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!