1. THE GOLDEN RATIO
By : Nicole Paserchia
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2. WHAT IS IT?
 Two quantities are in the golden ratio if the ratio of the sum of
the quantities to the larger quantity is equal to the ratio of the
larger quantity to the smaller one.
 The ratio for length to width of rectangles of 1.61803
39887 49894 84820
 This ratio is considered to make a rectangle most pleasing to
the eye.
 Named the golden ratio by the Greeks.
 In math, the numeric value is called phi (φ), named for the
Greek sculptor Phidias.

3. PLAIN AND SIMPLE
If you divide a line into two parts so that the longer
part divided by the smaller part is also equal to the
whole length divided by the longer part

4. THE PROOF
Then Cross Multiply to get:
First,
Make equation equal to zero:
Next, use the quadratic formula
=
Then,
By definition of φ,
So,
=
Simplify,
φ =

5. PHIDIAS
 Called one of the greatest Greek Sculptors of all time
 Sculpted the bands that run above
the columns of the Parthenon
 There are golden ratios
all throughout this
structure.

6. GREEK MATHEMATICIANS
 Appeared very often in geometry.

7. PYTHAGORAS
 Pythagorean’s symbol
Proved that the golden ratio was
the basis for the proportions of
the human figure.
He believed that beauty was
associated with the ratio of
small integers.

8. PYTHAGORAS
 If the length of the hand has the value of 1,then the
combined length of hand + forearm has the approximate
value of φ.

10. DERIVING Φ MATHEMATICALLY
φ can be derived by solving the equation:
n2 - n1 - n0 = 0
= n2 - n - 1 = 0
This can be rewritten as:
n2 = n + 1 and 1 / n = n – 1
The solution :
= 1.6180339 … = φ
This gives a result of two unique properties of φ :
If you square φ , you get a number exactly 1 greater than φ:
φ 2 = φ + 1 = 2.61804…
If you divide φ into 1, you get a number exactly 1 less than φ :
1 / φ = φ – 1 = 0.61804….

11. RELATIONSHIP TO FIBONACCI SEQUENCE
The Fibonacci sequence is:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ....
The golden ratio is the limit of the ratios of successive terms of
the Fibonacci sequence :
If a Fibonacci number is divided by its immediate predecessor in the
sequence, the quotient approximates φ.
When a = 1 :

12. DRAWING A GOLDEN RECTANGLE

13. THE GOLDEN RATIO IN MUSIC
Musical scales are based on Fibonacci numbers
• There are 13 different octaves of any note.
• A scale is composed of 8 notes, of which the 5th and 3rd note
create the basic foundation of all chords, and are based on whole
tone which is 2 steps from the 1st note of the scale.
• Fibonacci and phi relationships are found in the timing of
musical compositions.
•The climax of songs is at roughly the phi point (61.8%) of
the song.
•In a 32 bar song, this would occur in the
20th bar.
• Phi is used in the design of violins

14. http://www.youtube.com/watch?f
eature=player_detailpage&v=
W_Ob-X6DMI4Take a look
at this !

15. RESOURCES
 Radoslav Jovanovic. (2001 – 2003). Golden Section. Retrieved from :
http://milan.milanovic.org/math/english/golden/golden2.html
 (16 May 2012). Phi and Mathematics. Retrieved from:
http://www.goldennumber.net/math/
 Nikhat Parveen. Golden Ratio Used By Greeks. Retrieved from:
http://jwilson.coe.uga.edu/EMAT6680/Parveen/Greek_History.htm
 (2012). Golden Ratio. Retrieved from:
http://www.mathsisfun.com/numbers/golden-ratio.html
 Pete Neal. (2012). Golden Rectangles. Retrieved from:
http://www.learner.org/workshops/math/golden.html
 Gary Meisner. (2012) Music and the Fibonacci Series and Phi. Retrieved
from: http://www.goldennumber.net/
 Michael Blake. (15 June 2012). What Phi Sounds Like. Retrieved from:
http://www.youtube.com/watch?v=W_Ob-X6DMI4
See you
soon !