeamillionairequestion:istheman

inthepicturePlatonorAristote?

Ouransweris:heisneitherPlatonnorAristote.Hisnameis

Empedocle.HeisaphilosopherbeforethetimeofSocrates,

Platon,andAristote.

Empedocle,whoisoneofthefirstscientistsofmankind,held

uniquetheoriesonmetaphysics.

Acordingtohim,besidesthefourbasicelements:water,fire,

air,andearth,thereweretwootherelementsofnolesser

importance:loveandhate.Loveandhatesometimesbrought

water,fire,air,andearthtogetherandsometimestorethem

apart;suchact,hedeemed,wastheforcebehindcosmossperpetualmotion.

Andthenextquestionforwhomwantingtobeamillionaire:

whichsolidsdoloveandhatecorrespondto?


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THISistheanswervotedbythelargerpartoftheaudience.

Yetnotouranswer.Wesupposelovewouldnotcorrespondto

anyregularsolidbecauseitisneverwhollysymmetrical.


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Symmetry

Nowwewilllearnmoreaboutsymmetry.Thisfigureillustrates

twotypesofsymmetryofaregularrectangle.

Thefirstsetoffiguresillustrateaflipoverthealtitude

originatingfrompointBoftheregularrectangleABC.ThistypeofflippreservespointBandswitchthepositionsofpointA

andpointC.

Thesecondsetoffigureillustratea60-degreerotationofthe

trianglefromitscentroid.Thistypeoftransformationbrings

pointAtothepositionofpointB,BspositiontoCsandCsto

As.

Forhighschoolerssymmetryishardertograspthanpoint,

line,face,orsolidbecausethelatterareallinvariant

whilesymmetry,ingeneral,indicatesmotions.


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FortheregularrectangleABC,symmetrycanbeenvisionedin

twoways:discreteandcontinuous.

Thediscreteapproachpayssoleattentiononwhatsymmetry

doesforpointsA,B,andC.Weforgettheresidueofsuchtransformationonthetriangle,andonlynoteitseffectonthe

trianglespoints.Thisapproachonourtrianglecanbestated

conciselyasapermutationofthethreeletters,e.g.CBAwould

meanapermutationpreservingBandswitchingAandC.

Fromthediscreteapproachwecandrawaninapparent

conclusionthataregulartriangleonlyhasalimitedamountof

symmetries.Indeed,weonlyhavesixpossiblepermutationsofthreelettersA,B,andC;suchsixpermutationscorrespondto

sixsymmetriesoftheregularrectangleABC.

Infacteachindividualsymmetryisnotasimportantasthe

mannerthattheyinteractwitheachother.Acombinationof

varioussymmetricinteractionsformanactofpermuting.In

theillustrationweseetwopermuttedvaluescreateanew

value.SixpermuttedpositionsofthreelettersA,B,andC,plus


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theircollectivepermuting,formamathematicalstructure

calledset.

Setisacoredefinitionofmodernmathematics.Agroupofall

permutationsbythreelettersA,B,andC-onecomposedof6objects-isoneofthesimplestsets.Inmaths,thedefinitionof

setisanexpressionthatmathematicallyclarifiestheambiguity

ofsymmetry.Themoresymmetricalasolidis,themoregrand

andcomplexitssetsofsymmetriesare.

Fromthecontinousapproach,weimaginethemovementsof

two-dimensionalshapesonaplane,andthemovementsof

three-dimensionalobjectsinspace.Inbothcases,themovementsformrespectivesetsofmovements.

Incountlessnumberofmovementsinspaceonlyalimited

numberofmovementspreservetheshapeofaregularsolid.

Asetofallsymmetriesofaregularsolidisafinite,discrete

subsetofallmovementsinspaceofsuchsolid.

Anicosahedronhasexactly120symmetries.These120

symmetriesformasubsetofasetofallmovementsofthe

icosahedroninspace.Theicosahedronssetmightbethe

simplestthingthatsurpasseshumansabilitytoperceiveusing

observationandsensation.


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Platonadvocatedthatwhatcouldbeperceivedthroughsenses

wasfutileandimperfect.Hebelievedthatonlymathematical

realityexisted,inopposedtoitsshadowwhichwewereaware

ofthroughsensation.Foramathematician,thesolidswere

themselvesshadowsofamoreabstractreality,thesymmetric

sets.

Forme,itisthebeautyofmathematics.Itiswhenreasoning

comprehendscompletelyanexistencethatsensationcanonly

grasppartly.

Althoughsensesdonthelpusinunderstandingsymmetry,the

symmetriesinspaceandsoundhavetheiraffectsonus.


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ThisisthepaintingScreambyEdvardMunch,atypicalwork

withinexpressionismmovement.Inthispainting,familiar

symmetriesarebroken,causinganxietyandapprehension.


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LogisticsInstitutebuilding,atypicalarchitecturalconstruction

ofmilitarycamp.Diagonalsymmetryisaparticulartraitof

suchstyle.Theseconstruction,youmayagree,dontbudgeus

much,emotionally.Perhapsitsnotevenintendedbythe

architect.Wellbutwhileitdoesntmoveusemotionallyit

suggestsanotherfeeling....


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Yes,thatstheanswer:firm,reliable.


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ThisisapaintingbySolLewittinMOMA,NewYork.Istopped

toobserveitforaverylongtime,wonderingwhyitpullsmein

sostrongly,whychalklinesthatseemtobedrawnrandomly

havesuchconnectiontomymind,whyIfeelthattheselinesarebeautifulasawhole.

ThenIfigureditout.Therearehiddensymmetriesinthis

painting.

Lewittmadetheworkbydrawingthesquaresfirst,thenthe

otherlines;thenheerasedthesquarestoleavethelines

behindforus.

WhileIwaslookingatthepainting,mysubconsciousness

strovetofindthesquareagainanactthatmyreasoning

wasntawareof.Itseemslikewheneverweobserveanything,


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oursubconsciousnessalwaysansweredtoacalltolookfor

hiddensymmetries.

AccordingtoPlaton,thewordaswesensedwasmerelythe

shadowofanotherreal,eternalworld.

Artworksarejustshadowsfromtheworldofsenses.Whatwe

do,rootingfromreasoningandsubconsciousness,isthereal

comprehensionoftruthfromitsownshadow,orevenfromthe

shadowsshadow.

Thesubconsciousprocessoffindingtruthresemblesthe

feelingforaesthethics.

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