http://home.att.net/~vmueller/prop/theo.html
Proportions: Theorie and Construction
Golden Section or Golden Mean, Modulor, Square Root of Two
golden section or golden mean
Modulor
square root of 2
comparisons
applied proportions
golden section links
recommended reading
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 Nantes-Rezé 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 Naredi-Rainer'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 non-linear 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 non-equal 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 self-evident, 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)
(Purpose of proportioning schemes: Explanation of the perception of beauty?)
http://home.att.net/~vmueller/links/links.html#g_sec
Golden Section (Proportions) Links:
General Aspects of the Golden Section and Proportions:
My very own Proportion pages.
http://home.att.net/~vmueller/prop/theo.html#gs
Sacred Geometry, thoughts by Catherine Yronwode.
http://www.luckymojo.com/sacreddefined.html
Some articles from newsgroups about the divine proportion.
http://mathforum.org/~sarah/HTMLthreads/articletocs/divine.proportion.html
Phi in betting / gambling (some book for sale).
http://www.ozinet.com.au/phi.htm
Golden Section as Geometric Construction:
Animated geometric subdivision of a line in the proportion of the Golden Section.
http://home.att.net/~vmueller/prop/theo.html
Another method for the geometric subdivision of a line in the proportion of the Golden Section.
http://www.perseus.tufts.edu/GreekScience/Students/Tim/Golden.html
About the Mathematics of the Golden Section:
The Museum of Harmony and Golden Section.
http://www.goldenmuseum.com/
R. Knott's mother of all Fibonacci sites with pages on the Golden Section.
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html
The Golden Mean @ Favorite Mathematical Constants page.
http://pauillac.inria.fr/algo/bsolve/constant/gold/gold.html
Answers to Frequently Asked Question about the Golden Ratio by Dr. Math, Swarthmore College.
http://mathforum.org/dr.math/faq/faq.golden.ratio.html
The occurrence of the Golden Section in function graphs (y=x+1& y=x2).
http://www.vashti.net/mceinc/golden.htm
Timothy Relyga's page on Greek Mathematics.
http://www.perseus.tufts.edu/GreekScience/Students/Tim/Contents.html
About the Golden Section in Nature, Art, Architecture, Music, etc.:
Structures in Nature with their proportions.
http://ourworld.compuserve.com/homepages/Robert_Conroy/
The Golden Section in the human face, an article.
http://jwilson.coe.uga.edu/emt669/student.folders/banker.teresa/golden/goldbiol.html
The Golden Section in music analysis (in this case in section 19).
http://boethius.music.ucsb.edu/mto/issues/mto.96.2.5/mto.96.2.5.clendinning.html
The Golden Section in Mozart's Music.
http://www.americanscientist.org/template/AssetDetail/assetid/24551;jsessionid=baagIVue3KSs-w
The Golden Section in Debussy's Music.
http://www.music.indiana.edu/som/courses/rhythm/annotations/howat83.html
The ratio of the Golden Section in urban growth documented by the University of Michigan Institute of Mathematical Geography.
http://www-personal.umich.edu/~sarhaus/image/solstice/fonseca0.html
Michelangelo and the Golden Section in his work.
http://www.anselm.edu/homepage/dbanach/mich.htm
Galileo Galilei's proportioning compass in Firenze, Italy.
http://brunelleschi.imss.fi.it/esplora/compasso/dswmedia/simula/esimula2.html
A plate, patterned with Golden Section proportioning on: Plate 1 of the White House Collection at the National Museum of American Art.
http://americanart.si.edu/collections/exhibits/whc/whc-noframe.html?/collections/exhibits/whc/cox_fobject.html