Le nombre d'or

2) je ne comprend pas l'énoncé "²= 1+ et que 1/ = -1 " 3) Vous pouvez dire que le nombre d'or est une des deux solutions de l'équation ou "graphiquement" que le nombre d'or correspond à l'abscisse d'un des deux points correspondant aux interceptions avec l'axe des abscisses de la fonction f(x)=x²-x-1 4) Je crois qu'il manque une partie de l'énoncé "L/l = "

Bonjour. Désolé, mais vous démarrez très mal. Phi, nombre d'or, est ; . le résultat artithmétique de (V5 + 1) / 2, soit (très environ) 1,618... . le résultat algébrique de l'équation (Phi carré) - Phi - 1 = 0 ; . et le résultat géométrique, du rapport des côtés du rectangle d'or (facile à construire compas-équerre, mais un-peu compliqué à expliquer). Aimez-vous les suites de chiffres et nombres, et autres fibonnacci-teries? Je vous propose une magnifique formule, qui: . à la forme, possède toutes les vertus de l’élégance et d’indubitables qualités esthétiques… … Quant au fond [rapport entre Phi-d'or et TT (Pi)], je n’ai pas vérifié. HTTP://trucsmaths.free.fr/nombre_d_or.htm#pi Vous «parlez d’or»?. Pour perpétrer le «partage en extrême et moyenne raison» euclidien, les moines de Cîteaux avaient créé un compas dont les égales branches s’articulaient sur un rivet placé à la jonction de: «petit segment» – «grand segment», de longueurs correspondant respectivement aux deux dimensions du rectangle d’or. Si bien que quelle que soit l’ouverture, se trouvait immédiatement corrélée une dimension à l’autre extrémité, en «divine proportion» exacte de Phi, avec l’objet mesuré (et ce dans les deux sens). Bon. Récréation. Travaux manuels. Un petit truc pour se changer les idées: travaux pratiques sur les suites de Fibonnacci et la relation de Simson: . Découpez dans du contre-plaqué (marine) de 358, un carré de huit unités de côté. Superficie 64. (faute de contreplaqué, prenez un carton léger ou… ce que vous avez sous la main -carton à chaussures?) . Re-coupez (inexorablement-dedans) un rectangle 8 x 5. . Dans l’icelui, détaillez deux trapèzes 5 x 5 x 3. . Dans la «chûte» rectangulaire 8 x 3, diagonalisez-tranchez deux triangles (rectangles, évidemment!) 3 x 8. Tatouilllez vigoureusement le tout, et… Abracadabra, ajustez-assemblez : . Les triangles aux trapèzes, par leur côté commun 3; . puis les deux triangles (toujours rectangles, 5 x 13) formés, par leur hypothénuse, en un charmant rectangle 5 x 13. Terminé. Surface du rectangle (13 x 5) ??? SURPRISE !!! (de quoi rendre fous les lotisseurs-géomètres-promoteurs vendant le terrain au mètre carré…) … Et y’a pire… Mais avec des triangles seuls, cette fois. (plus difficile à découper, plus sophistiqué) (mais toujours Simson’s relation & Fibonnacci & Co. – Associés). … Et considérez la (grande) pyramide, de hauteur 147 mètres, et de côté à la base 230 mètres; le rapport de la hauteur au demi-côté approche (érosion) 1,2720196…, soit la racine carrée de 1,6180339… qui est Phi, nombre d’or : (V5+1)/2. Phi, étant également le rapport de l’apothème, au même demi-côté. Surabondamment, le demi-périmètre de Khéops sur sa hauteur, donne une valeur approchée de TT (Pi). Amusant, non ? Bien à vous.

Bonjour j y comprend rien et a quoi ça sert ? a bientot

Http://www.stefanides.gr/pdf/BOOK%20_GRSOGF.pdf http://www.stefanides.gr/pdf/PROPOSED_GEOMETRY_OF_THE_PLATONIC_TIMAEUS_GREEK.pdf.pdf PANAGIOTIS STEFANIDES http://www.stefanides.gr

The Most Beautiful Triangle - Timaeus Plato's SUMMARY This work[presented to various conferences] is a proposed interpretation of Plato's Timaeus triangle(s), by Panagiotis Stefanides.These are the scalene orthogonal and the isosceles orthogonal triangles.* Particularly it gives a novel interpretation of the scalene orthogonal triangle and of the solids structured by these triangles -formed by the four basic elements.It is noted, here, that a "similar"- "constituent part of this triangle" but "not the same", is the Kepler -Magirus triangle :http://en.wikipedia.org/wiki/Kepler_triangle The Most Beautiful Triangle - Timaeus Plato's HELLENIC MATHEMATICAL SOCIETY INTERSCIENTIFIC RESEARCH GROUP SOCIETY FOR THE HISTORY OF MATHEMATICS Postgraduate Seminar for Classical Greek Mathematics, of the National Technical University of Athens. Panhellenic Conference for the History and Philosophy of Mathematics (Dedicated to the Anniversary for the Completion of 2,200 Years from the Death of Archimedes) Subject : History and Philosophy of Classical Greek Mathematics Conference Proceedings Paper Issued by : P.C. Stefanides National Research Center Athens 2-4 March1989 31 March 1996 " Το πιό ωραίο τρίγωνο - ΤιμαίοςΠλάτωνος" "The Most Beautiful Triangle - Timaeus Plato's Πρώτον μεν δη πυρ και γη και ύδωρ και αήρ, ότι σώματά εστί............................................... τρίγωνα πάντα εκ δυοίν άρχεται τριγώνοιν.... προαιρετέον ούν αύ των απείρων το ΚΑΛΛΙΣΤΟΝ..... ΤΡΙΠΛΗΝ ΚΑΤΑ ΔΥΝΑΜΙΝ ΕΧΟΝ ΤΗΣ ΕΛΑΤΤΟΝΟΣ ΤΗΝ ΜΕΙΖΩ ΠΛΕΥΡΑΝ ΑΕI". In section 53, of PLATO'S "TIMAEUS", PLATO speaks about the triangular shapes of the Four Elemental Bodies, of their kinds and their combinations : These Bodies are the Fire (Tetrahedron) the Earth (Cube), the Water (Icosahedron), and the Air (Octahedron). These are bodies and have depth. The depth necessarily, contains the flat surface and the perpendicular to this surface is a side of a triangle and all the triangles are generated by two kinds of orthogonal triangles : the "ISOSCELES" Orthogonal and the "SCALΕΝΕ" Orthogonal. This is the probability of the beginning of the creation of the Fire and of the other Bodies. These bodies are four, are dissimilar but capable of each one of them being created from the other and also dissolved into others. If it is happening so we have the truth about the creation of the Earth, Fire and their other proportional media. We do not agree that there will be other Bodies more beautiful than those ones, each one of them with its own gender. From the two kinds of triangles the "Isosceles" Orthogonal has one nature. (i.e. one rectangular angle and two acute angles of 45 degrees), whereas the "scalene" has infinite (i.e. it has one rectangular angle and two acute angles of variable values having, these two acute angles, the sum of 90 degrees). From these infinite natures we choose one triangle "THE MOST BEAUTIFUL". Thus, from the many triangles, we accept that there is one of them "THE MOST BEAUTIFUL", and we leave those by which the equilateral triangle is constructed (i.e. by using six "scalene" orthogonal triangles, having 30 and 60 degrees their acute angles). Let us choose then, two triangles, which are the basis of constructing the Fire and the other Bodies : "Το μεν ισοσκελές, το δε τριπλήν κατά δύναμιν έχον της ελάττονος την μείζω πλευράν αεί." One of these two is the "ISOSCELES" orthogonal triangle, the other is the "SCALENE" orthogonal triangle, its hypotenuse having a value equal to the "CUBE" of the value of its horizontal smaller side and having its vertical bigger side the value of the "SQUARE" of its smaller horizontal side. The value of the smaller horizontal side is equal to the square root of the GOLDEN NUMBER, the ratio of the sides is equal, again, to the square root of the GOLDEN NUMBER (geometrical ratio) and the Tangent of the angle between the hypotenuse and the smaller horizontal side is also equal to the SQUARE ROOT of the GOLDEN NUMBER (Θ=51 49-38-15-9-17-19-54-37-26-24-0 degrees). This angle reaches the PYRAMIDAL one. The product of the smaller horizontal side and that of the hypotenuse is equal to the "SQUARE" of the bigger vertical side, of this triangle, and at the same time the "PYTHAGOREAN THEOREM" is valid. The values of the sides of this triangle are given by surd numbers, (solution of a fourth degree equation). Reorganizing this triangle, we get another one with the same angle values, which has its bigger vertical side equal to FOUR (4), its smaller horizontal side equal to FOUR divided by the SQUARE ROOT of the GOLDEN NUMBER, and its hypotenuse equal to FOUR multiplied by the SQUARE ROOT of the GOLDEN NUMBER. (Four divided by the SQUARE ROOT of the GOLDEN NUMBER is equal to 3.14460551) On the 30-9-1986, it was sent by the writer to W.I.P.O. in Geneva a text of one page, which contained the elements of one triangle having its vertical side equal to 4, its horizontal side equal to 1/0.3180049125 (equal to 3.14460551), its hypotenuse equal to 5.08878597 (4x1.27201965), and its angle between horizontal and hypotenuse equal to: θ = 51 49-38-15-9-17-19-54-37-26-24-0 with the name of "Tο ειδικό ορθογώνιο τρίγωνο" (i.e. The "Special Orthogonal Triangle") this "Special Orthogonal Triangle", was the result of studying TOM VALENTINE's book "THE MYSTERY OF THE GREAT PYRAMID" (Greek translation published by ORORA - 1981), in which it is stated that JOHN TAYLOR reached into the conclusion that the area of each of the four sides of the Great Pyramid, is equal to the square of its height (in the conference, it was stated that this relationship was given by "HERODOTUS"). Also, according to MAX TOTH, Best Seller 1988, "The Prophesies of the Pyramids"-Greek translation - and the pieces of information from "HERODOTUS" that the area of each side of the Great Pyramid is equal to the square of its height, the mathematicians have derived a triangle with vertical side equal to the square root of the GOLDEN NUMBER, the horizontal side equal to unity and the hypotenuse equal to the GOLDEN NUMBER". Also they derived a relationship which states that Pi is equal to four divided by the square root of the GOLDEN NUMBER and it is stated as being a useful formula for it (however, it is not derived as a proof). This study was undertaken by the writer after discussing with a certain researcher in Athens, certain philosophical mathematics and it was suggested to him to study the Great Pyramid. On 26-10-1987, the writer sent again to W.I.P.O. (World Intellectual Property Organization) a text with the title: "PLATO'S TIMAEUS" - "THE MOST BEAUTIFUL TRIANGLE". The text, was one page and it contained the triangle presented to this conference. Further down the Mathematical analysis is presented. It must be noted, also, that "PLATO" speaks about the construction of the Earth (cube), from the "ISOSCLELES" triangle, and the other three bodies from the "SCALENE". The base of the cube (Plane surface) from four (4) "ISOSCELI" orthogonal triangles, whereas the base of the other bodies from six (6) "SCALENE" of the 30 and 60 degree types. Also, "PLATO" speaks of a further fifth, structure which God used it for the design of the UNIVERSE. It is concluded here that by "THE MOST BEAUTIFUL TRIANGLE", PLATO correlates the four elements (UNIFIED THEORY) through the General Analogies of their sides (Fire, Air, Earth and Water), i.e. Fire/Air is equal to Air/Water is equal to Water/Earth, to T, where T is equal to the SQUARE ROOT of the GOLDEN NUMBER. T = SQR ((SQR.(5) + 1)/2) (ό τι περ πύρ προς αέρα, τούτο αέρα προς ύδωρ, και ό τι αήρ προς ύδωρ, ύδωρ προς γήν, ξυνέδησε...............ουρανόν, PLATO'S TIMAEUS SECTION 32). REF : * http://www.stefanides.gr/pdf/BOOK% 20_GRSOGF.pdfhttp://www.stefanides.gr/Html/GOLDEN_ROOT_SYMMETRIES.htm PANAGIOTIS STEFANIDES http://www.stefanides.gr