Solving the Vitruvian Man : Revised

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Newly revised and edited version of the Solving the Vitruvian Man paper uploaded in the Spring of 2011. This explores the the various Vitruvian Man geometries proposed throughout history, and corrects Le Corbusier's flawed geometries that create the Modulor System.

Transcript of Solving the Vitruvian Man : Revised

SolvingtheVitruvianManPatrickM.Dey&DamianPiLanningham

TheProblemandtheGeometry SolvingtheVitruvianMan1Problemseemstohaveeludedmathematiciansandgeometriciansforalmost 2000years.Someprobablyneverknewtherewasproblem.Itusesthatmysterious,andyetveryfamiliar numberphi(),anumberthatseemstolurkintheshadowsofallmathematics.Wewillshowhowthe human in the square and circle is created, and how the same geometries can account for all anthropometrics,whetherthehumanisinmotionornot.Furthermore,wewillilluminatehowthesame geometricprocessisrelatedtoothergeometries,suchasthepentagram,andhowitisanaturalself replicatinggeometry.Vitruviusdescribesthegeometricalproportionsofthehumanbodyas:Inthehumanbodythecentralpointisnaturallythenavel.Forifamanbeplacedflatonhis back, with his hands and feet extended, and a pair of compasses centered at his navel, the fingersandtoesofhistwohandsandfeetwilltouchthecircumferenceofthecircledescribed therefrom.Andjustasthehumanbodyyieldsacircularoutline,sotooasquarefiguremaybe foundfromit.Forifwemeasurethedistancefromthesolesofthefeettothetopofthehead, andthenapplythatmeasuretotheoutstretchedarms,thebreadthwillbefoundtobethesame astheheight,asinthecaseofplanesandsurfaceswhichareperfectlysquare. DeArchitectura,BookIII,1:3

AlthoughVitruviusdoesnotpresentthequestionathand,thequestionstillexists:howdoesthehuman body fit into both a circle and a square? This problem has been searched and worked out by many geometricians, such as Agrippa, Cesariano, Drer, and Da Vinci. The problem is somewhat of a latent probleminthatwecanmaptheproportionsandgeometriesofthehumanbodyanddiscovernumerous relationshipsandproportions.Butwemustaskthen:howisthisgeometrycreatedinthefirstplace? InlookingatLeonardDaVincisVitruvianMan,themostnotableinterpretationofVitruviussCanonof Proportions,wecanbegintounderstandsomegeneralproblems:

ThroughouthistorytheVitruvianManhasbeenastudyoftheidealmalebody.Thismakesgenderneutrality difficulttoworkintothispaper.Wewouldliketoapologizefortheuseofhisratherthanhis/her.Inthat respect,theproblemofidealizationandthefemininebodyisaddressedinthispaperonpages1213.1

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Thefirstindicationthatthereisageometricalproblemisthefactthattheupperverticesofthesquare donotmeettheboundaryofthecircle,butratheroverlapit.Thisisnomilderror,astheoverlapisin errorbyroughly1.5%ofthecirclesradius.Consideringthiserror,oneoftwothingsmustbeaccepted: eitherthehumanbodyisnotperfectandnatureisnotaperfectgeometrician,orthereisanaturaland geometrically perfect means of constructing the Vitruvian Canon. Since the former is antithetical to humanist ideals of the body being perfect, as well as the belief that nature is the most perfect (and beautiful) geometrician, we have to argue for the latter. But once again: attempted solutions need examination. Inordertotacklethisproblem,abitofhistoryandotherattemptstosolveitshouldbelookedat.This question was considered and attempted by the architect Charles Le Corbusier Jeanneret, one of the fathersofModernistarchitecture.From1943to1945LeCorbusierworkedwithayoungcollaborator, who is only known as Hanning, as well as Mlle Elisa Maillard. Le Corbusier, working with AFNOR on standardizingproducts,goods,andbuildings,presentedaproportioningproblemtoHanning:Takeamanwitharmupraised,220m.inheight;puthiminsidetwosquares110by110 meterseach,superimposedoneachother;putathirdsquareastridethesefirsttwosquares.This third square should give you a solution. The place of the right angle should help you decide wheretoputthisthirdsquare.2

Logically,thisprocessshouldincorporateandbedevelopedfromthegoldenmean,sincethispropertyis inherent in human proportions. The first proposal given by Hanning (Figure A) was to begin with a squareandthenproducegoldenrectanglea.Next,fromtheinitialsquaredrawroottworectanglebon the opposing side of the square from the golden rectangle. The resulting rectangle is almost two adjacentsquares(FigureB),thereforetheangle()oftheregulatinglines3oftheoverallrectangleis greaterthan90(FigureC):

FigureA FigureB FigureC Theproductoftheroottwosolutionrectangleisinerrorby1.6%,and,therefore,notaviablesolution. About two years later, meeting with Mlle Elisa Maillard, another solution was proposed. A golden rectangleisconstructedfromasquare.Nextalineisconstructedfrompointaofthegoldenrectangleto theopposingmidpointoftheinitialsquare,pointb.A90isdrawnfromlineabtointersectwiththe base of the geometry at point c (Figure D). The resulting rectangle is closer to two adjacent squares (FigureE),butstillinerrorby0.63%andproducesregulatinglinesthathaveanangle()greaterthan 90(FigureF):LeCorbusier.TheModulor.Trans.PeterdeFranciaandAnnaBostock.Basel,Switzerland:BirkhuserPublishers. 2004.P.37. 3 RegulatinglinesisatermusedbyLeCorbusiertodenotelinesthatintersectat90angles.2

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FigureD FigureE FigureF Thedifferenceintheerrorsoftheabovesolutionscanbeseenhere:

Although this system is in error, Le Corbusier used this geometric structure to devise his infamous Modulor System, which is a system of measurements built on the principle of anthropometrics that proportionatelygrow(ordiminish)fromtheinitialgeometricconstruct.Thiswashismeanstohumanize architecture. We must admit that the geometries are approximated, and at certain times completely falseandprobablyforced(forinstanceatangentiallineproducedfromtheregulatinglines,whichisnot tangential to its circle4). But given that these geometries were constructed with Tsquares, triangles, compasses,andpencils,itseemslogicalthatageometricalerrorwouldbeconsideredanerroronthe draftsmansbehalf.Moresurprisingly,though,isthefactthatLeCorbusierdidknowthatthisgeometry was flawed! In 1948, a Monsieur R. Taton informs Le Corbusier: Your two initial squares are not squares;oneoftheirsidesislargerbysixthousandthsoftheother[grossmiscalculation].LeCorbusier being economical considers: In everyday practice, six thousandths of a value are what is called a negligiblequantityitisnotseenwiththeeye.ButLeCorbusieralsobeingamysticfurtheradds:I suspect that these six thousandths of a value have an infinitely precious importance: the thing is not openandshut,itisnotsealed;thereisachinktoletintheair;lifeisthere,awakenedbytherecurrence of a fateful equality which is not exactly, not strictly equal And that is what creates movement (sic).5 We, on the other hand, argue that there is no error in the initial problem and that Le Corbusier just happens to be dead wrong (which is certainly not the first time he was dead wrong, but 20th Century paradigmsonurbanismdonotneedtobeaccountedhere).Asmathematiciansandgeometricianswe claimthatifthereisanerror,nomatterhowtrivialandsmall,itisanerror,andthereforeincorrect. 4 5

Ibid.,p.64,Fig.21. Ibid.,p.234.

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Furthermore,wearguethatthesolutionhastobeelegant.AnotherlookatLeCorbusiersgeometrywill showthatitisnotelegant.Forhisgeometrytobeinaccordancewiththehumanbody,theportionof the rectangle created by the regulating lines (established at point c) shown in Figure D has to be removed from its initial side of the square to be placed next to the golden rectangle (Figure G). This createsthedistancebetweenthetopoftheheadtothetipofthefingerswhenthearmisraisedabove the head (Figure H). The final problem with Le Corbusiers solution is that it only accounts for the proportionsofverticalhumanmeasurements,thatis,itonlyattemptstosolvetheheight.TheModulor nevertakesintoaccounthorizontalmeasurements,suchtheproportionsofthearmspantothewhole. Thiswewillaccountforshortly.

FigureG

FigureH (DrawingbyLeCorbusier, fromTheModulor)

LookingatLeCorbusiersModulorwefindthatitisnotonlyincorrect,butitisalsonotelegant.Natures geometries do not disassemble and then reassemble a construction. Humans are the ones, with their reason and logic, construct geometries and reassemble them arbitrarily in whatever manner suits their liking. On the other hand, the geometries of nature are built up from an initial geometry to construct the whole. Man disassembles and reassembles harmony to create beauty, while nature simplyconstructsharmonythatisbeautiful,naturejustexists.Nowwereturntotheinitialquestion,but withmoreconstraints: 1)Constructageometrythatcontinuouslybuildsuponitself 2)Thegeometryconstructsthehumaninsidebothacircleandsquareinaccordancewiththe VitruvianCanonofProportions 3)Thegeometryaccountsnotonlyfortheverticalanthropometrics,butalsoforthehorizontal measurementsandproportions 4)Itisdevisedfromthegoldenmean 5)And,finally,theconstructionofthegeometryhasanelegantsolution. In considering this problem, we find that it is justifiable that Le Corbusier would assume that starting with a square and constructing a golden rectangle that could inevitably produce two perfect squares throughsomeotherformofgeometrizing.Also,weassumehemustalsoberightinassumingthatthe critical dividing lines created by an overlapping third square must be found in accordance with the humanbody,suchastheheightofthehead.

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When we began approaching this problem, we segmented a square abcd with a line op so that the segmentsoflineabareagoldenmean(Figure1).Wedidthisassumingthatthegoldenmeanhadtobe presentwithasquare,butnotnecessarilyformingagoldenrectangle.Ifthelengthofsegmentopthat dividesthesquareabcdintoagoldenmeanissquared6,thisnaturallycreatesagoldenrectanglebcef forthewholegeometrythusfar(Figure2).Therefore,Figure1=Figure2,andthesquaresabcdand opfe are equal. Finally, if the initial square is squared, this naturally creates two perfect and adjacent squares,aswellasgivingthepositionofthethirdsquareoverlappingthetwoadjacentsquares(Figure 3).

Figure1 Figure2 Figure3 Although this does solve the problem of the two squares created from a golden mean and is rather elegant,itsimplyisnotelegantenough.Itisnotelegantenoughsimplybecausesacredgeometries,and, therefore, natural geometries, do not typically segment a square with a golden mean without being created initial