escher e a matemática
Transcrição
escher e a matemática
INTERFACE TECNOLÓGICA - v.5 - n.1 - 2008 ESCHER E A MATEMÁTICA Adriana Batista da SILVA* Cristina Aparecida ZAPATA** Luciana Aparecida FERRAREZI*** RESUMO O objetivo é mostrar a importância das obras de Escher, não só para os matemáticos, mas também sua utilização na matemática de Horma prática, possibilitando ao aluno visualizar os conceitos matemáticos. Assim, a contribuição de Escher é signiſcativa principalmente por que identiſcamos de Horma prática alguns conceitos matemáticos existentes em suas obras, que mostra que podemos utilizar outros recursos para tornar as aulas mais atrativas, com exemplos suſcientemente claros, Hacilitando o entendimento da matemática. PALAVRAS-CHAVE: Escher. Matemática. Arte. INTRODUÇÃO $RREVHUYDUDOJXPDVGDVREUDVGH(VFKHUSHUFHEHPRVDH[LVWrQFLDGHGLYHUVRVFRQFHLWRVPDWHPiWLFRV HP VHXV TXDGURV p QHVVH FRQWH[WR TXH VXUJH DOJXPDV SHUJXQWDV FRPR 6HUi TXH (VFKHU HUD XP HVWXGLRVRGD0DWHPiWLFD"6HUiTXHFRQKHFLDRVFRQFHLWRVPDWHPiWLFRVTXHHQYROYLDPVXDVREUDV" 6HUiTXH(VFKHUVDELDTXHVXDVREUDVHUDPHVWXGDGDVHDGPLUDGDVSHORV0DWHPiWLFRV" 2REMHWLYRGRWH[WRYHPGHHQFRQWURFRPHVVDVSHUJXQWDVQXPDWHQWDWLYDGHUHVSRQGrODV3DUDLVVR DSUHVHQWDUHPRVXPDEUHYHELRJUD¿DGH(VFKHUGHVWDFDQGRRVSULQFLSDLVSRQWRVGHVXDYLGDFRPR SRUH[HPSORRIDWRGHHOHQmRJRVWDUGHPDWHPiWLFDEHPFRPRXPEUHYHUHODWRVREUHDPDWHPiWLFD envolvida em suas obras e a divisão das obras principais. O presente artigo faz parte do trabalho de conclusão de curso de $GULDQD%DWLVWDGD6LOYDFRPFDUiWHUGHSHVTXLVDELEOLRJUi¿FDHFDUDFWHUtVWLFD principal de divulgação de suas obras. 1. A VIDA DE ESCHER 0RXULWV&RUQHOLV(VFKHUQDVFHXHPGHMXQKRGHHP/HXZDUGHQDRQRUWHGD+RODQGD6HXV pais eram Sarah Gleichman e Gorge Arnold Escher que era engenheiro civil. 6HJXQGR(UQHVW(VFKHUQmRJRVWDYDGHLUDHVFRODRTXHH[SOLFDRIDWRGHOHVHUXPDOXQR UHODWLYDPHQWHIUDFRHWHUVLGRUHSURYDGRSRUGXDVYH]HV$V~QLFDVDXODVGHTXHJRVWDYDHUDPDVDXODV VHPDQDLVGHGHVHQKR(VFKHUQmRFRQVHJXLXRGLSORPD¿QDOSRUQmRREWHUERDVQRWDV _______________________________ $OXQDGD/LFHQFLDWXUD3OHQDHP0DWHPiWLFD)$7(%%LULJXL(PDLOGU\NDBEV#\DKRRFRPEU 0HVWUHHP0DWHPiWLFD$SOLFDH&RPSXWDFLRQDO81(635LR3UHWR'RFHQWHGR,0(6)$),&$&DWDQGXYDH)$7(% %LULJXL(PDLOFULVWLQD#YLFHQ]]RFRPEU 0HVWUHHP(GXFDomR0DWHPiWLFD±81(635LR&ODUR'RXWRUDQGDGD)&/$U81(63$UDUDTXDUD±3URJUDPDGH (GXFDomR(VFRODU'RFHQWHGD)$7(&7DTXDULWLQJDH,0(6)$),&$&DWDQGXYD(PDLOOXDIHUUDUH]L#KRWPDLOFRP 143 INTERFACE TECNOLÓGICA - v.5 - n.1 - 2008 Figura - Auto retrato )RQWH(51(67S $LQGDFRPRGHVFUHYHRDXWRURSDLGH(VFKHUDFKDYDTXHR¿OKRGHYHULDID]HUXPFXUVRQDiUHDGDV FLrQFLDVH[DWDVSRUHVVHPRWLYRHP(VFKHUIRLSDUD+DDUOHPHVWXGDU$UTXLWHWXUDQD(VFRODGH $UWHV'HFRUDWLYDV0DVHVVDDYHQWXUDQmRGXURXPXLWRWHPSRSRLV6DPXHO-HVVHUXPGH0HVTXLWD TXHHUDVHXSURIHVVRUYLXTXH(VFKHUHVWDYDYROWDGRSDUDDV$UWHV'HFRUDWLYDVGRTXHSDUDRFXUVR GH$UTXLWHWXUDHDVVLPPXGRX(VFKHUGHFXUVRPDVTXHWDPEpPQmRREWHYHPXLWRVXFHVVRHP$UWHV Decorativas. 1RDQRGH(VFKHUDEDQGRQRXDHVFRODGH$UWHV1RSHUtRGRHPTXH¿FRXQRFXUVRGH$UWHV (VFKHUDGTXLULXXPDERDEDVHHPGHVHQKRHHPWpFQLFDVGHJUDYXUDVDUWtVWLFDVGRPLQDYDDWpFQLFDGH [LORJUDYXUDVDUWHGHHVFXOSLUHPPDGHLUD (P(VFKHUYLDMRXSDUDD,WiOLDSRUGXDVYH]HVHHPVXDVHJXQGDYLDJHPTXHGXURXDWpRLQtFLR GR SUy[LPR DQR IRUDP FULDGDV DV SULPHLUDV [LORJUDYXUDV GH SDLVDJHQV7DPEpP HP XPD GH VXDV YLDJHQVSDUDD,WiOLDFRQKHFHX-HWWD8PLNHUFRPTXHPVHFDVRXHP$VVLPFRPRDPmH-HWWD GHVHQKDYDHSLQWDYDHPERUDQmRWLYHVVHDGHYLGDSUHSDUDomR(5167 1HVVD PHVPD pSRFD VHJXQGR (UQVW (VFKHU H DOJXQV DPLJRV ¿]HUDP YiULDV YLDJHQV TXH TXDVH VHPSUH HUDP IHLWDV HP FRPERLRV HP QDYLRV RX DWp PHVPR D Sp (VVDV YLDJHQV FKHJDYDP DGXUDUPHVHVHTXDQGRUHWRUQDYDPSDUDFDVDHVWDYDPPDLVPDJURVHFDQVDGRVPDVVHPSUHFRP PXLWRVGHVHQKRVRTXHSDUDHOHFRPSHQVDYDWRGRRHVIRUoR(PXPDGHVVDVYLDJHQV(VFKHUIRLDWp &DVWURYDOYDRQGHIH]XPDGHVXDVPDLVEHODVREUDVGH/LWRJUD¿DHVSpFLHGHHVWDPSDGHSDLVDJHQV DIDPRVD&DVWURYDOYDSURGX]LGDQRDQRGH(VVDREUDIRLPXLWRHORJLDGDSRUYiULRVWLSRVGH FUtWLFRVDWpPHVPRRSUySULR(VFKHUH[DOWDDEHOH]DGHVVDSDLVDJHP 'XUDQWHDOJXPWHPSR(VFKHUGHSHQGHXGHVHXVSDLVSRLVQmRKDYLDIHLWRPXLWDVH[SRVLo}HVHQHP YHQGLDPXLWRVTXDGURV(PHOHFRPHoDYDDYLYHUEDVLFDPHQWHGHVXDVREUDVSRLVVRPHQWHQHVVH DQRYHQGHUDHVWDPSDVHHPPDLVRTXHOKHUHQGHXXPDTXDQWLDGHÀRULQVPRHGD TXHFLUFXODYDQDpSRFD0DVDHVVDDOWXUD(VFKHUMiHUDFRQKHFLGRQmRSHODVVXDVSDLVDJHQVPDVSHODV UHSURGXo}HVJUi¿FDVGHXPPXQGRPDWHPiWLFRTXHRIDVFLQDYDPHVPRVHPFRQKHFrORPXLWREHP 144 INTERFACE TECNOLÓGICA - v.5 - n.1 - 2008 $LQGDGHDFRUGRFRPDSHVTXLVDKLVWyULFDUHDOL]DGDSRU(UQVWHPHIRUDPODQoDGRV GRLV OLYURV RQGH DV LOXVWUDo}HV IRUDP IHLWDV SRU (VFKHU QHVVH SHUtRGR R VHX UHFRQKHFLPHQWR IRL DXPHQWDQGRQmRVyQD(XURSDPDVWDPEpPQD$PpULFDRQGHHPIRLSUHPLDGRFRPRWHUFHLUR OXJDUHPXPDH[SRVLomRUHDOL]DGDHP&KLFDJRFRPDOLWRJUD¿D1RQ]D (P SRU FDXVD GD SROtWLFD GD ,WiOLD (VFKHU H D IDPtOLD PXGDUDPVH SDUD D 6XtoD RQGH QmR SHUPDQHFHUDPSRUPXLWRWHPSR(VFKHUGL]LDTXHDSDLVDJHPJHODGDQmRHUDLQVSLUDGRUD (VFKHUHDHVSRVDYLDMDUDPGXUDQWHDOJXQVPHVHVHPXPQDYLRFDUJXHLURRTXDOIRLSDJRFRP HVWDPSDV GH VXDV REUDV 1HVVD YLDJHP (VFKHU H D HVSRVD SDVVDUDP SRU )LXPH GH RQGH SDUWLUDP SDUD 9DOrQFLD H *rQRYD (VVD YLDJHP IRL PXLWR SURYHLWRVD SDUD HOH SRLV IRL TXDQGR REVHUYRX R IXQGDPHQWR EiVLFR GD GLYLVmR UHJXODU GH VXSHUItFLHV IXQGDPHQWRV HVVHV TXH VHULDP DSOLFDGRV HP suas obras de grandes sucessos. (P(VFKHUHDIDPtOLDPXGDUDPVHSDUD8NNHOQD%pOJLFDPDVORJRHVWRXURXDJXHUUD (P(VFKHUSHUGHXRSDLRTXDOMiHVWDYDFRPDQRVHQRDQRVHJXLQWHHPSHUGH WDPEpPDPmHHHQWmRHPUHVROYHXPXGDUFRPDIDPtOLDSDUD%DDUQQD+RODQGDSRLVOiD escola secundária tinha um bom nome e também era seu país de origem. $SDUWLUGH(VFKHUSHUFHEHTXHHVWiQRiSLFHGHVHXWUDEDOKRSRLVYrVHXQRPHHVWDPSDGRHP várias revistas internacionais como 7LPH e The Studio. (VFKHUWDPEpPIH]YiULDVH[SRVLo}HVHQWUHHODVXPDHP$PHVWHUGmRDTXDORFRUUHXGXUDQWHXPD &RQIHUrQFLD,QWHUQDFLRQDOGH0DWHPiWLFDHSRVWHULRUPHQWHHP:DVKLQJWRQ(VFKHUWDPEpPSXEOLFRX XPWH[WRVREUHDGLYLVmRUHJXODUGHSODQRTXHVHWRUQRXPXLWRIDPRVR $SDUWLUGHVVHPRPHQWRDVREUDVGH(VFKHUWRUQDPVHXPHORHQWUHDFLrQFLDHDDUWHHQWUHD$UWHHD Matemática. (QWUHRVDQRVGHH(VFKHUDGRHFHHSDVVDSRUYiULDVFLUXUJLDVPHVPRHPUHFXSHUDomR HPHOHFULDVXD~OWLPDREUDD;LORJUDYXUD6HUSHQWHVTXHVHWRUQRXXPDGHVXDVPDLVIDPRVDV obras. 3DUDFRPHPRUDURDQLYHUViULRGHDQRVGH(VFKHUHPR0XVHXGD&LGDGHGH+DLDSURPRYHX XPDJUDQGHH[SRVLomRUHWURVSHFWLYDGHVXDREUDHWDPEpPSRGHPRVGHVWDFDUTXHHVVDH[SRVLomRWHYH XPJUDQGHS~EOLFR $V REUDV GH (VFKHU LQVSLUDUDP XP FRPSRVLWRU TXH MXQWDPHQWH FRP D 2UTXHVWUD )LODUP{QLFD GH 5RWHUGmR¿]HUDPWUrVUHFLWDLVFRPDDSUHVHQWDomRVLPXOWkQHDGDVREUDVGH(VFKHUGHVWDFDPRVTXH WRGDVDVDSUHVHQWDo}HVWLYHUDPFDVDFKHLDHWDPEpPTXHDPDLRUSDUWHGDSODWpLDHUDFRPSRVWDGH jovens. (PODQoDUDPXP¿OPHVREUH(VFKHU¿OPHHVVHTXHIRLHQFRPHQGDGRSHOR ministro dos negócios 145 INTERFACE TECNOLÓGICA - v.5 - n.1 - 2008 HVWUDQJHLURVGRV3DtVHV%DL[RV 7DPEpPHP(VFKHUPXGRXVHSDUDD&DVDGH5RVD6SLHUFDVDSDUDDUWLVWDVLGRVRVHP/DUHP DRQRUWHGD+RODQGDPDVPHVPRPXLWRGRHQWHIRLSXEOLFDGRROLYUR7KH:RUOGRI0&(VFKHU (O PXQGRGH0&(VFKHU(VFKHUPRUUHXHPGH0DUoRGHQRKRVSLWDO+LOYHUVXP 2. 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A estrutura da VXSHUItFLH VHJXQGR (UQHVW VHUYLX FRPR EDVH SDUD FULDU WUrV QRYRV JUXSRV 0HWDPRUIRVH &LFORVH$SUR[LPDomRDRLQ¿QLWR - Metamorfose )LJXUD0HWDPRUIRVH, )RQWH(51(67S 147 INTERFACE TECNOLÓGICA - v.5 - n.1 - 2008 &LFORV $SUR[LPDomRDRLQ¿QLWR )LJXUD5pSWHLV )LJXUD/LPLWHFLUFXODU,9 )RQWH(6&+(5S )RQWH(6&+(5S 5HSUHVHQWDomRSLFWyULFDGDUHODomRHQWUHHVSDoRHVXSHUItFLHSODQD 1HVWDFDWHJRULDpGHVWDFDGRRFRQÀLWRYLYLGRSRU(VFKHUTXDQGRSURGX]LDXPD¿JXUDHVSDFLDORX VHMDWUrVGLPHQV}HVWrPTXHVHUUHSUHVHQWDGDVHPXPHVSDoRELGLPHQVLRQDOSRUpPQHPVHPSUHDV ¿JXUDVFULDGDVSRU(VFKHUSRGHULDPH[LVWLUQRHVSDoR 1HVVD WHUFHLUD FDWHJRULD QRYDPHQWH GHVWDFDVH D FULDomR GH WUrV QRYRV VXEJUXSRV VHQGR HOHV$ HVVrQFLDGDUHSUHVHQWDomRFRQÀLWRHVSDoR± VXSHUItFLH3HUVSHFWLYDH)LJXUDVLPSRVVtYHLV $HVVrQFLDGDUHSUHVHQWDomR 3HUVSHFWLYD )LJXUD'UDJmR )RQWH(6&+(5S )LJXUD3URIXQGLGDGH )RQWH(6&+(5S 148 INTERFACE TECNOLÓGICA - v.5 - n.1 - 2008 - Figuras impossíveis )LJXUD%HOYHGHU )RQWH(6&+(5S 'XUDQWH VXD YLGD DUWtVWLFD SHUFHEHVH TXH (VFKHU HP GHWHUPLQDGRV SHUtRGRV WUDEDOKD FRP WHPDV GLIHUHQWHVHSRULVVRSRGHVHD¿UPDUTXHVXDVREUDVVHGLYLGHPHPTXDWURJUDQGHVSHUtRGRV 3HUtRGRVGDVREUDVGH(VFKHU ±3HUtRGRGDVSDLVDJHQV 3DUD (UQHVW QHVVH SHUtRGR D PDLRU SDUWH GDV ¿JXUDV GH (VFKHU UHSUHVHQWDP SDLVDJHQV GH SHTXHQDVFLGDGHVGRVXO,WDOLDQRHGHUHJL}HVFRVWHLUDVGRPHGLWHUUkQHRVmRGHVWDFDGRVWDPEpPQHVVH SHUtRGRDOJXQVUHWUDWRVSODQWDVHDQLPDLV $ JUDQGH REUD GHVVH SHUtRGR IRL D SDLVDJHP &DVWURYDOYD $ ~OWLPD JUDYXUD FRP R WHPD SDLVDJHPIRLDOLWRJUD¿D7UrV0XQGRV 149 INTERFACE TECNOLÓGICA - v.5 - n.1 - 2008 )LJXUD&DVWURYDOYD )RQWH(6&+(5S )LJXUD7UrVPXQGRV )RQWH(6&+(5S ±3HUtRGRGDVPHWDPRUIRVHV &RPHoRXFRPDJUDQGHREUD0HWDPRUIRVHPDVRJUDQGHPRPHQWRGHVVHSHUtRGRIRL'LDH1RLWH LVVRSRUTXHDOpPGDPHWDPRUIRVHREVHUYDPVHFLFORVHDWUDQVIRUPDomRGRELGLPHQVLRQDOHP tridimensional. 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ABSTRACT 7KH REMHFWLYH LV WR VKRZ WKH LPSRUWDQFH RI WKH ZRUNV RI (VFKHU QRW RQO\ IRU WKH PDWKHPDWLFLDQV EXWDOVR\RXUXVHLQWKHPDWKHPDWLFVLQDSUDFWLFDOZD\PDNLQJSRVVLEOHWKHVWXGHQWWRYLVXDOL]HWKH PDWKHPDWLFDOFRQFHSWV/LNHWKLVWKHFRQWULEXWLRQRI(VFKHULVPDLQO\VLJQL¿FDQWZK\ZHLGHQWL¿HG LQDSUDFWLFDOZD\VRPHH[LVWHQWPDWKHPDWLFDOFRQFHSWVLQ\RXUZRUNVWKDWLWVKRZVWKDWZHFDQXVH RWKHUUHVRXUFHVWRWXUQWKHPRVWDWWUDFWLYHFODVVHVZLWKH[DPSOHVVXI¿FLHQWO\FOHDUIDFLOLWDWLQJWKH XQGHUVWDQGLQJRIWKHPDWKHPDWLFV KEYWORDS: (VFKHU0DWKHPDWLFV$UW 153 INTERFACE TECNOLÓGICA - v.5 - n.1 - 2008 REFERÊNCIAS (51(67%(VSHOKRPiJLFRGH0&(VFKHUHG637DVFKHQ 7$6&+(1%0&(VFKHUJUDYXUDHGHVHQKRHG+RODQGD7DVFKHQ 7$6&+(1%Caleidociclos de M. C. Escher.HG+RODQGD7DVFKHQ 154
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