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editor : alpha [82.250.252.26] 2013/08/20 17:02:45 {div {@ style=" text-align:center; font:bold 3em georgia; text-shadow:0 0 8px black; color:white; "} modulor, feet & inches} {center [[modulor_0]] | [[modulor_1]] | [[modulor_2]] | [[modulor_3]] } _h3 1) the modulor {center {show {@ src="data/modulor/modulor_2.jpg" height="250" width="600" title="Usual dimensions with Le Corbusier"}} {show {@ src="data/modulor/modulor_1.jpg" height="250" width="600" title="Série Rouge et Série Bleue from Le Corbusier"}} {show {@ src="data/modulor/modulor4-640x284.jpg" height="230" width="1000" title="Usual dimensions with Le Corbusier"}} } _p The "Série Rouge" and "Série Bleue" imagined by Le Corbusier follows the Fibonacci serie. Generating the numbers of [[Fibonacci|http://fr.wikipedia.org/wiki/Suite_de_Fibonacci]] is simple : {b F(n) = F(n-1) + F(n-2)} and this sequence has a remarquable property : the ratio of two successive Fibonacci numbers tends to the Golden Number : {pre Ø = (1+sqrt(5))/2 = {/ {+ 1 {{lambda (x) return Math.sqrt(x)} 5}} 2}, the root of x{sup 2} + x - 1 = 0 which is equivalent to (a+b)/a = a/b when a=1 and b=x : « two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to their maximum » } _p The [[pascal]] triangle generates the Fibonacci numbers through diagonal sums : {pre 1 -> 1 = 1 1 1 -> 1+1 = 2 1 2 1 -> 1+2 = 3 1 3 3 1 -> 1+3+1 = 5 1 4 6 4 1 -> 1+4+3 = 8 1 5 10 10 5 1 -> 1+5+6+1 = 13 1 6 15 20 15 6 1 -> 1+6+10+4 = 21 1 7 21 35 35 21 7 1 -> 1+7+15+10+1 = 34 1 8... -> 1+8+21+20+5 = 55 1 } _p There are close relations between the pascal triangle and fundamental mathematics, physics and biologic laws, about equilibrium, minimum energy, and so on. The "Golden Rectangle" - whose sides are in the golden ratio - is considered as an harmonic figure. Playing with squares generates a beautiful "spiral curve" : {center {show {@ src="data/modulor/suitefibonacci.jpg" height="180" width="1000" title="Fibonacci Spiral in a Golden Rectangle"}} {show {@ src="data/modulor/tournesol.jpg" height="180" width="1000" title="Fibonacci Spiral in Nature"}} {show {@ src="data/modulor/soft_fibonacci_spiral.jpg" height="210" width="1000" title="Fibonacci Spiral in Fractals (http://fractalfiend.deviantart.com/art/Soft-Fibonacci-Spiral-v1-267963431)"}} {show {@ src="data/modulor/ksoa_esc.jpg" height="210" width="500" title="A Fibonacci Spiral could be seen in a perspective view of a standard spiral stair | Alain Marty architect"}} } _p Le Corbusier built the "Série Rouge" on the value 1.13cm which was supposed to be the umbilic's height of an "ideal" 6 feet heigth man. And he built the "Série Bleue" on the value 2.26cm (2*1.13cm) which is the height reached by his hand when the arm is raised up. Why not !! The goal was to have in hands a set of "magic" numbers which were supposed to help the conceptors. « It's a language for proportions which makes ugliness difficult and the beauty easy. » had said Einstein about The Modulor. _p Below is a subset of these numbers rewritten in an approximation of feet and inches. {pre {@ style="float:left;width:120px;margin:5px;"} FIBONACCI 1: 2/1 3: 3/2 5: 5/3 8: 8/5 13: 13/8 21: 21/13 34: 34/21 55: 55/34 89: 89/55 144: 144/89 233: 233/144 SERIE ROUGE sur m = 1.13cm * m Ø Ø Ø * m Ø Ø * m Ø m / m Ø / m Ø Ø / m Ø Ø Ø SERIE BLEUE with M = 2.26cm * M Ø Ø Ø * M Ø Ø * M Ø M / M Ø / M Ø Ø / M Ø Ø Ø } {pre {@ style="float:left;width:160px;;margin:5px;"} {/ 2 1} {/ 3 2} {/ 5 3} {/ 8 5} {/ 13 8} {/ 21 13} {/ 34 21} {/ 55 34} {/ 89 55} {/ 144 89} {/ 233 144} SERIE ROUGE {* 1.13 1.618 1.618 1.618} {* 1.13 1.618 1.618} {* 1.13 1.618} 1.13 cm {/ 1.13 1.618} {/ 1.13 1.618 1.618} {/ 1.13 1.618 1.618 1.618} SERIE BLEUE {* 2.26 1.618 1.618 1.618} {* 2.26 1.618 1.618} {* 2.26 1.618} 2.26 cm {/ 2.26 1.618} {/ 2.26 1.618 1.618} {/ 2.26 1.618 1.618 1.618} } {pre {@ style="float:left;width:260px;;margin:5px;"} -> Ø = {/ {+ 1 {{lambda (x) return Math.sqrt(x)} 5}} 2} 1' = 30.48cm, 1" = 2.54cm ≠ 30cm ≠ 2.5cm ≠ 4.775 = 16' - 1" ≠ 16' = 4.80 ≠ 2.950 = 10' - 2" ≠ 10' = 3.00 ≠ 1.825 = 6' + 1" ≠ 6' = 1.80 ≠ 1.125 = 4' - 3" close to 1.20 ≠ 0.700 = 2' + 4" close to 0.75 ≠ 0.425 = 1' + 5" close to 0.45 ≠ 0.250 = 1' - 2" close to 0.30 ≠ 9.575 = 32' - 1" ≠ 32' = 9.60 ≠ 5.900 = 20' - 4" ≠ 20' = 6.00 ≠ 3.650 = 12' + 2" ≠ 12' = 3.60 ≠ 2.250 = 7.5' ≠ 1.400 = 5' - 4" close to 1.50 ≠ 0.850 = 3' - 2" close to 0.90 ≠ 0.525 = 2' - 3" close to 0.60 } _p The "Série Rouge" and "Série Bleue" give a set of numbers rather impredictibles and difficult to memorize and to use. The fact is that these series are actually unused ! It's interesting to know that, contrary to Fibonacci, Le Corbusier had copyrighted his "discovery". Open Source didn't exist ! No regret. In the third column, the values of theses series has been compared to some values coming from another approach, which will be analyzed in the following section : [[modulor_2]].