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history/modulor_1/20130910-220217.txt
editor : alpha [82.255.178.5] 2013/09/10 22:02:17 {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 _p At all times the architects tried to build conceptual tools helping them to compose harmonious spaces. The architect {b Le Corbusier} developped a system of measurement supposed to replace advantageously the ancient Imperial System and the modern Metric System. The two fundations of his system was the Human size and the Golden Ratio. {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"}} } _h5 the golden ratio _p Le Corbusier imagined two sequences of numbers, called "La série Rouge" and "La série Bleue" built on the the Fibonacci numbers. Generating the numbers of [[Fibonacci|http://fr.wikipedia.org/wiki/Suite_de_Fibonacci]] is simple : {b F(0) = 1, F(1) = 1 and F(n) = F(n-1) + F(n-2)} for n >1 : {pre 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 ...} _p This sequence has a remarquable property : the ratio of two successive Fibonacci numbers tends to the Golden Number named « {b Ø} » which is defined this way : {b « Two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to their maximum »} {pre This can be expressed by this algebraic equality : (a+b)/a = a/b. Rewriting a = 1 and b = x, this equality can be expressed like a quadratic equation : x{sup 2} + x - 1 = 0 This equation has two roots : x1 = (1+sqrt(5))/2 = 1.618033988749895 x1 = (1-sqrt(5))/2 = -0.618033988749895 Ø is given the first value : 1.618033988749895 } _p The Fibonacci numbers can be generated by the [[pascal]] triangle 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 The Pascal triangle has a lot of properties. 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"}} } _h5 the human size _p As everybody knows people have different sizes ! But architects must agree on median values. A known default of the ancient Imperial System based on the {b foot} was that, in the past, every country gave it a different value. Another default was in the choice of the inch equal to 1/12 of a foot, making difficult arithmetics on these numbers. The Metric System was supposed to solve these two defaults : a common unit, the meter and the use of the decimal notation. In geodesy and astronomy it was a great progress. In architecture, it was a great loss : despite the rational principles enlighted by the "French Revolutionaries", the actual human body is easier to describe using feet and inches. _p In order to find a workaround between these two systems, Le Corbusier choose to define two sequences of numbers : _ul "La série Rouge" starting on the value 1.13cm which was supposed to be the umbilic's height of an "ideal" 6 feet heigth man. _ul "La série Bleue" starting on the value 2.26cm (2*1.13cm) which is the height reached by his hand when the arm is raised up. _p 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. They are rewritten approximatively in meters/centimeters, feet and inches, in the hope it would help to find a logic in these sequences. {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 } _h5 and so what ? _p The "Série Rouge" and "Série Bleue" were supposed to help the concepors. Actually, as it can be seen in the two first columns, the result is a set of numbers rather impredictibles and difficult to memorize. The fact is that these series are actually unused ! It's noticeable to know that, contrary to Fibonacci, Le Corbusier had copyrighted his "discovery". Open Source didn't exist in the years 1940/1950 ! No regret. _p In the third column, the values of theses series has been compared to some values coming from another approach. This approach will be now analyzed in the following section : [[modulor_2]].