Computers in education final

THE GOLDEN RATIOBy : Nicole Paserchia

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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.

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

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 =

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.

GREEK MATHEMATICIANS

Appeared very often in geometry.

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.

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.

DERIVING Φ MATHEMATICALLY

φ can be derived by solving the equation:n2 - n1 - n0 = 0

= n2 - n  -  1  = 0This 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….

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 :

DRAWING A GOLDEN RECTANGLE

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

http://www.youtube.com/watch?feature=player_detailpage&v=W_Ob-X6DMI4Take a look

at this !

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-X6DMI4See you

soon !