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The Golden Section andArchitecture
Do we find it pleasing?
Incorporated into Buildings forMillennia
• Egyptians used it in the design of their religiousbuildings
• Had many proponents in Ancient Greece• Explosion in art and architecture during the
Renaissance• Psychologists have been examining its aesthetic
qualities since the 1870s• Today many architectural firms have it on their
web sites• Is it incorporated into Ross
What is the golden section?
Suppose that we were to divide a linesuch that the ratio of the shorter
segment to the longer segment equalsthe ratio of the longer segment to the
entire line.
Mathematically, this can be shown asa/b = b/(a + b)
Where a, b represent the lengths ofthe shorter and longer segments,
respectively, and a + b = 1 Phi 1
--------------|-------- 1 phi
phi = 0.618Phi = 1.618…
In Book 6, Proposition 30, Eucliddiscusses division of a line to obtain
the golden section 1
A G B-------------------------------
g 1 – g
GB = AG , 1 – g = gAG AB g 1
What are some of the propertiesof Phi?
• Phi = 1.618…, phi = 0.618…• Phi = 1 + phi• Phi = 1/phi• Phi2 = Phi +1• Phi = (sqr5 +1)/2
How can the golden section beconstructed?
Also can be found in ratios ofFibonacci numbers
1,1,2,3,5,8,13,21,…1/1,1/2,2/3,3/5,5/8,8/13,13/21,…
1,.5,.67,.6,.625,.615,.619,…
Which Structures HaveIncorporated It?
Egyptians used the goldensection in their pyramids
In Greece, the most famousbuilding incorporating phi is
….
Also popular in medieval andRenaissance architecture
Or Notre Dame in Paris
Many art books talk aboutcentering objects about 1/3 of the
way across
Many artists incorporated it intotheir works
Famous modern day architectsdiscussed its beauty
Incorporated into buildings suchas the UN in NYC
Le Corbusier’s Villa Stein inGarches
I.M. Pei’s Bank of China
Is it aesthetically pleasing?
Controversial
• Many studies have been done• Conduct of studies has been controversial•
Early Psychological Research
• Fechner (1871) “Zur experimentalenAesthetik”
• Subjects given 10 rectangles of equal areaand asked to choose most pleasing
• 35% expressed a preference for the goldensection
• 21% expressed a preference for the 1.5:1rectangle
Criticism
• Golden rectangle was in middle ofproportion ranges (7th)
• Subjects not randomly selected• Subjects may have been cognizant of
Fechner’s hypothesis• Preference is a function of associations• Association being the human body which
all subjects had
Angier and Division of a Line1903
• Asked 9 subjects to divide a line 72 times“at the most pleasing place.”
• Only two subjects chose the goldenproportion with great regularity
• Mean proportion was 0.6
E.L. Thorndike 1917
• Subjects given 12 rectangles of varyingproportions, but with same height
• Height to width ratios varied from 1.3: 1 to 3.75:1• Asked to rank in terms of “which do you like the
looks of most?”• The golden rectangle and the next two most
elongated rectangles were ranked 1st, 2nd and 3rd
Similar Studies Kept Being Done
• Studies by age groups• Preschoolers had no particular preference• 3rd and 6th graders preferred wider triangles• Post secondary students preferred rectangles
0.55, 0.60 and 0.65 ratios• Elderly subjects preferred wider rectangles
Studies by Culture
• Canadian versus Japanese• 9% of Canadians preferred it versus 5% of
Japanese• Americans versus Europeans• Substantial differences showed up between
these two groups
Result is a “mass of conflictingdata and conclusions.”
(Green 1995)
• “It may simply be that the psychologicalinstruments we are forced to use in studyingthe effects of the golden section are just toocrude.” D.E. Berlyne, 1971
According to you folks…
Dimensions:
A 0.4:1B 0.5:1C 0.92:1D 0.62:1E 0.85:1
Thank You!