The Beauty of the Golden Ratio Presented by: Nikki Pizzano Danielle Villano Tanya Hutkowski Pedagogy Conference April 2009

Beauty or Beast? A B Pedagogy Conference April 2009

Activote Activity A B Pedagogy Conference April 2009

Let’s Get Started Width of the face Top of the head to the chin Pedagogy Conference April 2009

Width of the nose Length of lips Pedagogy Conference April 2009

Nose tip to chin Lip to chin Pedagogy Conference April 2009

Analyze the Results Let’s take a closer look at these measurements in Excel. Pedagogy Conference April 2009

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. Pedagogy Conference April 2009

Finding the Gold Pedagogy Conference April 2009 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 Pedagogy Conference April 2009

The Beauty Test http://www.intmath.com/Numbers/mathOfBeauty.php 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 Pedagogy Conference April 2009

How Phi is their Face? Pedagogy Conference April 2009

How Phi Are They? Pedagogy Conference April 2009

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. 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. Pedagogy Conference April 2009 13

d i a m e t r to circumference Pedagogy Conference April 2009 14

y x Pedagogy Conference April 2009 15

1 x 1 1 Pedagogy Conference April 2009 16

We can solve this quadratic equation using the Quadratic Formula. Pedagogy Conference April 2009 17

1 +1 -1 where a = 1 b = 1 c = -1 Pedagogy Conference April 2009 18

Pedagogy Conference April 2009 19

…which can be substituted into the ratio : Pedagogy Conference April 2009 20

How did the ancient Greeks use the Golden Mean? Pedagogy Conference April 2009

Size your compass to be the length of one side of the square 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 Pedagogy Conference April 2009

Place the point of your compass on point B and repeat. Pedagogy Conference April 2009

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. Pedagogy Conference April 2009

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. Pedagogy Conference April 2009

Repeat, creating segment BF Pythagorean Theorem: Pedagogy Conference April 2009

Pythagorean Theorem: c b a b c a Pedagogy Conference April 2009

Use your ruler to extend segments AB and CD. Pedagogy Conference April 2009

Adjust your compass so that it is the length of segment CE Make a mark on the extension of segment AB. Pedagogy Conference April 2009

Place the point of your compass on point F. Make a mark on the extension of segment DC. Pedagogy Conference April 2009

Create points GH and finish the rectangle by using your ruler to create segment GH. Pedagogy Conference April 2009

The rectangle you have created is the golden rectangle. Segment DE = 1/2 Segment EF = Segment DF = Pedagogy Conference April 2009

This rectangle is proportional to our original rectangle. If we make a square in the remaining rectangle, you will again see one rectangle left. This rectangle is proportional to our original rectangle. Pedagogy Conference April 2009

We could continue doing this over and over in a never ending pattern. Pedagogy Conference April 2009

phi = 0.618033988… (Smaller/Larger) Phi = 1.618033988… (Larger/Smaller) 1 1 2 3 5 8 13 … Pedagogy Conference April 2009 35

Pedagogy Conference April 2009 36

Pop Quiz!! a. in the human face b. in the Parthenon c. in DNA Where can the golden ratio be found? a. in the human face b. in the Parthenon c. in DNA d. all of the above Pedagogy Conference April 2009

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