THE GOLDEN RATIOBy : Nicole PaserchiaFollowMe !!
WHAT IS IT? Two quantities are in the golden ratio if the ratio of the sum ofthe quantities to the larger quantity is equal to the ratio of thelarger quantity to the smaller one. The ratio for length to width of rectangles of 1.6180339887 49894 84820 This ratio is considered to make a rectangle most pleasing tothe eye. Named the golden ratio by the Greeks. In math, the numeric value is called phi (φ), named for theGreek sculptor Phidias.
PLAIN AND SIMPLEIf you divide a line into two parts so that the longerpart divided by the smaller part is also equal to thewhole length divided by the longer part
THE PROOFThen 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 abovethe columns of the Parthenon There are golden ratiosall throughout thisstructure.
GREEK MATHEMATICIANS Appeared very often in geometry.
PYTHAGORAS Pythagorean’s symbolProved that the golden ratio wasthe basis for the proportions ofthe human figure.He believed that beauty wasassociated with the ratio ofsmall integers.
PYTHAGORAS If the length of the hand has the value of 1,then thecombined length of hand + forearm has the approximatevalue of φ.
DERIVING Φ MATHEMATICALLYφ 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 – 1The solution := 1.6180339 … = φThis 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 SEQUENCEThe 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 ofthe Fibonacci sequence :If a Fibonacci number is divided by its immediate predecessor in thesequence, the quotient approximates φ.When a = 1 :
THE GOLDEN RATIO IN MUSICMusical 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 notecreate the basic foundation of all chords, and are based on wholetone which is 2 steps from the 1st note of the scale.• Fibonacci and phi relationships are found in the timing ofmusical compositions.•The climax of songs is at roughly the phi point (61.8%) ofthe song.•In a 32 bar song, this would occur in the20th bar.• Phi is used in the design of violins
http://www.youtube.com/watch?feature=player_detailpage&v=W_Ob-X6DMI4Take a lookat 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. Retrievedfrom: http://www.goldennumber.net/ Michael Blake. (15 June 2012). What Phi Sounds Like. Retrieved from:http://www.youtube.com/watch?v=W_Ob-X6DMI4See yousoon !