Divina proportio Index De calculatione | In arte | Notae | Nexus interniTabula navigationisDr. Johann Samuel Traugott Gehler, Physikalisches Wörterbuch oder Versuch einer Erklärung der vornehmsten Begriffe und Kunstwörter der Naturlehre. Leipzig 1796

ArsGeometria plana EuclideanaInventionesNotiones mathematicaeNumeri irrationales


ratio metallicaphinumerus irrationalisartificesarchitectiaestheticesaeculo XIX1835Martinus Ohmnumeri FibonaccianiLeonardus VinciusLucas Pacioli






Proportio inter longitudinem totam a+b et segmentum a est aequa proportioni inter segmentum a et segmentum b.


Divina proportio[1] sive sectio divina[2] sive ratio aurea[3] sive sectio aurea[4] sive ratio Fibonacci est ea proportio quae inter numeros a et b, aequa est proportioni inter a+b et a; prima ratio metallica, circa 1.6180339887, phi (φdisplaystyle varphi ) Graeca littera designata.


a+ba=ab=φ.displaystyle frac a+ba=frac ab=varphi ,.

Solutio est numerus irrationalis:


φ=1+52≈1.6180339887...displaystyle varphi =frac 1+sqrt 52approx 1.61803,39887,...

Multi artifices et architecti divina proportione utantur, praesertim aureo rectangulo, cuius breve et longum latus in divina proportione consistunt. Aureus rectangulus placere aesthetice dicitur.


Nomen Latinum quod est "sectio aurea" invenitur in saeculo XIX anno 1835 in libro Germanice scripto, cuius titulus erat "Die Reine Elementarmathematik Bd II: Die ebene Raumgrößenlehre". Ibi Mathematicus Martinus Ohm mentionem fecit huius sectonis nomine Germanico "Goldener Schnitt".[5]




Index





  • 1 De calculatione


  • 2 In arte


  • 3 Notae


  • 4 Nexus interni




De calculatione |


Duo numeri a et b sunt in divina proportione φdisplaystyle varphi si


a+ba=ab=φ;displaystyle frac a+ba=frac ab=varphi ,;

sic φdisplaystyle varphi , definiatur.


Cum dextra aequatio monstrat a=bφdisplaystyle a=bvarphi , hac aequatione substituta, sinistra fit:


bφ+bbφ=bφb.displaystyle frac bvarphi +bbvarphi =frac bvarphi b,.

Quae deleto b fit


φ+1φ=φ.displaystyle frac varphi +1varphi =varphi .

Quae multiplicatis ambis partibus per φdisplaystyle varphi , fit:


φ2−φ−1=0.displaystyle varphi ^2-varphi -1=0.

Itaque φdisplaystyle varphi est prima ratio metallica.


Una positiva solutio est:


φ=1+52≈1.6180339887...displaystyle varphi =frac 1+sqrt 52approx 1.61803,39887,...

Ratio maioris numeri Fibonacciani ad proximum minorem est, quo maiores sunt eiusmodi numeri, eo proprior rationi divinae: haec ratio accedit ad φdisplaystyle varphi .



In arte |




Aurea rectangula in pictura Mona Lisa, a Leonardo Vincio.


Leonardus Vincius cum picturas fecit in libro a Lucas Pacioli De Divina Proportione, dictus est divina proportione in clara pictura Mona Lisa usus esse: utrum hoc sit concursatio annon, scit nemo.



Notae |



  1. ↑ Lucas Pacioli, De divina proportione.


  2. ↑ Apud Keplerum.


  3. ↑ [1] .mw-parser-output .existinglinksgray a,.mw-parser-output .existinglinksgray a:visitedcolor:gray.mw-parser-output .existinglinksgray a.newcolor:#ba0000.mw-parser-output .existinglinksgray a.new:visitedcolor:#a55858
    (Germanice)


  4. ↑ google Dr. Johann Samuel Traugott Gehler, Physikalisches Wörterbuch oder Versuch einer Erklärung der vornehmsten Begriffe und Kunstwörter der Naturlehre. Leipzig 1796
    (Germanice)


  5. ↑ Martin Ohm, Die Reine Elementarmathematik Bd II: Die ebene Raumgrößenlehre, Jonas Verlagsbuchhandlung. Berlin 1835


Nexus interni


  • Aesthetica

  • Fractales

  • Numerus constans

  • Numeri Fibonacciani

  • Phi

  • Sectio Aurea (grex)

  • Triangulus rectus praecipuus








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