pedagogy conference april 2009 the beauty of the golden ratio presented by: nikki pizzano danielle...

Pedagogy Conference April 2009

The Beauty of the Golden Ratio

Presented by:

Nikki PizzanoDanielle Villano

Tanya Hutkowski


Beauty or Beast?

A B


Activote Activity

A B


Let’s Get Started

• Width of the face

• Top of the head to the chin


•Width of the nose

•Length of lips


• Nose tip to chin

• Lip to chin


Analyze the Results

• Let’s take a closer look at these measurements in Excel.


Linking Beauty To Phi• 1.618 0339 887...

• Also known as the golden number

• Phi is an irrational number which is found in all aspects of nature, and supposedly using it as a proportion leads to the most beautiful shapes.

• Many of the ancient Greek sculptors and builders

used to use it.


Finding the Gold

a = Top-of-head to chin

b = Top-of-head to pupil

c = Pupil to nose tip

d = Pupil to lip

e = Width of nose

f = Outside distance between eyes

g = Width of head

h = Hairline to pupil

i = Nose tip to chin

j = Lips to chin

k = Length of lips

I = Nose tip to lips


The Beauty Test

• This mask of the human face is based on the Golden Ratio. The proportions of the length of the nose, the position of the eyes and the length of the chin, all conform to some aspect of the Golden Ratio.

• http://www.intmath.com/Numbers/mathOfBeauty.php

http://www.intmath.com/Numbers/mathOfBeauty.php


How Phi is their Face?

€

127

65=1.954

€

32

24=1.333

€

32

17=1.882

€

152

96=1.583

€

47

28=1.679

€

52

31=1.677


How Phi Are They?

€

137

66= 2.076

€

36

29=1.241

€

30

22=1.364

€

92

56=1.643

€

31

19=1.632

€

22

14=1.571


The Fibonacci series appears in the foundation of many other aspects of art,

beauty and life. Even music has a foundation in the series, as 13 notes make the octave of 8 notes in a scale, of which the 1st, 3rd, and 5th notes create the basic foundation of all chords, and the whole tone is 2 steps from

the root tone.

The Greeks knew this as the Golden Section and used it for beauty and

balance in the design of architecture.

The Renaissance artists knew this as the Divine Proportion and used it for beauty

and balance in the design of art.


πto

circumference

diameter


x

yϕΦ

x

y

y

x


1 1

x1ϕΦ 1

x= 1+ x1


1 2= +x x

0 12= + −x x

1 1

1x

x=

+

We can solve this quadratic equation using the Quadratic

Formula.


xb b 4ac

2a

2

=− ± −

where a = 1b = 1c = -

1

x x2 1 0+ − = 1 +1 -1

x2 x =0

ax bx c 02 + + =


xb b 4ac

2a

2

=− ± −

x =− ±1 5

2x =

− +1 52

x =0618033988. ...

x =− ± − −( ) ( ) ( )( )

( )1 1 4 1 1

2 1

2

x =− ± − −1 1 4

2( )

ϕ


…which can be substituted into the ratio :

1 1

1x

x=

+Φ1618033988. ...

1

0 618033988

1 0 618033988

1. ...

. ...=

+

Φ

longshort

⎛⎝⎜

⎞⎠⎟


How did the ancient Greeks use the Golden

Mean?


• Given square ABCD.

• Size your compass to be the length of one side of the square

• Make a small arc above segment AB and one below segment AB

QuickTime™ and aAnimation decompressor

are needed to see this picture.

€

1

€

1


• Place the point of your compass on point B and repeat. QuickTime™ and a

Animation decompressorare needed to see this picture.

€

1

€

1


• Place your ruler on the intersections of the arcs.

• Make a mark on segment AB and segment CD.

• These are the midpoints of the segments.



€

1

€

1


• Mark points E and F at the midpoints of segments AB and CD.

• Use your ruler to create a segment from point C to point E.






€

1

2

€

1

Pythagorean Theorem:

€

a2 + b2 = c 2

aa

bbcc

Repeat, creating segment BF


€

1

2

€

1Pythagorean Theorem:

€

a2 + b2 = c 2

aa

bbcc

bbaa cc

€

1

4+1 = c 2

€

1

2

⎛

⎝ ⎜

⎞

⎠ ⎟2

+ (1)2 = c 2

€

€

5

4 = c 2

€

5

2= c


Use your ruler to extend segments AB and CD.



€

5

2

€

1

2


• Adjust your compass so that it is the length of segment CE

• Make a mark on the extension of segment AB.



€

1

2€

5

2


• Place the point of your compass on point F.

• Make a mark on the extension of segment DC.



€

5

2

€

1

2


• Create points GH and finish the rectangle by using your ruler to create segment GH.



€

1

2€

5

2


• The rectangle you have created is the golden rectangle.

• Segment DE = 1/2

• Segment EF = • Segment DF =•

€

5

2

€

1

2+

5

2=

1+ 5

2

€

5

2

€

1

2


• If we make a square in the remaining rectangle, you will again see one rectangle left.

• This rectangle is proportional to our original rectangle.


We could continue doing this over and over in a never ending pattern.


ϕ

Phi = 1.618033988…

phi = 0.618033988…

(Larger/Smaller)

(Smaller/Larger)

Φ1 1 2 3 5 8 13 …


Pop Quiz!! Where can the golden ratio be found?

a. in the human faceb. in the Parthenon c. in DNAd. all of the above


Question 2 Which is the closest approximation of the Golden Ratio?

a. 0.1680339887…b. 3.14159…c. 1.6180339887…d. 3:00

pedagogy conference april 2009 the beauty of the golden ratio presented by: nikki pizzano danielle...

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