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{div {@ style="
text-align:center;
font:bold 3em georgia;
text-shadow:0 0 8px black;
color:white;
"} modulor, feet & inches}

{div {@ style="color:blue; text-align:center; font-style:italic;"} ( This (long) page is enhanced in 3 (shorter) pages : see [[MODULOR|?view=modulor_0]]. )}

_p Following the strange and complex (and unused) [[Modulor|http://ww3.ac-poitiers.fr/arts_p/b@lise14/pageshtm/page_7.htm]], this is a personal reflexion about usual dimensions based on simple "human" combinations of two values coming from the ancient days : the foot, (≠30cm) and the inch (≠2.5cm) ?
_p

_h3 1) the modulor
{center
{show {@ src="data/modulor/modulor_1.jpg" height="250" width="600" title="Usual dimensions with Le Corbusier"}}
{show {@ src="data/modulor/modulor_2.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 is analyzed now.

_h3 2) feet and inches
_p Beside the Metric System, the Imperial System and the Modulor System, it appears that in the architectural domain, usual dimensions can be easily reached by a simple combination of feet and inches in the approximation {b 1 foot = 30.48cm ≠ 30cm} and {b 1 inch = 2.54cm ≠ 2.5cm}. Examples :

_ul a seat : 45cm = 1.5'
_ul a table : 75cm = 2.5'
_ul a kitchen work desk : 90cm = 3'
_ul a tall man : height 180cm = 6', idem for the width between fingers
_ul a standard door's dimensions : 210cmx90cm = 7'x3'
_ul ceiling's heights : 225cm = 7'6", 240cm = 8', 250cm = 8'4", 270 = 9'
_ul a little room : 270cmx270cm = 9'x9' and a better one : 360cmx360cm = 12'x12'
_ul passage unit : 60cm = 2' (which is also the step of the man who walks)
_ul stairs formula : 2h+g in [60,65] : for g = 30cm = 1' h must be between 15cm = 0.5' and 17.5cm = 0.5' 1"
_ul thickness of a standard plasterboard : 12,7mm = 1/2". Other thickness are 4/12" and 8/12". Two plasterboards makes one inch (1/2"+1/2"), and improve the acoustic behaviour of the partition. With two plasterboards with different thickness (4/12" + 8/12"), the frequencies of resonance will be different and the filter better. Thinking to this in the Metric System is very difficult, not in the Imperial System !
_ul ducts diameters 15mm = 1/2", 25mm = 1", ... 300mm = 12"
_ul and so on ...

{center {show {@ src="data/modulor/modulo.jpg" height="400" width="600" title="Usual dimensions with the Imperial System ... and Alain Marty's 5%"}}}

_p So, with this simple set of "metricized" feet and inches : {b [30|2.5]}, it is possible to generate a serie of useful related values :
{pre 
first downwards :
30 / 2 = 15 / 2 = 7.5 / 3 = 2.5 (or 1/2'=6", 1/4'=3", 1/12'=1")
then upwards :
2.5 | 5 | 7.5 | 10.0 |12.5 | 15 | 17.5 | 20.0 | 22.5 | 25 | 27.5 | 30
and beyond that, for instance mixing with 15cm = 1/2' = 6"
45 | 60 | 75 | 90 | 105 | 120 | 135 | 150 | 165 | 180 | 195 | 210 |..
and also the metric series using 2" = 5cm or 4" = 10cm :
50 = 45 + 5 = 1.5'2", 100 = 90 + 10 = 3'4", .....
}
_p As it has been shown before, some of these values are more or less close to values coming from the Modulor. Nothing prevent to limit the choice to these values and to get the benefit of the Modulor's golden ratio properties supposed to lead to harmonious proportions. Comparing both approaches in the sketch of Le Corbusier showing 8 figures :
{pre
.                      MODULOR    cm, feet & inches
a low seat :           0.27       0.30 = 1'
a chair :              0.43       0.45 = 1.5'
a table :              0.70       0.75 = 2.5'
a kitchen work desk :  0.86       0.90 = 3'
at a bar (ombilical) : 1.13       between 1.05 = 3.5' and 1.20 = 4'
under the shoulders :  1.40       1.50 = 5'
man's height :         1.83       1.80 = 6'
man's hand up :        2.26       2.25 = 7.5'
}
_p And playing with other bigger values of the Modulor series :
{pre
4.78 can be usefully replaced by 4.80 = 4 x 1.20 = 4 x 4 x 0.30
2.96 can be usefully replaced by 3.00 = 6 x 0.60
9.57 can be usefully replaced by 9.60 = 9.00 + 0.60 = 8 x 1.20 
5.92 can be usefully replaced by 6.00 = 20 x 0.30 = 5 x 1.20
3.66 can be usefully replaced by 3.60 = 2 x 1.80 = 4 x 0.90
}
_p Even if these values slightly deviate from the ideal propotions supposed to be given by the "golden ratio', their simplicity makes easy the perception of the scale and of the relations between the dimensions. Architecture is not only "Geometry", it is {b Geometry dot dimension}.

_p So, in any case, all the values made of a combination of the couple {b [30|2.5]} are easy to memorize and combine, and are compatible with the values coming from the Metric System (12.5, 25.0, 100, ...). This system coming with a ligth approximation of the true foot and inch {u makes the junction} between the "Imperial System" and the "Metric System" in respect to the old measures of the ancient days ...

_h5 the ancient days
_p These considerations may seem ridiculously "Middle Age" and useless ! If so, why are the foot and the inch used in modern technologies such Computer Technologies ? The dimensions of a screen, of a pixel, of a font, of a page, are given in inches and not in centimeters ! Is it only because of the predominance of the USA in these technologies, a country in which the Metric System didn't penetrated ? Whatever it may be, my experience as an architect confirmed the fact that the couple {b [30|2.5]} gives the basis of a very good tool for conception in architecture, even if the dimensions are finally written in centimeters. In a lot of countries, the conceptual background of buildings dimensions is made of feet and inches. The Modulor was an attempt to fight against the Metric System, considered to be inappropriate in the building construction. But it was too complex, the Le Corbusier's "Série Rouge" and "Série Bleue" are not easy to use and so, they are never used today.

_p So, {i IMHO}, I think that the idea of a "little smart modulor" made on gentle combinations of feet and inches could be seen as an acceptable and useful successor of the Modulor. Don't you ?

_h3 3) architecture
_p This "little modulor" can be applyed to architectural conception. As it can be seen more thoroughly here : [[RISC/GRAINS|http://marty.alain.free.fr/risc/?view=grains]]. Drawing with a pencil on a paper sheet or drawing with a mouse on a digital screen follows the same principle : "choose a grain according to the desired precision". The scale is irrelevant in this process, it can be 1/100, 1/50, 1/20 or even 1/1. It is notable that this process, traditional in free-hand drawing, can be transposed in the "Computer Aided Design". {u Rather than using a keyboard to position and dimension the elements of a drawing, the best grain (30cm, 2.5cm or another "relevant" one) has to be chosen according to the desired precision, i.e. the "grain" will be the GCD (Greatest Commun Divisor) of all the dimensions. The cursor's diplacements are constrained to finite steps and with the help of the mouse's feedback, precise drawings can be done easily, in a hand drawing style.} 
_p An old drawing style transposed in modern tools, it is as simple as that! I give a few examples below.

_h6 an office building
_p The conception of the buildings (1989) shown below has been based on the simple set {b [30|2.5]}. As it can be seen on the 5 levels office building, the picture on the left shows a first draft drawn on a 30cm ≠ 1' grain, and the picture on the center shows a more precise step drawn on a 2.5c= ≠ 1" grain. The picture at the right shows how a Fibonacci spiral can be inserted in the building's facade, with a little help of a hammer ... 
{center
{show {@ src="data/modulor/trame.jpg" height="120" width="1000" title="Using 30cm ≠ 1 foot / Alain Marty architect"}}
{show {@ src="data/modulor/detail.jpg" height="120" width="1000" title="Using 2.5cm ≠ 1 inch / Alain Marty architect"}}
{show {@ src="data/modulor/palacio_fibonacci.jpg" height="120" width="1000" title="The Fibonacci spiral inserted with a hammer in the building's facade. The owner was my Apple Macintosh 128 dealer ..."}}
}
_p Note that the building was designed in year 1988 on an Apple Macintosh+ (µProc at 8MHz, 1024kb ram, 2 floppy disks 400kb, screen 512/342 N&B, ImageWriter II A4 format) with Fullpaint (a kind of MacPaint+). No Autocad on a powerful CAD station ! Nothing but a painting tool for a child ... and the {b [30|2.5]} method.

_h6 an individual house
_p Another example : [[NYLS|http://marty.alain.free.fr/am/?view=nyls]], a small house (2001) composed on the 30cm ≠ 1' modulus, a very simple plan and section. A child could have design this house with [[LEGO blocks|http://creative.lego.com/fr-fr/default.aspx?icmp=COFRFR17BricksMore]].

{center
{show {@ src="data/modulor/nyls_plan_coupe.jpg" height="190" width="1000" title="A small house composed on the 30cm ≠ 1' modulus, a very simple plan and section | Alain Marty architect"}}
{show {@ src="data/modulor/nyls_1.jpg" height="190" width="1000" title="The house, a pool and the Catalan Canigou mountain in the distance"}}
}

_h6 a (virtual) temple

_p The set {b [30|2.5]} is one of several other choices. It could be : {b [52.5|2.5]}, as it is shown below for the project of temple based on big limestone blocs ([[Pierre du Gard|http://www.pierredupontdugard.com/]]) whose dimensions are H = 210cm ≠ 7', L = 105cm ≠ 3.5' ≠ 3'+6", E = 52.5cm ≠ 1'+9", or 4M/2M/1M with M = 52.5cm. The Sketchup 3D model has been easily and quickly built on this "grain" of 52.5cm. More can be seen [[here|http://ensam.wiki.free.fr/projetenchantier_20120206/?view=temple]].

{center
{show {@ src="data/modulor/2_2_mix.jpg" height="130" width="1000" title="Project of a temple based on H:210/L:105/E:52.5 limestone blocs (Pierre du Gard)"}}
{show {@ src="data/modulor/2_3_mix.jpg" height="130" width="1000" title="Project of a temple based on H:210/L:105/E:52.5 limestone blocs (Pierre du Gard)"}}
}

_h6 the Glenn Murcutt's Marika house

_p Another example higlights the use of another couple {b [2.5cm|0.5cm]} to "decrypt the complex dimensionning of a wooden house built by the great architect [[Glenn Murcutt|http://fr.wikipedia.org/wiki/Glenn_Murcutt]]. Note that the sub-division under {b 2.5cm} is: {b 0.5cm} and not {b 1cm} ; do you see why ? After this work, the student was able to draw by hand (paper and pencil) the plan, a section and any perspective, straight from his mind. 
{center
{show {@ src="data/modulor/ly_grain_de_25_plan.jpg" height="130" width="1000" title="The couple [2.5cm|0.5cm] was used to give a limit to the complex dimensionning of this wooden house, and to discover its basic composition."}}
{show {@ src="data/modulor/ly_3D_Terrasse.jpg" height="130" width="1000" title="The Sketchup 3D model is the result of a true and precise dimensionning, not a quick and blurred sketch"}}
}
_p More to see about this study here : [[Lune Yann|http://ensam.wiki.free.fr/projetenchantier_20120206/?view=LUINE%20Yann]].

_h3 what about curves ?
_p OK, OK, such a strict orthogonal drawing system based on a grid looks fine on ... strict orthogonal architecture. But what about more complex and curved architectures ? That can't work ! Where could be found any grid in such a shape :

{center
{show {@ src="data/modulor/2.jpg" height="300" width="1000" title="A project from Serero architect ; I think this project was never built."}}
}
_p I will give a short answer given on a slightly simpler shape below :
 
_ul 1) the most important thing to know is that {u the grid is not to be found in the shell's surface itself, but in its control points !} 
_ul 2) in the picture to the left below, a 10cm thick shell is built in Sketchup 3D ( to be precise,  with a Ruby/pForms plugin, see page [[pForms]] ) on {b 15} control points positionned on a 30cm grid ; the {b 15} points are actually {b 5} parabolas controlled by {b 3} points ; 
_ul 3) given its 3 control points, each parabola is defined through a recursive process based on a simple {b division by 2}, as it can be seen in the picture to the right (with planks instead of cords).

{center
{show {@ src="data/modulor/shell_coque_1_.jpg" height="190" width="1000" title="In Sketchup 3D, with the Ruby/pFormes plugin, example of a 10cm thick shell built on 15 (5x3) control points distributed on a 30cm grid ; note that, contrary to the triangular represntation) the quadrangles are flat (no folded into two triangles) and so easier to assemble"}}

{show {@ src="data/modulor/pL3_realisation.jpg" height="190" width="600" title="There is always this moment when you have to draw in site, far from Autocad, with simple and sharable gestures, with planks, cords and some stakes."}} 
}
_p If you are able to find the middle of a cord stretched between two stakes, you are able to build this shape in space ! Witout AutoCad, straight from your mind !

_p More to see here : [[RISC/POLES|http://marty.alain.free.fr/risc/?view=poles]].

_h3 words under images
_p In this page I tryed to show briefly :
_ul 1) how a complex dimensionning system, like the MODULOR, can be replaced by a simpler one, that can be shared with anybody in the true life. No mathematics !
_ul 2) how complex shapes can be mastered in simple gestures with simple tools, a cord and stakes, that can be shared with anybody in the true life. No mathematics !

_p This page was conceived and composed inline using {b simple words} written with a simple tool : alphawiki. No complex page editor, no M$WORD, no [[latex]], nothing but a small wiki easy to install (about 100 kb) and working anywhere in a standard browser. As it can be seen in page [[help]], a wiki is a collaborative tool, but first of all - {b provided the syntax allows it} - it is a tool for composing and coding inline web pages with a rich dynamic content. It's the goal of the lambdaway project whose alphawiki is the last child. And this page is an example of what can be done with it.

_p Some reflexions about the future of such a tool here : [[RISC/WORDS|http://marty.alain.free.fr/risc/?view=mots]].

_p Alain Marty 18/08/2013

{center {show {@ src="data/modulor/iproc_sketch.jpg" height="500" width="500" title="The small cube hanged on the man's neck is a multi-layered block of silicium enriched to be a complete computer with CPU, memory, battery, two cameras, two video-projectors, wifi ; one video-projector displays on any surface (i.e. on a wall) the content of the memory (a document), another displays a keyboard on another surface (i.e. on a table) ; a camera follows the fingers on the keyboard, another the hands on the wall ; and the wifi holds the connection with the internet and the cloud where are stored the data. The OS is reduced to a simple browser connected to an α-wiki. Et voilà !"}}}

{center {show {@ src="data/modulor/perfect_piggy.jpg" height="330" width="700" title="Perfect piggy, after Marcus Vitruvius Pollio and Leonardo da Vinci"}}}