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Proportions: Theorie and Construction
Golden Section or Golden Mean, Modulor, Square Root of Twogolden section or golden mean Modulor square root of 2 comparisons applied proportions golden section links
The Golden Section or Golden Mean is derived with
simple geometric constructions, its ratio expressed in numbers
is, however, irrational
((Square root of five)  1) : 2
= .618034 : 1 (= 1 : 1.618034). 
Starting
with the line that needs to be subdivided as the longer side, a rectangle
of the proportion 2 : 1 (two squares) is constructed. The
diagonal through the rectangle is drawn next. One of the short sides is
subtracted from the diagonal by drawing an arc with that side as radius
and a corner intersected by the diagonal as center. The intersection
divides the diagonal into two segments. The longer of the segments is
rotated onto the adjacent long side of the rectangle subdividing it so
that the ratio of the shorter subdivision and the longer subdivision is
the same as the ratio of the longer subdivision and the whole long side.

The
beauty of the golden section may be indicated by the fact that a golden
section rectangle subdivides into a square and another, smaller golden
section rectangle. This process can be continued ad infinitum, and
similarly inversed by adding a square over the longer side of a golden
section rectangle, thus establishing a proportional relationship over the
entire imaginable scale of human artifacts. 
In architecture the application of the golden section may afford the
integration of the whole scope of a design from the site to the minutest
detail. Charles Edouard Jeanneret, also known as Le Corbusier, develops
two sequences of measurements ("blue" and "red" sequence) in his "Le Modulor". He takes the  assumed average size of a
human, and subdivides and expands it, closely based on the golden section
relationship. In the sequel to "Le Modulor",
"MODULOR 2", Corbu describes how after development of
the measurements Jose Luis Sert and other modern architects, and of
course, he himself, applied them in designs.
Reliefs depicting the scheme,
usually in connection with an abstracted human, raising one arm to the 226
cm low ceiling, and resting the other hand on an 86 cm high desk (shown
here in an early version, still hiding the lower hand), can be found on
some of the buildings that Le Corbusier designed, for example on the Unité
d'habitation in Berlin, and the units in NantesRezé and Marseilles,
France. The culmination is probably the hand  dove of peace in
Chandigarh, India, turning part of the icon of the Modulor into a monument
of unrelentless modernist hope for the achievement of human betterment
through better architecture . . .
There is additional significance and deeper meaning to the golden section in combination of
pentagons and pentagrams, I recommend reading up on that in Paul
von NarediRainer's "Architektur und Harmonie". 
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.).
Now, does
that constitute any understandable reason to connect golden mean
proportioning inseparable with beauty? Without doubt: No. Because of the
nonlinear nature of the golden section, as clearly demonstrated in the
Modulor derivations, it is possible to find some base length and some
subdivisions close enough to the ratio of the golden section in anything
that may be perceived as beautiful. But that may have to do with the
underlying structuring into nonequal divisions that establish scale and
generate more interest because of the increased amount of detail that is
generated or that is cause of the inequal divisions. 
Another
proportioning system is the ratio of
(Square root of 2) : 1. The simplicity of the
derivation (square root of 2 is the diagonal through a square of side
length 1) is paralleled by the ease of maintaining the proportion through
division or multiplication of the proportioned rectangles. The sum of two
rectangles of proportion (Square root of 2) : 1
long side by long side is (Square root of 2) : 2.
Divided by the square root of two we arrive at
1 : (Square root of 2), the same ratio as the two
rectangles that were added together, only with a change of orientation.

The prevalent
paper formats in Germany are defined by the DIN 476 (DIN is the
German Insitute for Standardization, comparable to ANSI in the U.S.A, ISO
internationally, etc.). The sequence A of sizes is based on the ratio
(Square root of 2) : 1. This of course means that
pasting two DIN A pages together at their long sides yields the next
larger DIN A formatted page. Similarly, cutting a DIN A page
into halves by division of the longer side, yields two pages of the next
smaller DIN A formatted page. 
The advantage of sizing paper that way is selfevident, and without
doubt XEROX would have preferred that kind of paper formatting over
"letter" and "legal" format. (Ever wondered why for the longest time
copying machines seemed to have 141% and 70% as zooming limits?)
From the Golden Section to the
(Square root of 2) : 1 ratio there is clearly a
reduction in variation. While the Golden Section rectangle by definition
includes a square and another Golden Section rectangle, i.e. in fact two
different modules, or proportioning schemes, the
(Square root of 2) : 1 ratio contains only
itself, therefore, is by character closer to the square. The difference
here is, that the former scheme is derived by subdivision into halves,
while latter is subdivided into quarters. The scale steps in the latter
scheme are larger (instead of 1.414 and 0.707 it is 2.0 and 0.5), meaning
it is even less flexible. 
A
comparison of some of the most common proportions shows how little
they differ. Of the proportions that are shown here, the letter format is
the widest, and the golden section is the longest. 3 x 4 is the
aspect ratio of the common TV, 5 x 7 is a popular photo print
format, and the 35mm film format (24 x 35) is the basis for most
photography. 
The proportion of 3 x 4 has the additional significance that
its diagonal is 5. (According to Pythagoras the sum of the squares over
the kathetes equals the square over the hypotenuse, i.e.
9 + 16 = 25, with the square root of 25 being 5. qed)


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