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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]].