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Le Corbusier's use of the Golden Section begins by 1927 at the Villa Stein in Garches, whose rectangular proportion in ground plan and elevation, as also the inner structure of the ground plan, approximately show the Golden Section . Le Corbusier himself calls his Unité d'Habitation in Marseille (1945-52) a demonstration of his Modulor system. Indeed the real measures considerably differ from the theory: real length 140 m, instead of Modulor 139,01 m; width 24 m, Modulor 25,07 m; height 56 m, Modulor 53,10 m . To address these differences, attempts have been made to trace back the building's dimensions not to the Modulor but to the exact relations of the Golden Mean, but with little success.
B. From Peter Brinkworth, The Place of Mathematics http://www.maths.adelaide.edu.au/people/pscott/place/pm16/pm16.html Golden section and Fibonacci sequence (I) To understand
Le Corbusier’s Modulor, we need to begin with two of its elements: the
golden section (or golden ratio) 1/
1, 1, 2, 3, 5, 8, 13, 21, 35, ... However, a general Fibonacci sequence, beginning with any two numbers is: u0, u1, (u0 + u1), ( u0 + 2u1), (2u0 + 3u1), (3u0 + 5u1), ... The coefficients of u0 and u1 are terms of the basic Fibonacci sequence.
Now consider a
sequence of rectangles beginning with a golden rectangle (1) of unit width.
Along the length of this rectangle, add a square (2) to produce a new
rectangle of length 1 +
Prove that the
lengths of this sequence of rectangles form a Fibonacci sequence.
Furthermore, show that the terms of this sequence form a geometric
progression with first term 1 and common ratio
The Modulor (I)
Le Corbusier
wanted to design mass housing for the post-World War II reconstruction which
was modularised, relatively cheap and yet inhabitable. To achieve this, he
argued, the proportions needed to be based on the proportions of the human
body so that people would feel ‘at home’, and the measurements compatible
with each other to facilitate the modular construction. He appealed to an
idea of the Ancient Greeks (and Egyptians), that the navel divides the
upright body in the ratio
The Modulor (II) Having demonstrated the constructibility of the proportions for his modular grid, Le Corbusier looked for a standard height upon which to base the Modulor. He settled on 183 cm, the height of a ‘six-foot detective’, giving 113 cm as the navel height (the dimension of the unit square), 226 cm as the height of the fingertips of the upraised arm, and 86 as the height where the hand rests. He then defined a Red
Series (Rn) : 5, 11, 16, 27, 43, 70, 113, 183, 296, 479,
775, 1254, ... Blue Series (Bn) : 1, 6, 7, 13, 20, 33, 53, 86, 140, 226, 366, 592, ... – Fibonacci sequences containing the key dimensions 113 and 183, 86 and 226. Allowing for rounding off, the system worked well in metric and imperial units. Using the values in the two Series, Le Corbusier was easily able to demonstrate that any square or rectangular region whose dimensions corresponded to those values could be dissected in seemingly limitless numbers of ways into smaller regions whose dimensions also took values from the Series. Here was the proportional method which would allow modularisation of components for building yet provide needed variety. He was proud to boast that in his Unité d’Habitation he only needed 15 standard measurements for the whole project, including dimensions of windows, doors, furniture etc. The Modulor system was successfully used in a whole range of other design projects, including factories, sculpture, typography and offices.
C. From Volker Müller. Proportions: Theorie and ConstructionGolden Section or Golden Mean, Modulor, Square Root of Two
http://home.att.net/~vmueller/prop/theo.html
Clearly the golden section proportion is closely connected with the square, the most neutral rectangular proportion (1 : 1) imaginable. (The "Modulor" books are square!) Compared with other proportions, the golden section rectangle is relatively long. That creates a certain tension between golden section and square, which may contribute to the interest that this proportioning scheme can maintain (see Corbu's Modulor), especially when compared to schemes that use the square as only proportioning scheme (see O.M.U.).
D. Excerpt from Michael J. Ostwald, "Review of Modulor and Modulor 2 by Le Corbusier (see sources below)
http://www.nexusjournal.com/reviews_v3n1-Ostwald.html
In 1954 the first English language edition of Le Corbusier's (Charles Edouard Jeanneret's) seminal work The Modulor was published by Faber and Faber. This first English edition, translated by Peter De Francia and Anna Bostock from the 1948 French edition, has since been reproduced in facsimile form by a variety of publishers. The most recent edition from Birkhäuser contains both the 1954 volume of The Modulor as well as the 1955 Modular 2 (Let the User Speak Next) in a reduced facsimile format inside a common slipcase featuring the graphic colour representation of the Modulor and its twin red and blue series. Le Corbusier developed the Modulor between 1943 and 1955 in an era which was already displaying widespread fascination with mathematics as a potential source of universal truths. In the late 1940s Rudolf Wittkower's research into proportional systems in Renaissance architecture began to be widely published and reviewed. In 1951 the Milan Triennale organised the first international meeting on Divine Proportions and appointed Le Corbusier to chair the group. On a more prosaic level, the metric system in Europe was creating a range of communication problems between architects, engineers and craftspeople. At the same time, governments around the industrialised world had identified the lack of dimensional standardisation as a serious impediment to efficiency in the building industry. In this environment, where an almost Platonic veneration of systems of mathematical proportion combined with the practical need for systems of co-ordinated dimensioning, the Modulor was born. For Le Corbusier, what industry needed was a system of proportional measurement which would reconcile the needs of the human body with the beauty inherent in the Golden Section. If such a system could be devised, which could simultaneously render the Golden Section proportional to the height of a human, then this would form an ideal basis for universal standardisation. Using such a system of commensurate measurements Le Corbusier proposed that architects, engineers and designers would find it relatively simple to produce forms that were both commodious and delightful and would find it more difficult to produce displeasing or impractical forms. After listening to Le Corbusier's arguments Albert Einstein summarised his intent as being to create a "scale of proportions which makes the bad difficult and the good easy."[2] A more mundane motive might also partially explain this endeavour. Le Corbusier saw that such a system could be patented and that when it became universally recognised and applied he "would have the right to claim royalties on everything that will be constructed on the basis of [his] measuring system."[3] Like Vitruvius and Alberti before him, Le Corbusier sought to reconcile biology with architecture through the medium of geometry. Just as Vitruvius describes the human body pierced with a pair of compasses and inscribed with Euclidean geometry, as an allegorical connection between humanity and architecture, so too Le Corbusier uses a Euclidean geometric overlay on the body for similar purposes.[4] After much experimentation Le Corbusier settled on a six foot tall (1.828m) English, male, body with one arm upraised. The French male was too short for the geometry to work well [5] and the female body was only belatedly considered and rejected as a source of proportional harmony.[6] According to Le Corbusier, the initial inspiration for the Modulor came from a vision of a hypothetical man inscribed with three overlapping but contiguous squares. Le Corbusier advised his assistant Hanning to take this hypothetical "man-with-arm-upraised, 2.20m. in height; put him inside two squares 1.10 by 1.10 metres each, superimposed on each other; put a third square astride these first two squares. This third square should give you a solution. The place of the right angle should help you to decide where to put this third square." [7] In this way Le Corbusier proposed to reconcile human stature with mathematics. To solve Le Corbusier's conundrum, Hanning started with the central (overlapping) square and then generated a golden section arc (from a diagonal of half the square) in one direction and another arc (from the diagonal of the full square) in the opposite direction. These arcs then generate two new contiguous squares which are also defined by a right angled triangle with its right angle passing through the common boundary between the two newly formed squares. The idea being that the resulting form can be used to create a series of Golden Section rectangles at multiple scales; except that it doesn't work geometrically. The final "squares" generated by the golden section and the arc are rectangles not squares; they are very close to being square (sufficiently close to fool amateur geometers) but are not equal sided as the mathematician Taton pointed out to Le Corbusier in November 1948. [8] A few weeks later, in December 1948, Mlle Elisa Maillard proposed an alternative solution for Le Corbusier's problem. Maillard's solution initially produces a golden section from the starting square to generate the second square and then uses the diagonal of the newly produced golden rectangle (the two overlapping squares) to form one edge of the right angle triangle. The remainder of the triangle generates the second square. Le Corbusier rapidly simplified Maillard's geometric solution (it had too many circles and thus looked too feminine) to the three square problem and replaced the human figure at its centre. He then used the vertical dimensions or heights generated by these three squares (which now overlap creating Golden rectangles) to produce measures which are proportional to the human body and which reflect the Golden Section. Despite now being geometrically valid, Le Corbusier's proportional system had another problem. Specifically the divisions between the ideal dimensions were too widely spaced to be useful or practical. Le Corbusier solved this problem by producing two parallel syncopated strips of dimensions, one based on the unit 108cm, the other on double that unit, 216cm. After further development the first sequence, now called the red series, became 1.130m (height of the navel of "man-with-arm-upraised") and the second sequence, now called the blue series, became 2.260m (height of the tip of his upraised fingers). Thus Le Corbusier finally defined the Modulor as "a measure based on mathematics and the human scale: it is constituted of a double series of numbers, the red series and the blue." [9] Le Corbusier's Modulor represents a curious turning point in architectural history. In one sense it represents a final brave attempt to provide a unifying rule for all architecture - in another it records the failure and limits of such an approach. Le Corbusier is quite open when he notes that the Modulor has the capacity to produce designs that are "displeasing, badly put together" or "horrors." [10] Ultimately he advises that "[y]our eyes are your judges" [11] and that the "Modulor does not confer talent, still less genius." [12] He also completely abandons the Modulor when it does not suit and persistently reminds people that since it is based on perception then its application must be limited by practical perception. Large dimensions are impossible to sense with any accuracy and so Le Corbusier does not advocate the use of the Modulor for these scales. Similarly construction techniques render the use of the modular for very small dimensions impractical. This proviso is important to remember and it is in part responsible for the way in which Le Corbusier eventually applied the rule. Having developed the Modulor and used it selectively in a few designs it then became largely invisible (and also immeasurable) in Le Corbusier's later works where it instinctively informed his eye as a designer but did not control it. Ultimately the two books of The Modulor represent a maddening and enthralling description of the trials and tribulations of an architect trying to find a universal solution to the problems of human proportion. The maddening aspects include a complete lack of consistency in geometric conventions or descriptions and a blatant ignorance of actual human proportions. None of this is helped by the erratic index in each volume or the occasionally inaccurate and misleading cross-reference. Those unfamiliar with Le Corbusier's circuitous prose will also find much of this famous work irritating and exasperating. Many dozens of pages are filled with self-congratulatory notes, ramblings about Indian mysticism and obscure literary perambulations into territories best left to the Rabelaisian characters Gargantua and Pantagruel.[13] Le Corbusier's hand-drawn illustrations are also occasionally geometrically difficult to understand or replicate and he is quite unconcerned about the mixture of degrees of accuracy he uses to support his thesis. A lengthy section at the end of the first volume even uses inaccurate measurements (taken while he was studying in Turkey some 40 years before) to trace the presence of the Modulor in several famous ruins and buildings! Yet, for all of these obvious flaws, Le Corbusier also helpfully records some of the dimensions which don't work as well as those that do. In this way he infers that certain proportions that are close to the Modulor are as beautiful as those that are more precisely Modulor. Despite all of these criticisms, Le Corbusier's two volumes on the Modulor are landmark works on the relationship between architecture and mathematics. These volumes describe the grand and quixotic search for a universal system; they record the practical and metaphysical problems with such an approach and they show how difficult it is to meld the human form with geometry and with architecture. NOTES FOR FURTHER READING.
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E. From: http://www.ac-nancy-metz.fr/Pres-etab/CollJLagneau/BLUM_CIBP/13ors/condor.html The Modulor is a system of measures based on the golden number and the series of Fibonacci. Construction du Modulor:
· * Je trace deux carrés juxtaposés ABCD et CDEF de côtés 113. · * Je trace le cercle de diamètre [DC]. · * La diagonale [AF] du double carré ABFE coupe le cercle en S. · * Je trace l'arc de cercle de centre F et de rayon FS; il coupe (FB) en G. · * Je construis un 3ème carré GJIH de côté GJ=AB. · * L'angle droit BDF coupe les côtés [HG] et [IJ] du 3ème carré en K et L. · * (KL) coupe [EF] en M. · * Je trace (PL) perpendiculaire à (DL) en L. · * On prouve que l'angle LPM est un angle droit. · * Les cercles centrés sur (KL), passant par H et I, I et E...dessinent la "spirale rouge". · * Les cercles centrés sur (YS), symétrique de (KL) par rapport à (AE), passant par R et D, D et P...dessinent la "spirale bleue". La Section d'Or 113,70 fournit la série "rouge": ...11,16,27,43,70,113,183,296,479,775... La Section d'Or 140,86 fournit la série "bleue": ...13,20,33,53,86,140,226,366,592,958... F. Conversion to feet and inches per Corbusier Red series Blue series 5 cm (2.5”) 1 cm (.5”) 11 cm (4”) 6 cm (2.25”) 16 cm (6.5”) 7 cm (2.75”) 27 cm (10.5”) 13 cm (5”) 43 cm (17” = 1’ 5”) 20 cm (8”) 70 cm (27.5” = 2’ 3.5”) 33 cm (13” = 1’ 1”) 113 cm (44.5” = 3’ 8.5”) 53 cm (21” = 1’ 9”) 183 cm (72” = 6’0”) 86 cm (34” = 2’ 10”) 296 cm (116.5” = 9’ 8.5”) 140 cm (55” = 4’ 7”) 479 cm (188.5” = 15’ 8.5”) 226 cm (89” = 7’ 5”) etc. etc.
We have studied the theories on which architects have based proportional systems for much of architectural history, and we have learned how to use arithmetic progression, geometric progression, harmonic mean and the Golden Section in the design of a series of rooms. Now it is time to consider the proportional/modular system developed by Le Corbusier in the period after World War II. Before beginning, it is important to remember these points: 1. Le Modulor is based on an approximation of a double square, which is made more interesting by use of an approximate golden section. 2. Le Modulor is completely an approximation, rather than a precise application of exact proportions. 3. The most interesting aspect of the Modulor is its use of two different Fibonacci series that interconnect with reference to the original double square. 4. There is one major difference between Le Modulor and Renaissance proportional systems. Architects like Palladio applied their proportions in specific ways….as in making the height of a room the harmonic mean between the width and the length. In contrast, Corbusier used the numbers of the Modulor in whatever way they seemed helpful, although some of the numbers have specific references to the spaces needed by the human body. In this way, it is more accurate to call the Modulor a system of modular numbers instead of a true system of proportions.
There is one more interesting aspect of the Modulor. We remember that the Platonists developed two different progressions of numbers, represented as we have seen in the form of the Greek letter lambda: 1 2 3 4 9 8 27
It is interesting that Corbusier also used two different progressions, the red and blue series: 1 5 6 11 7 (red) 16 13 (blue) … … However, whereas the Platonists used a static system that stopped after reaching the cubes (8 and 27), Corb developed a dynamic system of numbers that is ever-expanding. I believe this is significant in that it reveals a fundamentally different point of view between the 16th and the 20th centuries. Where the Renaissance saw a world of simple, unchanging order, we find ourselves in a cosmos that is changing and dynamic and complex.
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,
and the Fibonacci sequence. The golden section may be defined conveniently
in terms of the golden rectangle, which has the property that the
ratio of the length of the smaller side to the greater is equal to the ratio
of the length of the greater side to the sum of the lengths of the two
sides. The sides of the golden rectangle are in the ratio 
5)/2
= 1.618... The Ancient Greeks, who first defined this ratio, and Le
Corbusier (among many others) believed that the golden rectangle has
proportions which are most appealing to the eye.
