nombre d’or rectangles d’or divine proportion …
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Nombre d’OR Rectangles d’Or Divine proportion …. Avertissement. - PowerPoint PPT PresentationTRANSCRIPT
Le Nombre dOR
Nombre dOR
Rectangles dOr
Divine proportion AvertissementOn na pas toujours pu dterminer de faon certaine si les proportions observes dans les uvres architecturales et les ouvrages dart rvlaient une intention plus ou moins consciente de l'artiste, ou si ce ntait qu'une grille de lecture place a posteriori sur une uvre ( Il faut dans ce domaine rester modeste et ne pas vouloir tout prix faire apparatre le nombre d'or partout )Ces limites tant poses, on peut nanmoins prsenter quelques exemples d'oeuvres o le nombre d'or semble jouer un rle important
Partons donc la dcouverte du Nombre dOr
Un petit test : Regardez simplement chacun des rectangles de la diapositive suivante et retenez celui que vous jugez le plus harmonieux Retenez bien le n choisi
145362Refaisons le mme test 51326 4Les rectangles d'or sont respectivement les nos .
Il parat (*) que ces rectangles sont le plus souvent choisis... Leurs proportions donnent une belle impression d'harmonie .
(*)
Les rectangles d'or sont respectivement les nos 3 et 4 !
Daprs une tude du Philosophe allemand Gustav Feshner en 1876
longueur Le rapport ------------- largeur
vaut peu prs 1,62
On dsigne gnralement le nombre dor par la lettre grecque en hommage au sculpteur grec Phidias ( 490 430 avant J.C. ) qui dcora le Parthnon Athnes.La Section dore est une appellation qui remonte 1830 .
Elle tait appele par les Grecs partage dun segment en moyenne et extrme raison Principe :Ce principe est sens raliser en architecture , en peinture, en sculpture, les proportions les plus quilibres , les plus harmonieuses
Dans un ensemble compos de 2 parties, le tout est la plus grande comme celle-ci est la plus petite .m partage le segment [ a , b ] selon ce principe si abmm partage le segment [ a , b ] selon ce principe si
abmLe toutm partage le segment [ a , b ] selon ce principe si
abmLe toutLa plus grandeLa plus grandem partage le segment [ a , b ] selon ce principe si
abmLe toutLa plus grandeLa plus grandeLa plus petite Un peu de math
abmLe toutLa plus grandeLa plus grandeLa plus petite =
abmLe toutLa plus grandeLa plus grandeLa plus petiteor
=
abm
On a donc =
abm
=
abm
=
Rduisons au mme dnominateur
= 1 +
= 1 +
2 - - 1 = 0 Une simple quation du 2eme degr
Lquation 2 - - 1 = 0
possde deux solutions car
Lquation 2 - - 1 = 0
possde deux solutions car
Lquation 2 - - 1 = 0
possde deux solutions car
1 = 2 = Lquation 2 - - 1 = 0
possde deux solutions car
1 = 2 = Seule la 1re solution correspond un point m
Constructions du Nombre dOrUne construction simple
c12ab
Thormede PythagoreTraons la parallle [ab] par le milieu de [ bc] c12ab
c12ab
prenons notre compas c12ab
c12ab
c12ab
Et voil le Nombre dOR !c12ab
Et voil le Nombre dOR !c12ab
Variante
A nouveauPythagore
+
Rectangles dOr
b aConsidrons un rectangle dOr
Rectangles dOr
b aInscrivons-y le plus grand carr possible
Rectangles dOr
b a b a - bCarr
b a b a - b
CarrExaminons le rapport des dimensions du rectangle obtenu
b a b a - b
Le nouveau rectangle obtenu est donc un rectangle dOrCarr
En inscrivant successivement le plus grand carr aux rectangles obtenus Rectangle dorOn obtient une succession de rectangles dOr CarrRectangle dorOn obtient une succession de rectangles dOr CarrRectangle dorOn obtient une succession de rectangles dOr CarrRectangle dorOn obtient une succession de rectangles dOr CarrRectangle dorOn obtient une succession de rectangles dOr On obtient une succession de rectangles dOr On obtient une succession de rectangles dOr
Spirale des rectangles dOr La spirale des rectangles d'or est une fausse spirale parce qu'elle est constitue d'arcs de cercles au lieu d'avoir une variation continue du rayon. Cependant les raccordements des arcs sont parfaits car les centres des arcs sont chaque fois situs sur la mme droite et il y a une unique tangente chaque point de raccordement .Elle tend rapidement vers un centre Z . Le segment de droite qui joint le centre Z un point de la courbe crot en progression gomtrique. La longueur du rayon vecteur est multiplie par le nombre d'or chaque fois que sa direction tourne d'un quart de tour.
Cette spirale se rencontre beaucoup dans la nature Nautile
Nautile
Nautile
Nautile
Nautile
Nautile
Nautile
Nautile
Nautile
Nautile
Nautile
Nautile
modle mathmatiqueQuelques repres HistoriquesIl y a 10 000 ansPremiers signes de la connaissance par lhomme ( Temple dANDROS dcouvert sous la mer des Bahamas )2800 Avant J C : Pyramide de Kheops
Selon la lgende , les prtres gyptiens disaient que le carr construit sur la hauteur verticale galait exactement la surface de chacune des faces triangulaires
ahEncore un peu de mathH = a.h HahDaprs HerodoteHahH = a.h HahH = a.h HahH = a.h HahH = a.h HahH = a.h HahH = a.h HahH = a.h HahH = a.h HahH = a.h HahH = a.h HahH = a.h HahH = a.h HahH = a.h HahH = a.h HahH = a.h HahH = a.h HahH = a.h HahH = a.h HahH = a.h h - a = a.h
En utilisant nouveau le thorme de PythagoreHahH = a.h h - a = a.hDivisons les deux membres par ah :
Hah
H = a.h h - a = a.hDivisons les deux membres par ah :
Hah
H = a.h h - a = a.hDivisons les deux membres par ah :Posons
Hah
Posons
On retrouve lquation qui nous a permis de trouver le nombre dOrdonc la proportion entre la hauteur ( h) d'une face triangulaire et la moiti (a) du ct de la base est gale au nombre dor
ah
Pythagore (-580;-500)
mathmaticien et philosophe grec tait passionn par l'harmonie et les proportions. Son trait sur la musique est clbre. On lui doit la dcouverte de l'irrationalit de certains nombres : et que l'on trouve dans le nombre d'or et le pentagone rgulier Pythagore (-580;-500)
Les Pythagoriciens voyaient dans les nombres les principes de toute chose
Le pentagramme tait le symbole des pythagoriciens.
Pentagones et dcagones rguliersC572Pentagone rgulierC572Pentagone rgulierC572Pentagone rgulierC572Pentagone rgulierC572Pentagone rgulierC5Pentagone toil ( Pentagramme )E5C5Pentagone toil ( Pentagramme )E5C5Pentagone toil ( Pentagramme )E5C5Pentagone toil ( Pentagramme )E5C5Pentagone toil ( Pentagramme )E5abC10367272ordcDcagone rgulierabC10367272orcDcagone rgulierabC10367272ordcDcagone rgulierabC10367272ordcDcagone rgulierabC10367272ordcDcagone rgulierabC10367272ordcDcagone rgulierabC10367272ordcDcagone rgulierabC10367272ordcDcagone rgulierabC10367272ordcDcagone rgulierabC10367272ordcDcagone rgulierabC10367272ojrdcabC10367236oConstruisons la bissectrice aj de obj36rdcabC10367236o j3672rdcabC10367236o j3672rdcabC10367236o j3672S aob ajbrdcabC10367236o j3672S aob ajb
oaajabjbrdcabC10367236o j3672
oaajabjbrr
dcabC10odcR
Le nombre dOr est donc prsent dans le dcagone rgulierC5E5
On montre aussi par les triangles semblables que447-432 av.JCLe sculpteur grec Phidias utilise le nombre dOr pour dcorer le Parthnon Athnes, en particulier pour sculpter la statue dAthna Partnos .
Le Parthnon sinscrit dans un rectangle dor
=
ABFGCDE
=
Sur la toiture , on a aussi
anbentre autre
Statue dAphroditeLe thatre dEpidaure ( IVme sicle av.J-C )
les rapports et sont
Retenons bien les nombres 21 , 34 et 55 On y reviendra dans quelques instants !III me sicle av. J-C. Premires traces crites :Dans ses Elments , Euclide explique le partage dun segment en extrme et moyenne raison
Une droite est dite coupe en extrme et moyenne raison quand, comme elle est toute entire relativement au plus grand segment, ainsi est le plus grand relativement au plus petit (Euclide , Elments, livre IV , 3eme dfinition )Lonard de Pise, plus connu sous le nom de Fibonacci, est l'origine du premier modle mathmatique de la croissance des populations. Il tudia du point de vue numrique la reproduction des lapins. L'unit de base est un couple de lapins, il considre qu'un couple de jeunes lapins met une saison devenir adulte, attend une deuxime saison de gestation, puis met au monde un couple de jeunes lapins chaque saison suivante
En supposant que les lapins ne meurent jamais , on obtient donc le schma suivant :
Lorsque l'on compte le nombre de couples de lapins chaque saison, cela donne... la suite de Fibonacci 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610............ pour laquelle on a
un = un-1 + un-25 + 8 = 138 + 13 = 2113 + 21 = 3421 + 34 = 55Etc
Et revoici le Nombre dOr on remarque que le rapport entre un nombre de la suite et son prcdent s'approche de plus en plus du nombre d'or: ~ 1,61803398 21/13 ~ 1,615
34/21 ~ 1,619
144/89 ~ 1,617
610/377 ~ 1,618
La pomme de pin montre clairement les spirales de Fibonacci : 8 vertes dans un sens, 13 rouges dans l'autre sens.Fibonacci dans la natureLa fleur de tournesol forme deux sries de spirales tournant en sens contraire: 13 spirales partent de son centre dans une direction, 21 dans l'autre
Ananas :13 spirales droite et 8 spirales gauche
Escargot : la spirale de baseLes btisseurs de cathdrales Aux XIe et XIIe sicles ,hritire des traditions anciennes, lorganisation ouvrire du compagnonnage o les savoirs se transmettaient oralement de matre compagnon , a fait des rgles du nombre dor le principe du savoir-faire des btisseurs
Notre-Dame de Paris
Notre-Dame de Paris
Cathdrale de Strasbourg
abcdpyo
Le clbre Taj Mahl, immense monument funraire construit en Inde par un architecte persan assist de nombreux compagnons de nationalits diffrentes, a t construit galement selon les proportions du nombre dor
Les coupoles de plusieurs basiliques et mosques clbres furent proportionnes selon la Section d'Or et elles reposent sur une base cubique, comme, par exemple, celles de la Basilique Sainte Sophie Istanbul, de la Basilique Saint Pierre RomeLes btisseurs de cathdrales Les btisseurs de cathdrales utilisaient une pige constitue de cinq tiges articules, correspondant chacune une unit de mesure de lpoque, relatives au corps humain : la paume, la palme , lempan, le pied et la coude Les btisseurs de cathdrales Les btisseurs de cathdrales utilisaient une pige constitue de cinq tiges articules, correspondant chacune une unit de mesure de lpoque, relatives au corps humain : la paume, la palme , lempan, le pied et la coude Coude
PaumeempanpalmeCoudePiedempan
palmePaumeLes longueurs taient donnes en lignes ( une ligne mesurant 2,247 mm )Paume347,64palme5512,63empan8920Pied14432,36Coude23352,36Lignes cmCoudePiedempan
palmePaumePaume347,64palme5512,63empan8920Pied14432,36Coude23352,36Lignes cmOn retrouve les termes de la suite de Fibonacci !! Fra Luca Pacioli, un moine professeur de mathmatique , crit
De divina proportionne Fra Luca Pacioli, un moine professeur de mathmatique , crit
De divina proportionne Lonard de Vinci (1452-1519) a dessin les polydres pleins ou creux qui illustrent le trait de Luca Pacioli, son matre de gomtrie .
DECDans ce tableau de Jicopo de Barbari, o Fra Luca Pacioli explique un thorme , on a
= 1,62
icosadredodcadre
Pour Platon , le Dodcadre ne symbolise rien moins que lUniversLonard de Vinci , philosophe humaniste de la renaissance, comme plusieurs autres peintres clbres a utilis la proportion dOr dans ses toiles
Le visage de sa clbre Joconde est inscrit dans un rectangle dor
Le Saint Jrome de Leonard de Vinci est aussi parfaitement intgr dans un rectangle dor
LAnnonciation ( Leonard de Vinci )abmc x y
Leonardo de Vinci est aussi clbre par ses observations du corps humainL'Homme est inscrit dans un cercle. Quand il lve les bras et a les jambes cartes, le centre du cercle correspond au nombril. S'il se tient jambes serres et bras l'horizontale, il s'inscrit dans un carr.
Le nombril divise la hauteur de l'homme en deux segments qui sont dans le rapport d'Or
Le rapport entre la distance comprise entre l'extrmit de la main droite et l'paule gauche et celle comprise entre l'paule gauche et l'extrmit de la main gauche correspond au Nombre d'Or.
Dessin de Leonard de Vinci ( sans doute autoportrait ) certains des rectangles de la grille sont des rectangles dOr
Lhomme parfait dAggripa Von Nettesheim ( 1553) (Mdecin, philosophe et alchimiste allemand )
Bien que les peintres aient souvent travaill en se laissant guider par un sens inn de lharmonie des volumes et des formes , la construction gomtrique de la peinture dans certaines uvres nest pas le fruit dune spculation, mais bien une ralit.
L'tude de lillustration des Grandes Chroniques de France peintes par Jean Fouquet a t loccasion de vrifier lexistence dun trou de compas dans une scne mettant en uvre le pentagone rgulier
Souvent, le peintre, place llment, le personnage ou lvnement dans la section dore du tableau pour que le regard du spectateur y soit naturellement attirVoici quelques autres exemples
La Naissance de Venus ( Boticelli )
La Naissance de Venus ( Boticelli )
La Naissance de Venus ( Boticelli )
LAdoration des Mages ( Velasquez )
Le format du tableau correspond un rectangle dOr Le tableau s'organise autour de la diagonale. Les visages de Marie et du personnage qui est ses cts s'inscrivent galement dans un Rectangle d'or. Plus contemporains
Pablo Picasso
La Parade ( Seurat )
g f e ahi
Le Sacrement du Dernier Repas de Salvator Dali (1904-1989) est peint lintrieur dun rectangle dor et des proportions dores auraient t utilises pour positionner les personnagesAB
AB
une partie dun gigantesque dodcadre symbolisant lUnivers surplombe la table
Autres exemples dutilisation de la proportion dOr en architecture
Santa Maria Novella ( Florence )Renaissance Italienne
Tempietto de Bramante ( Rome )
Villa Farnese ( Rome )
La Villa Farnese est btie suivant un plan pentagonalUn dessin de Francesco Giorgio Martini associe la forme du corps humain et le plan d'une glise
le rapport des dimensions est analogue celui de San Spirito Florence conue par Brunelleschi (1377-1446)
Rectangle
Rectangle dOrSan Spirito (Florence )
Chteau de ThoiryLarchitecture de Philibert de LOrme dans ce chteau est entirement rgle par les nombres. Les proportions des ailes, fentres, chemines sont toujours bases sur la longueur du chteau selon des sous-multiples des entiers fondamentaux 1 , 2 , 3 et 5
le nombre dor y joue un grand rle ; le vestibule a les proportions de la chambre funraire de la pyramide de CheopsChteau de Thoiry ( Philibert de LOrme )
Un plan de porteRectangle dorRectangle 2 Cette proportion fut tudie l' poque moderne puisque Le Corbusier, architecte franais dorigine suisse (1887;1965) l'a immortalise dans Le Modulor.
Prsent en avril 1947 par le Corbusier, Le Modulor est un systme de mesure bas sur les proportions du corps humain
Dessin et photo de la maison dAmde Ozenfant par Le CorbusierChapelle de Ronchamps ( Le Corbusier )
Faut-il voir le nombre dOr partout ???
Collection "Pratique du dessin et de la peinture", Bordas, 1973.
Tir du semestriel des "Amis de Herg" n 6du mois de dcembre 1987
Recherches ralises par les lves de 5LM-SA au cours de Math complmentaires de Mr Colin
BilalFatimaSoufiane
ChanthimLaetitiaYouness
FarahIkram
Nadir RaniSuleyman AyoubImadSources : http://trucsmaths.free.fr/nombre_d_or.htm
http://www.bib.ulb.ac.be/coursmath/rectangl.htm
http://www.sciences-en-ligne.com/momo/chronomath/chrono1/Fibonacci.html
http://www.rtsq.qc.ca/aiguillart/projet/rech/artmath/no_or/intropas.htm
http://pedagogie.ac-aix-marseille.fr/etablis/lycees/craponne/or/partheno.htm
http://users.hol.gr/~helen/index.files/LE%20NOMBRE%20DOR%202.htm
http://www.ac-nice.fr/artsap/fichedocumen/nombredor.html
http://www.thoiry.tm.fr/thfhchno.htm
http://www.tintin.be/fr/doss_fr/regl1_fr.html
Ville de Bruxelles - Athne Lon Lepage rue des Riches Claires 30 1000 Bruxelles 02 /548 .27 .10 Courriel: [email protected] http://www.brunette.brucity.be/lepage/index.html
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Feuil1 (2)
yYX
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Feuil1
yYX
Feuil2
Feuil3
MBD0007604D.unknown
MBD0005B89E.unknown
Graph400.32425017250.64716405680.9346547121.15291617561.27200628931.26938521591.13303312430.86377672810.47650013680-0.5246367791-1.0471114439-1.512271324-1.8654183722-2.0581061761-2.0538652793-1.8332475951-1.3975907461-0.7709772213-00.84886230861.69422631622.44685500223.01824692613.33001579293.32315402182.96619460892.26130182731.24744114410-1.3734592153-2.7412581796-3.9590113935-4.8835235265-5.3879655529-5.3768632073-4.7993028772-3.6587863565-2.0183597711-02.22225701044.43535573476.40568043477.90154106598.71772826468.69976466947.76527205535.91991632483.2657061096
yYX
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Feuil1 (2)
yYX
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Feuil1
yYX
Feuil2
Feuil3
MBD0007604D.unknown
MBD0005B89E.unknown
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yYX
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Feuil1 (2)
yYX
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Feuil1
yYX
Feuil2
Feuil3
MBD0007604D.unknown
MBD0005B89E.unknown
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yYX
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Feuil1 (2)
yYX
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Feuil1
yYX
Feuil2
Feuil3
MBD0007604D.unknown
MBD0005B89E.unknown
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yYX
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Feuil1 (2)
yYX
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Feuil1
yYX
Feuil2
Feuil3
MBD0007604D.unknown
MBD0005B89E.unknown
Graph400.32425017250.64716405680.9346547121.15291617561.27200628931.26938521591.13303312430.86377672810.47650013680-0.5246367791-1.0471114439-1.512271324-1.8654183722-2.0581061761-2.0538652793-1.8332475951-1.3975907461-0.7709772213-00.84886230861.69422631622.44685500223.01824692613.33001579293.32315402182.96619460892.26130182731.24744114410-1.3734592153-2.7412581796-3.9590113935-4.8835235265-5.3879655529-5.3768632073-4.7993028772-3.6587863565-2.0183597711-02.22225701044.43535573476.40568043477.90154106598.71772826468.69976466947.76527205535.91991632483.2657061096
yYX
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Feuil1 (2)
yYX
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Feuil1
yYX
Feuil2
Feuil3
MBD0007604D.unknown
MBD0005B89E.unknown
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yYX
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Feuil1 (2)
yYX
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Feuil1
yYX
Feuil2
Feuil3
MBD0007604D.unknown
MBD0005B89E.unknown
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yYX
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Feuil1 (2)
yYX
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Feuil1
yYX
Feuil2
Feuil3
MBD0007604D.unknown
MBD0005B89E.unknown
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yYX
Feuil1 (2)trxy01100.314161.049300.997940.324250.628321.101020.890740.647160.942481.155300.679070.934651.256641.212250.374611.152921.570801.272010.000001.272011.884961.33471-0.412451.269392.199111.40051-0.823201.133032.513271.46954-1.188890.863782.827431.54199-1.466520.476503.141591.61800-1.618000.000003.455751.69776-1.61467-0.524643.769911.78145-1.44123-1.047114.084071.86927-1.09873-1.512274.398231.96142-0.60611-1.865424.712392.05811-0.00000-2.058115.026552.159560.66734-2.053875.340712.266021.33193-1.833255.654872.377721.92362-1.397595.969032.494932.37282-0.770986.283192.617922.61792-0.000006.597342.746982.612530.848866.911502.882392.331901.694237.225663.024481.777742.446867.539823.173570.980693.018257.853983.330020.000003.330028.168143.49417-1.079763.323158.482303.66642-2.155072.966198.796463.84716-3.112412.261309.110624.03680-3.839231.247449.424784.23580-4.235800.000009.738944.44461-4.22707-1.3734610.053104.66371-3.77302-2.7412610.367264.89361-2.87639-3.9590110.681425.13484-1.58675-4.8835210.995575.38797-0.00000-5.3879711.309735.653571.74705-5.3768611.623895.932263.48690-4.7993011.938056.224705.03589-3.6587912.252216.531556.21187-2.0183612.566376.853536.85353-0.0000012.880537.191376.839402.2222613.194697.545886.104744.4353613.508857.917864.654006.4056813.823018.308172.567377.9015414.137178.717730.000008.7177314.451339.14747-2.826728.6997614.765499.59840-5.641807.7652715.0796410.07156-8.148075.9199215.3938010.56805-10.050813.26571
Feuil1 (2)
yYX
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Feuil1
yYX
Feuil2
Feuil3
MBD0007604D.unknown
MBD0005B89E.unknown
Graph400.32425017250.64716405680.9346547121.15291617561.27200628931.26938521591.13303312430.86377672810.47650013680-0.5246367791-1.0471114439-1.512271324-1.8654183722-2.0581061761-2.0538652793-1.8332475951-1.3975907461-0.7709772213-00.84886230861.69422631622.44685500223.01824692613.33001579293.32315402182.96619460892.26130182731.24744114410-1.3734592153-2.7412581796-3.9590113935-4.8835235265-5.3879655529-5.3768632073-4.7993028772-3.6587863565-2.0183597711-02.22225701044.43535573476.40568043477.90154106598.71772826468.69976466947.76527205535.91991632483.2657061096
yYX
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Feuil1 (2)
yYX
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Feuil1
yYX
Feuil2
Feuil3
MBD0007604D.unknown
MBD0005B89E.unknown
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yYX
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Feuil1 (2)
yYX
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Feuil1
yYX
Feuil2
Feuil3
MBD0007604D.unknown
MBD0005B89E.unknown