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{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} developed 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 0: 1 1: 1.1 2: 1.2.1 3: 1.3.3.1 4: 1.4.6.4.1 5: 1.5.10.10.5.1 6: 1.6.15.20.15.6.1 7: 1.7.21.35.35.21.7.1 8: 1.8.28.56.70.56.28.8.1 9: 1.9.36.84.126.126.84.36.9.1 10: 1.10.45.120.210.252.210.120.45.10.1 11: 1.11.55.165.330.462.462.330.165.55.11.1 12: 1.12.66.220.495.792.924.792.495.220.66.12.1 13: 1.13.78.286.715.1287.1716.1716.1287.715.286.78.13.1 14: 1.14.91.364.1001.2002.3003.3432.3003.2002.1001.364.91.14.1 15: 1.15.105.455.1365.3003.5005.6435.6435.5005.3003.1365.455.105.15.1 } _p Note that each number of the pascal triangle is the sum of two numbers : _ul 1) the number of the previous row in the same column, _ul 2) the number of the previous row and the previous column. {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 28 56 ... -> 1+8+21+20+5 = 55 1 9 36 84 ... -> 1+9+28+35+15+1 = 89 1 } _p Long before the XVIIth century French philosopher and mathematician Blaise Pascal, Omar Khayyam, an Iranian mathematician in the XI/XIIth century and Yanghui, an XIIth century Chinese mathematician, had studied the properties of this triangle of numbers ; but we will still call it "Pascal Triangle". 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"}} } _p One can guess the power of the Fibonacci and the Pascal triangle's numbers in the composition of spaces. But Architecture is not only "Geometry", it is also "Dimension" related to the human size. _h5 the human size _p As everybody knows {b 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]].