Futurism And the Forth Dimension

8/12/2019 Futurism And the Forth Dimension

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talian uturism n dT h e ourth Dmension

LindaDalrympleHendersonDuringhefirst hreedecades fthe wentiethcentury, rtists n nearly verymajormodernmovement ere nfluencedyahighly opularconceptknown s the ourth imension. nthisperiod, the ourth imension ignifiedhigher, nseen imensionfspacewhichmightholdarealityruerhan hat fvisual erception.Linked losely o the philosophicaldealismwhichdominatedhe era,belief n a fourthdimension ncouraged old,formal xperi-mentationy iberatingrtistsrom hedomi-nationof three-dimensionalisualreality. fsome artistic dvocatesf a fourth imension,suchas theCubists, idnotreject isualper-ception ompletely,upka, alevich, ondrian,and VanDoesburgound upportn the deafortheircreation fa totally bstract rt.TheFuturists occioni ndSeverinihus oinedadistinguishedist of artistsattractedo thefourth imension, group angingromAna-lytical ndSyntheticubistsswellasDuchamp,Picabia, ndKupka,o Russian uturistsndSuprematists,Americanmodernistsn theStieglitzndArensbergircles,Dadaists, em-bersof DeStijl, ndeven ertain urrealists.'Thepopular, arly wentieth-centuryiewthatspacemightndeedhavemore han hreedimensions as anoutgrowthf thedevelop-ment,during he firsthalf of thenineteenthcentury,of geometries f more than threedimensions,nownsn-dimensionaleometry.2Although igherdimensions f spacewerefirstbroughto publicattentionn the laternineteenthentury ydebates bout henatureofgeometricalxioms nd henature fspace,the ourthdimension uickly cquired u-merousnon-geometricssociationsswell. nEngland, or example, he authorCharlesHowardHinton eveloped system f hyper-space philosophy, 3asedon his belief hattheanswero theevilsofpositivismndmate-rialismwas forman odevelop ispowers fintuitionn order operceive fourth imen-

Fig. 1 PabloPicasso, ortraitfAmbroiseVollard,910,oil oncanvas.Moscow,PushkinMuseum.sion of space, the truereality.Thefourthdimension lso took on specificallymysticalqualities henadopted ycertain heosophistsandspiritualists;tfunctionedn thiswayn thewritings f themajor yperspacehilosophersdescendedromHinton-P. D. OuspenskynRussia ndClauderagdonntheUnited tates.Thefirst artisticapplicationsf a spatialfourthdimensionweredevelopedn Parisntheyears round910,and heCubistiteraturein which heyweresetforthhadan mpact nall subsequentheorizingn thesubject.Al-thoughPicasso'sCubismvolvedromhisin-terest n CezannendAfricanrt,hissteptomatureAnalyticalubism asunquestionablyencouragedby his era's beliefin a realitybeyondmmediateisual erception.he on-

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Fig. 2 Perspectiveavaliere,from.Jouffret,TraitelementaireegeometriequatredimensionsParis, 903),Fig.41.ceptualnature f African rtwas thespecificfactorhat ncouragedicasso, shesaid, topaintobjectsas I think hem,not as I seethem. 4Forthe theoristsMetzinger, leizes,andApollinaire,ntheotherhand,hefourthdimension erved sthemajor ationaleoraCubist ainter'sreedom o distort r deformobjectsaccordingo a higheraw,aswell asforhisrejectionfperspective.5sApollinairewroteonthesubject fformn 1912, It s tothefourth imensionlone hatwe oweanewnormof theperfect. 6Apollinaire'seferencenLesPeintresCub-'istesof 1913 operspectives thatmiserabletrickyperspective,hatfourthdimensionnreverse 7onfirmshespecific onnectionfthe ourth imensionithCubistictorialpaceas wellaswith orm.Theambiguouspace faworksuchas Picasso's ortrait fAmbroiseVollard f 1910offersa striking aralleloaWinter1981 317

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contemporaryllustrationn a textbook nfour-dimensionaleometryyE.Jouffret,ub-lished in Paris n 1903 (Figs. I and 2).Although ot the sourceof Picasso'sCubiststyle, uch llustrationsouldhave onfirmedthedirectionn whichCubism asdeveloping.Both mages eektoavoid traditionalhree-dimensional eadingof objectsand space.Moreover,multiple iewpointsf the objectare juxtaposed,ustas HenriPoincare adsuggestedn his textLaScience tl'hypothese,(1902),in adiscussionf apossiblemeans orepresent four-dimensionalbject. n fact,theexplanationyMetzingerndGleizesnDuCubisme 1912)that heCubistaintermovesaround issubject, athering ultipleiews fit inorder oproduce truermage,s basedonPoincare'sonnection ftactile ndmotorsensationsith hepossible erceptionfhigherspatial imensions.8Themethod f theCubistainter,hen,wasanalogous o that of a geometer trivingoachievea perceptionf a higherdimensionalobject. f motion n timewasinvolvedn theprocess, t wasonly ncidentalo theartist'spursuit f four-dimensionalpace.Yetwithinthe tradition f a spatial ourthdimensiontherehaddeveloped, ythe end of the nine-teenthcentury, secondapproacho higherdimensions, n approachn which imeandmotion layed positiveole.However,uthorssuch as Hinton,whoincluded ime n theirexplanationsf the fourthdimension,werenot returningo thesuggestionmade n theeighteenth enturyhattimeitselfcould bedefined s a fourth imension.9hatdeawasto gain widespread cceptance nce againonly after 1919 with the popularizationfEinstein'sGeneralTheory f Relativityasedona four-dimensionalpace-timeontinuum.'0Duringhenineteenthndearlywentiethen-turies, owever,he nterpretationfthe ourthdimensions timewasalmostompletelyver-shadowedyenthusiasmbout higher patialdimension, erpendicularothe hreedimen-sionsweknow.Thus,Hintonelievedirmlynthe existence f a fourthdimension f spaceandarguedhat imeandmotionwere ncom-pletely nderstood anifestationsfthat ddi-tional patial imension.Hinton'sheoreticalritingsentered boutthefour-dimensionalypercube,rtesseract,thehigherdimensionalnalogue f the cube(Fig. 3, lowerright).Justas a cubecan begenerated ythemotion f a two-dimensionalplane hrough thirddimension,hypercubewouldbe tracedby the motionof a three-dimensionalube ntoa new, ourth erpen-dicular irection.Givenhedifficultyfvisual-izing hefour-dimensionalppearancef thisfigure, ven na three-dimensionalerspectiveprojectionFig. 3, lower eft), a reversal fthis processwasmoreeffectiveor Hinton'spurposes.Asa result,heconcentratedn thethree-dimensionalectionswhichwouldbeproduced s afour-dimensionalbject assed

318 ArtJournal

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la quatriemeimension, hichappearednthe ront age f ComoedianMay0,1912.15Justas Kupka'smotionstudiesmayhaveacquired four-dimensionalssociation, isyoungadmirerMarcelDuchampmayhaveconnected is ownexperimentsn the staticrepresentationf movement '6ith hegen-eration fhigher imensionalorms.Duchamplater alked f hisSadYoungManon a Train(1911)and heNudeDescending Staircase(1912) (Fig. 8) as examplesof his earlyinterest in elemental/elementaryarallel-ism '7andthe parallelmultiplicationfthen-dim'lcontinuum o formthe n+1 dim'lcontinuum. '8 etDuchamp, ho knew hegeometryfJouffrets wellas Hinton'sdeas,mustquickly ave ealizedhatheresnothingfourdimensionalbouthemotion fanobjectunless tmovesoff ntoanew ourth irection(asHinton'sypercube as ohavedone).AsDuchampaterexplained, Themovementfform ntime nevitablyshered s intogeome-

Fig. 6 Etienne-JulesMarey,Fencer, 882. Paris,CinemathequeJranqaise.

Fig. 7Frantisek upka, ntitledrawing,ca. 1910,graphiteencilonpaper.Paris,Collectionarl linker.oriented tyle.Kupka'sainterlyxperiments,inspired y hechronophotographyf Etienne-JulesMareyFig. 6), appearo have aken na four-dimensionalssociationorhimoncehe was awareof Hinton's heories.'3Mostlikelyntroducedo Hinton'sdeas hroughheParisianTheosophical ublicationsn whichtheywere ummarized,'4upka ay ave ometoconsider he orms eneratedn works uchas Woman ickingFlowers MuseeNationald'ArtModerne,aris)anda related rawingfabout1910(Fig. 7) asanalogouso Hinton'sgenerationf thehypercubeymotionntime.

Fig. 8 Marcel uchamp, udeDescendingStaircase,o.2,1912,oil oncanvas.Philadelphia useumfArt,The ouise ndWalterrensbergollection.The notationon the drawing,whichKupkamusthave ddedn1912,confirmsisconcernwith these issues: Displacementn threedimensionsakesplace nspace,while hatnfourdimensionsy heexchangef atoms.Butto fixgesture, movementn thespaceof thecanvas,-arresteveraluccessive ovements.This tatements drawnirectlyrom segmentofGaston e Pawlowski'sale Voyageupays e

Fig. 9 KazimirMalevich,heKnife rinder,1912,oil on canvas.NewHaven,YaleUniversityrtGallery.tryandmathematics. '9ith hisrealization,Duchamp aveuphismotion tudies aswellas conventionalil painting)o carry utthemost erious tudy ffour-dimensionaleom-etryundertakenyany arly wentieth-centuryartist,a studywhichculminatedn TheBrideStrippedBarebyHerBachelors,Even(TheLargeGlass)of 1915-1923.Asimilarejectionfmotionnthree-dimen-sionalspacewas madeby KazimirMalevichafterhepainted isoneexamplenthismode,TheKnife Grinderof 1912 (Fig.9).20 Male-vich'sexplorationfsequentialmotionn thisworkwas mostlikely nfluenced y Leger'sEssaipour les troisportraitsof 1911 (Mil-waukeeArtCenter) hichwasexhibitedn theJack of Diamonds xhibitionn MoscownFebruary 912. Leger's wnsubtleexplora-tion of motion n thispaintingmaywell havebeen nspired y hechronophotographyash-ion, reinforced ythe UnanimisteoetJulesRomains'sbelief in the interpenetrationfobjectsnnature,s wellasbyHenri ergson'sdescriptionf a realityn fluxwithin heflowofduration.21et,Bergsonimself ad ejectedWinter1981 319



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thechronophotographnd hecinematograph.For himsuch snapshotiew[s]of a transi-tion 22were antitheticalo the continuous,pure imeofduration.hus,hechronophoto-graphicallynspired ecords f the motion fthree-dimensionalbjectspainted y Kupka,Duchamp,Malevich,ndLegerwerenot our-dimensionalor, f onewere ontentopursueBergsonian uration lone,were they trulyBergsonian.Enter the Futuristpainterand sculptorUmbertoBoccioni. n thesecond echniquesuggested yhyperspacehilosophyi.e.,thevisualizationf a four-dimensionalbjectbymeans of its successive hree-dimensionalsections),Boccioniwas ofindan alternativeapproachotheproblem f the ourth imen-sion. This olution, ebelieved,was arsupe-rior to theCubists'uregeometrical ethodas well as to the chronophotographiciewsthateven his countrymaniacomo allahadexplored n such worksas Dynamism f aDogon a Leashof 1912 (Albright-KnoxrtGallery, uffalo).AlthoughBoccioni's arlierwritingshadregularlyncludednumerous ositive efer-ences to mathematicsndscience,23nd al-thoughtalywasoneof themost rolificourcesof scholarlyiterature n the fourthdimen-sion,24tseems hatBoccioni'srtisticnterestin the topicwas arousedonly throughhisconnections ith heartworldof Paris.Mari-netti'sfrequentpresence n Parisfrom the1890sonwardndhisacquaintanceithAlfredJarryandApollinaire25ighthaveprovidedan nitial ntroductiono the dea.Morempor-tant,however,was heposition f theFuturistSeverininthemidst f theCubistvant-garde.WhenBoccioni ndCarraraveledo ParisnOctober 911,as apreludeo theirBernheim-Jeuneexhibition fFebruary912, hestudiovisitsSeverinirrangedncludedheatelier fMetzinger, ho particularlympressedBoc-cioni.26 fBoccioni id nothearofthefourthdimension tthat ime,hisreturnripsoParisin FebruaryndNovember912and nJune1913for hissculpturexhibition ouldhaveaffordedmple pportunityo discuss heno-tion.Forexample, occioniwrote ohis riendNinoBarbantiniromParisnFebruary912,describing he goal of his new paintingntermsof a spiritualizationwhich]will begivenby puremathematicalalues,bypuregeometrical imensions.... 27 n addition,the irstmajortatementf Cubistiews n thefourth imension assoon o bepublished yApollinairen anarticleof April1912inLesSoirdesdeParis,a text hat ormed he basisforonechapterfhisLesPeintresCubistes fMarch 913.Boccioni'snlymajor iscussionf theroleof the ourth imensionnFuturistrtwaswrit-tenandpublishednitiallynDecember913.28Itwas hen ncluded, ithadditions,nhistextPitturasculturafuturistedinamismolas-tico), of 1914.In the finalversionBoccioni

Fig. 10 Umbertooccioni,UniqueormsfContinuitynSpace,913,bronze.NewYork, heMuseumfModernrt.wroteas follows,ntroducingis ideaswithacritique fearlier ttemptstexpressingyna-mism nart:

Itseemslearome hat his uccessionsnot obe oundnrepetitionf egs,arms,andfaces,asmanypeople ave tupidlybelieved,ut s achievedhroughhe n-tuitiveearchor heuniqueformwhichgivescontinuitynspace. t isthe orm-typewhichmakes n objective n theuniversal. herefore,nstead f the old-fashionedonceptfsharp ifferentiationofbodies,nstead fthemodernonceptofthe mpressionistsith heirubdivision,their epetition,heir oughndicationsfimages,wewould ubstituteconcept fdynamic ontinuitysuniqueorm.Andit is notbyaccidenthatI say orm nd

not ine, incedynamicforms aspeciesof fourth imensionnpaintingnd culp-ture,which annotxistperfectlyithoutthecompleteffirmationf thethree i-mensionshich etermineolume:eight,width, epth.I rememberaving ead hatCubismwith its breaking p of the object ndunfoldingf theparts ftheobjectntheflatsurface f thepicture pproachedhefourth imension....However,hispro-cedure s nothingbut the transcriptiononto the surface f the canvas, f theplanes f theobjectwhichtsaccidentalposition reventss from eeing. t is arationalrocedurehich xistsnrelativ-ity,notin an intuitive bsolute. he n-tegral otion ftheobjectxists,with hisprocedure,nthe hreeonceptsfheight,320 ArtJournal



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:f .:- : ::....C1?Fig. 11 Umbertooccioni,MuscularDynamism,913,charcoalnpaper. ewYork,TheMuseumfModern rt.

width, epth,hus repeat,ntherelative,in the finiteof mensuration.f with heartisticntuitiont is ever ossibleoap-proachheconceptfafourthimension,it is we Futurists ho aregettingherefirst. nfact,with heuniqueormwhichgives ontinuitynspacewecreate formwhichs the umof hepotentialnfoldingofthe hree nownimensions.herefore,wecannotmake measurednd initefourth imension,ut athercontinuousprojectionfforcesnd ormsntuitedntheirnfinitenfolding.n act,heuniquedynamicormwhichweproclaims noth-ingother han he uggestionfaformnmotionwhich ppearsoramomentnlyto be lost n the nfiniteuccessionfitsvariety.In conclusion, e Futuristsivethemethodorcreating conception oreabstractnd ymbolicfreality,utwedonotdefinehe ixed ndabsolute easurewhich reatesynamism.29Boccioni'sunderstandingf the fourth i-mensionwasobviouslyarmoredynamichanthatoftheCubists, hosegeometricpproachhe specifically ejected.Dynamismnd, byimplication,motionn timeare essential le-ments of his interpretation.et,Boccioniscareful o separate imself rom heanalysesof linear equentialmotion ainted yKupka,Duchamp,ndhisfellowFuturistalla, swellas from hechronophotographicxperimentsof the Italianphotographer ragaglia, hoattemptedo embodyFuturistenets n hisfotodinamismo.30Beyondhis criticisms f the mistakenp-proaches f others ohigher imensions,oc-cioni's textprovideseveralndirect luesto

his own definition f the fourthdimension.Since dynamicorm s a speciesof fourthdimension orBoccioni, isview f thenotionmustbe closelyrelated o theunique ormwhichgives ontinuitynspace, he itleofhismost famous culptureFig. 10).31 A four-dimensionalorm for Boccioni s a form-type which ranscendsrtificial ivisionsnspace; n otherwords, t gives ontinuity.Boccioni's ourthdimensions an absolute,unmeasurable,nfiniteoncept, sopposedoCubism's measurednd inite ourth imen-sionwhich s onlyrelativen Boccioni's iew.Intuition ndnota rationalrocedure sthemeansbywhich heFuturistrtist anapproachthis fourthdimension,which s thesumofthe potentialunfolding f the threeknowndimensions. inally,ncreating more b-stract ndsymbolic onceptionfreality,heFuturistwillwiselyavoid ryingo defineordelimit the ixed ndabsolutemeasure hichcreatesdynamism.ToexplainBoccioni'serydifferentnder-standingof the fourthdimension, is priorartisticconcernsand his choiceof differentsourceson the ourth imensionmustbecon-sidered. nthe evolution f hisartisticheoryBoccionihadbeenparticularlynfluencedythephilosophyfBergson.asicohis hinking,then,was a belief n a realityn constantlux,to be intuitedby each individual ithin heflow of duration.32WhenBoccionibegan oconsider fourth imension,enaturallyre-ferrednterpretationsnvolvingime, hebasicelement in Bergson's heories.Hyperspacephilosophy,with ts recognitionf timeas ameans ohigher patial imensions,as deallysuited to Boccioni'sneeds.Bergsonhimselfhad inkedimewith pace, eferringo homo-geneous ime s a fourth imensionfspaceinhisEssai ur es donnees mmediates e aconscienceof 1889.33 ignificantly,owever,Bergson adused he ermnanegativeense,criticizinghemind'sendencyocontaminatethepure lowofdurationyspatializingt.Yet, or Boccioni he fourth imensionidimplyhigher patialdimensions ndwasnotsimplyimealone(asitwouldhavehad o beforaBergsonianurist). hroughoutoccioni'swritingsof the periodfrom1912 to 1914,therearefrequenteferenceso anew nnerreality, 34an deal,superior lane, 35ndtheneedof the artist o renderhe invisiblewhichstirsand ivesbeyondnterveningb-stacles. 36 occioni'sellowFuturisteverini,who was later o theorize xtensivelyn thefourthdimensionn thecontext f SyntheticCubism, rotenthe allof1913 hat wemustforget xterior eality ndourknowledgefitinorder ocreate henewdimensions.. . 37AndBoccionihimself evealed debt oApol-linaireandCubism'spatialourth imensionin his assertion hat the fourthdimensioncannot xistperfectlywithout hecompleteaffirmationf the hree imensionshich eter-mine olume -an echoofApollinaire'state-

mentof 1912that he fourthdimension isengenderedy he hree nown imensions. 38What hen was the relationshipf higherspace o time ndmotionn Boccioni'synamicinterpretationf the fourthdimension? isapproacho hyperspace hilosophy asnotthatof Kupka r Duchamp,who sought ogeneratea higherdimensionalormby themotion f a three-dimensionalbject hroughspace. nstead,fBoccioni asatall nfluencedbythehyperspacehilosophyf Hinton 9oran Italian arallel,twasthereverse rocess,the passageof a higherdimensionalormthrough ur space,that nterested im. Thepurpose f Hinton'system fcubeshadbeento educate reader's space ense n orderfor him o visualizehisprocess.Althoughhereaderwould ctuallyerceivenly hesectionsof thehypercubeitseight ndividualoloredcubes n succession),he wouldhopefullyx-perience ome ntuition f thegreater ealityofthewhole our-dimensionalbject.

Fig. 12 Intersectionfa Spiralnda Plane,fromHinton,TheFourth imension1904),p.27.An analogyHintonhad used as earlyas1888,andwhichwaswellknownnthisperiod,was henotion fa handwith ts five ingertipsplacedon a table.40 two-dimensionaleingin theplaneof thetablewouldbe unawarefthe single uniqueorm (thethree-dimen-sionalhand) giving thefiveseparate in-

gerprints] ontinuityn [three-dimensional]space. n otherwords, hehandwould unc-tionas ahigher imensionalorm,ranscendingtheartificialivisions etweenbjectsnspace,divisionswhichBoccioni onsistentlyecried.Thus,Boccioni'stridingigure, s depictednsketchesFig. 11), preparatoryodels, ndafinal culpture,uggestshepassagehroughurspaceof a four-dimensionaligure a uniqueform),whosesuccessivetatesmaterializenddematerializeefore ureyes.As the RussianmysticOuspensky ouldlaterargue, imeandmotionn three-dimen-sionalspacemaybe consideredllusionshatWinter1981 321



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resultonlybecause four ncompleteercep-tionofspace.41 occioni,he ollower fBerg-son,wouldneverhaveacceptedhisextremeview,however.nstead,e assertedhepositivevalueoftimeandmotion sthemosteffectiveindicationfahigher, ynamiceality. urthercomparison ithHintons suggestedyBoc-cioni'snterestnthespiralorm san nnatelydynamichapepossessing absolute otion,a concernmanifestednhiswritingsnd n hisDevelopment f a Bottle n Spaceof 1912(TheMuseumf ModemArt,NewYork).42nTheFourthDimensionHintonhad demon-stratedhewaynwhichhe llusion f circularmotion n a planecouldresultwhena spiralpassedhrough planeFig.12). Inrealityherelativemotion f thepointwouldbe sub-sumedwithinhe absoluteotion fthe piral(itsverticalmovementnathirddimension).ForBoccioni, hen,the fourthdimensionhadboth patial nd emporalmplications,sthe fixed ndabsolutemeasure hich at hesame ime]createsdynamism. evertheless,Boccioni'sourthdimension iffers adicallyfrom hatof Cubism,ecause f itsemphasison timeandmotion.t s a continuingrojec-tion of the forcesandformsntuitedn theirinfinite nfolding. occioni'soal nUniqueForms f ContinuitynSpacewasasyntheticdepiction f motion, syntheticontinuityasopposedothe analyticaliscontinuity 43of Kupka,Duchamp,ndthe 1912worksofBalla, r theCubists'imultaneousresentationof multiple iewsof an object.Undoubtedly,muchof Boccioni'stylistic evelopmentc-curred ndependentlyf aninterestnhigherdimensions.However, henhe attemptedobringhis own heoriesnto inewith hewide-spread ontemporaryeliefnafourth imen-sionofspace,Boccioni ound suitablenter-pretationnoneaspect fhyperspacehiloso-phy.HisUnique orms fContinuitynSpaceis atributeohigher imensionalpace swellas to thedynamiceality fBergson.Intheend, the ourth imension as arless integralo the artand heory f Boccionithanit had been for the FrenchCubists rwouldbe for artists uch as DuchampndMalevich. evertheless,t is a measure ftheimportance f the idea in this periodthatBoccioni elt thathe mustclaim the ourthdimension or Futurism nd even turnitagainst is Cubist ivals.LindaDalrympleHenderson s anassistantprofessorfthehistory fartattheUniversityf Texas tAustin.Shehasdoneextensive esearchn artandnon-Euclideangeometry nd scholars reindebtedoherarticle n TheArtQuarterlyn 1971.Notes1Thepresentssays drawnromheauthor'sforthcomingtudy f the ourth imensioninearlywentieth-centuryrtand heoryobepublishedyPrincetonniversityress. hat

text is a revised ndconsiderablyxpandedversionf The rtist,The ourthimension,'andNon-Euclideaneometry900-1930:ARomancef ManyDimensionshenceforthreferredo as TheArtist... ),a doctoraldissertationompletedtYale niversityn1975.2By he aternineteenthentury,notherypeofgeometry,nown snon-Euclideaneome-try,also ontributedopublicnterestnalter-native inds fspace. on-Euclideaneometry,formulatedy hemathematiciansobachevsky,Bolyai,ndRiemann,tudiespacesfpositive,negative,r variableurvaturend s thusatotallyeparatereaofstudyromn-dimen-sionaleometry.lthoughon-Euclideaneom-etry evernjoyedhewidespreadopularityfthe ourthimension,t didhave n mpacton several rtists,ncludingMetzingerndGleizes, uchamp,ndElLissitzky.3The erm hyperspacehilosophysnotHin-ton'sbutmyown. twas nventedn orderodistinguishhewritingsf ndividualsuch sHinton,heRussian.D.Ouspensky,nd heAmericanlauderagdon,llofwhomxplorethephilosophicalmplicationsf higher i-mensions,rommore traightforwardxposi-tions fageometricourth imension.inton'smajor heoreticalextsareA New EraofThoughtLondon:wan onnenscheinCo.,1888)andTheFourthDimensionLondon:SwanSonnenscheinCo.,1904;NewYork:JohnLane,1904);n addition,epublishednumerousrticlesnd hort toriesased nthe ourth imension.4 Picasso, s quotedn Ram6nG6mez e laSera, CompletaveridicaistoriaePicassoelcubismo,nRevistaeOccidenteMadrid),vol.xxvJuly 929), . 100.5Foraninitialdiscussionf Cubistnterestnthe ourth imension,eeHenderson,ANewFacet f Cubism:The ourth imension'nd'Non-Euclideaneometry'einterpreted,heArtQuarterly,ol.xxxivWinter971),pp.410-33. The ignificancefhigher imen-sions orCubistrtand heorys discussednfargreateretailnHenderson,The rtist...,Ch. iiI.6 Guillaumepollinaire,La eintureouvelle:Notes 'art, esSoireeseParis,no.3 (April1912), p.90-91.7Guillaumepollinaire,editationssthetiques:LesPeintres ubistesParis: ugeneiguiere,1913),p.68.8 On the importancef Poincare'sdeas orGleizes ndMetzinger,eeHenderson,ANewFacet fCubism, swellasHenderson,TheArtist.. ., Ch.II.9 D'Alembertublishedhisidea n his 1754article n dimensionn theEncyclopedieedited yhimself ndDiderot.agrangere-sented similar iewn 1797n hisTheoriedesfonctionsnalytiques.10Einsteinegano attractublic otice nlynNovember919,whenhe esultsfexperimentsduring solar clipseonfirmedisassertionintheGeneralheoryhatightwavesrebent

by the massof the sun.See,forexample,Henderson,ANewFacetof Cubism, p.414 19;andRonaldW.Clark,instein:heLife nd TimesNewYork:Worldublishing,1971), p.227 66.11SeeEdwin bbottbbott,latland:Romanceof ManyDimensionsya SquareLondon:Seeley&Co.,1884).12In contrasto Bragdon'sidactic iagram(Fig. 4), a sphere assing erpendicularlythrough planewouldactually roducesuccessionf ncreasingnddecreasingirclesaroundsingle ointntheplane. he learestanalysisftherelationshipetweenpace ndmotionntime s obe oundnPeter emiano-vich Ouspensky'sertiumOrganium: heThird anonfThought,Keyo heEnigmasof the World 1911),trans. rom2nd ed.(1916)byClaude ragdonndNicolas es-saraboff,ndAmer.d.rev. NewYork: lfredA.Knopf),hs.ii-vi. Ouspensky'shilosophyandparticularlyis elaborationf thetwo-dimensionalnalogyCh.vi)is summarizedin Henderson,TheMergingf TimeandSpace:The ourth imension'nRussiaromOuspenskyoMalevich, he tructurist,o.15/16 1975/1976),p.97-108.13OnKupkandMarey,eeMargit owell,AMetaphysicsfAbstraction,nFrantisekupka1871-1975: A RetrospectiveSolomon .Guggenheimuseum,ewYork, 975),pp.49- 67.AsRowelloints ut,Mareyctuallyexperimentedith the generationf three-dimensionalirtualolumesyphotographingarotatingormp.56).14See,forexample,C.W.Leadbeater,'AutreCote elamort Paris:ditionsheosophiques,1910),whichncludednaccountfHinton'sideas.

15Rowell AMetaphysicsf Abstraction,nFrantisekupka,.66) irst otedhe imilar-ity of Kupka'snscriptiono the followingpassageromPawlowski'sVoyageupays elaquatriemeimension Comoedia,0May1912, . 1):Whereasnthree-dimensionalisplace-ment heatomsonstitutingbodyrepushed sideandreplacedy otheratomsormingnotherody.. dis-placementnthecountryf the ourthdimensions enactedywhat neusedtocalla transmutation.heworldfthe ourth imensioneingontinuous,no movementntheordinaryense fthe word anbe produceds in themobile orldf hree imensions.here-forea displacements made hroughanexchangefqualitiesetweeneigh-boringtoms.... When neentershecountryf the ourthimension,ove-ment uchas weknowt, no longerexists;herere nly ualitativehangesandweremainmmobile,nthecom-mon ense ftheword.16Duchamp,squotedn ElevenuropeansnAmericaInterviewsyJamesJohnsonweeney),322 ArtJournal



8/8

TheMuseum f ModernArtBulletin, ol.xIII/4-5 (1946),p. 20. Rowell ontrastshechronophotographicallynspiredpproachfbothKupka nd Duchampo that of theFuturistsn Kupka,uchamp,ndMarey,Studionternational,ol.CLXXXIXJanuary-February975), p.48-51.17Duchamp,squotednPierre abanne,ia-logueswithMarcelDuchamp,rans.RonPadgettNew ork:ikingress,971), p.29,34.18Marceluchamp,l'infinitifTheWhiteox),inSaltSeller: heWritingsfMarceluchamp(Marchand u sel), ed. Michel anouillet(New ork: xfordniversityress,973),.92.19Duchamp,squotednCabanne,ialogues,p.31.20Malevich'suprematistitlesuch sMovementofPainterlyMassesntheFourth imensionconfirmhe continuedolemotion layednhis thinkingbouthigher imensions.ow-ever,Suprematism'seometricanguagendits infinitewhite pacefreerom ravityndany dentifiablehree-dimensionality)ere arremovedromTheKnifeGrinder'simplisticsequentialotion.eeHenderson,TheMergingof Time ndSpace, p.104-5, althoughfar moredevelopediscussionf thesubjectoccursntheauthor'sorthcomingook.21On herelationshipfLeger'sssaipourestroisportraits o the intellectual ilieuofParis, eeChristopherreen, eger nd theAvant-GardeNewHaven:aleUniversityress,1976), p.22 27.22HenriBergson, reative volution1907),trans.ArthurMitchellNewYork: .Holt&Co.,1911), .302.23TheTechnical anifestofFuturistPaintingof 11April 910hadreferredo the vivifyingcurrentfscience. eeFuturistManifestos,ed. UmbroApollonio,rans.Robert rain,R.W. lint, .C.Higgitt, arolineisdallNewYork: ikingress, 973), . 18.Similarly,hetext ncludedntheBernheim-JeuneatalogueofFebruary912, asednalecturef29May1911by Boccioni,pokeof a lawof ourinteriormathematicsFuturistManifestos,ed.Apollonio,. 50).Marinetti'satermani-festo,GeometricndMechanicalplendourand the NumericalSensibility18 March1914) s a furtherestamento theFuturist'sgeneral redilectionorreferenceso mathe-matics. eeFuturistManifestos,d.Apollonio,pp.154-60. On hestrain fpositivismni-tially videntmongheFuturistainters,eeMarianne .Martin,uturistArtndTheory1909-1915 (Oxford:larendonress, 968),pp.45,46.24 DuncanM.Y. ommerville'sibliographyfNon-Euclideaneometry,ncludingheThe-oryofParallels,heFoundationsfGeometry,andSpacefn-DimensionsLondon:arrison&Sons,1911) istsa totalof322books ndarticlesn the new eometriesublishednItalybetween 901and1910.Italyrankedthird ehindGermanyndFrancenSommer-

ville'snumericalnalysisfpublicationsndcountedmong erprominentcholarsfthenew eometriesugenioeltrami,iuseppeVeronese, inoLoria,Roberto onola, ndFederigonriques.25Jarry'snterestn four-dimensionalndnon-Euclideaneometriess evidentnhiswritingsof the late1890s,uchas Gestestopinionsdu Docteur austroll,ataphysicienParis:Eugene asquelle,911)and Commentairepour ervir la constructionratiquee lamachine explorere temps MercureeFrance, ol.xxixFebruary899], p.387-96).SeeHenderson,The rtist.., pp.93-100.OnMarinettinParis ndhisconnectionswithApollinairendJarry,eeMartin,uturistArtandTheory,p.28 37,43.SeealsoGinoSeverini,uttaa vitadi unpittore Milan:Garzanti, 946),pp.96-98. OnMarinettiand thepoetsof theAbbayee Creteil,eeMarianneMartin, Futurism,nanism ndApollinaire, rt ournal,vol. xxviiiSpring1969),pp.258- 68.26 SeeMartin,uturistArtndTheory,. 110.27LetterromBoccionio NinoBarbantini,2February912,nArchivielFuturismo,d.Maria rudi ambillondTeresaioriRome:DeLucaEditore,1958]), .40.OtherettersreproducedyGambillondFioridocumentBoccioni'sresencenParis n 12November1912 nd21June1913.28For he nitial, horterersionf this ext, eeBoccioni, Plastic ynamism Lacerba,5December913),nFuturistManifestos,d.Apollonio,p.92- 95.29Boccioni,Pitturaculturauturistedinamismoplastico), oesiaMilan),914, p.196-99.30OnBragaglia,eeMartin,uturist rtandTheory,. 179,aswellas the extbyGiovanniListalsewherenthis ssue.31ThatBoccionidentifiedhephrasend itlewith the fourthdimensions confirmedyGino everininhisarticleLa einture'avant-garde, Mercurede France,vol. cxxI 1June1917), p.451 68.Severinirotehat Boc-cioni,when efininghat ecallsdynamism,'in connection ithourearlieresearchesnmovement,lludedo akind ffourth imen-sionwhichwould e 'theuniqueorm ivingcontinuitynspace' p.459).32In his discussionf the fourthdimension,Boccionilsoreflectshe nfluencefBergsonin hispreferenceor he intuitivebsoluteoverCubism'srationalrocedurehich xistsinrelativity.nCreativevolutionergson,theadvocatef intuitionersusntellect,adassertedhat in the absolute e liveandmove ndhaveourbeing.Theknowledgeepossess fit is incomplete,odoubt, utnotexternalrrelative p.199).For urther videncef Boccioni'sarefulstudy fBergson,eeBrian etrie, Boccioniand Bergson, heBurlingtonMagazine,vol. xviMarch974), p.14047. InadditiontoBergson'seneral hilosophicalnfluence,

Petrie stablishespecificourcesnBergson'swritingsorBoccioni'sdeas nthe nterpene-trationof forms,on absolute nd relativemotion, ndonforce-lines.33Henri ergson,ssai ur esdonneesmmedi-ates de la conscienceParis:FelixAlcan,1889), p.82-83.34Boccioni,The lastic oundationsfFuturistSculpturendPainting Lacerba,5March1913),nFuturistManifestos,d.Apollonio,p.88.35Boccioni, PlasticDynamism,n FuturistManifestos,d.Apollonio,.94;also,Pitturasculturafuturiste,.201.36 The xhibitorsothePublic, nLes einturesfuturistestaliens,xh. at. Galerieernheim-Jeune&Cie,Paris,-24 February912), .6;textbasedon a lecture f 29 May1911byBoccioni.37Severini,The lastic nalogiesfDynamism-FuturistManifestounpub.),eptember-October 913,in FuturistManifestos,d.Apollonio,. 118.38Apollinaire,LaPeintureouvelle, . 90.39Inhis article f 1917, LaPeinture'avant-garde pp.461, 464), SeverinimentionsbothMaurice oucher'sssai url'hypere-space(Paris: elixAlcan, 903),whichhaddiscussedhework fHinton,nd he heoriesof A. de NoircarmehosebookQuatriemeDimension Paris:Editions heosophiques,1912)had inked time-orientedyperspacephilosophyo Theosophy.f Severini newtheseworks eforehewar,hemaywellhavedirected occionio themandevenbeyondthem o Hinton.40Hinton,NewEraofThought,.65.41Had e knownfit,Boccioniould lsohave

rejectedhe nterpretation,oicedmost learlyby OuspenskyTertiumOrganum, . 102),whichsuggestshatbeyondheillusions ftimeandmotionour-dimensionalealitysstatic.Pawlowskihared uspensky'siewsnthissubject,s thequotationn n. 15abovedemonstrates.42Boccioniirstdescribeduturistculpturesspiral rchitecturen his prefaceor thecataloguereExpositionesculpturefuturistedu peintreet sculpteuruturisteBoccioni(Galeriea Boetie,Paris,20June- 16July1913). eeModern rtistsnArt, d.Robert.HerbertEnglewoodliffs, .J.: rentice-Hall,1964),p. 48.Thespirals alsodiscussednPlastic ynamism,nFuturistManifestos,ed.Apollonio,.94.Althoughoccioniresentedisdual efini-tions fmotionnthe ataloguerefacef1913,his distinctionsetween absolute otionand relative otion reclearestn histextAbsoluteotion Relative otion Dyna-mism Lacerba,5March914),nFuturistManifestos,d.Apollonio,p.150-51.43Rowell,Kupka,uchamp,ndMarey, .48.

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Futurism And the Forth Dimension