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The Mathematics of Phi It’s a “phi-nomenon!” By Geoff Byron, Tyler Galbraith, and Richard Kim The History of Phi WHAT IS PHI? Phi can sometimes be misunderstood because it is known by so many different names: Ex: mean and extreme ratio, golden proportion, golden mean, golden section, golden number, divine proportion, φ, or sectio divina Phi is most often known as the golden ratio VALUES FOR PHI Two quantities are said to be in the golden ratio, if “the whole is to the larger part as the larger part is to the smaller part.” This can be demonstrated by: ab a a b Phi is equal to the following quadratic equation: 1 5 2 Therefore, we have Phi take on the values of 1.618 and .618, which are often written as Phi = 1.618 and phi = .618 THE GOLDEN MEAN From the graphic above we can derive the following about Phi: A is 1.618 times B and B is 1.618 times C. Alternatively, C is .618 of B and B is .618 of A. WHO FOUND PHI? There is debate over when and by who Phi was actually discovered. Egyptians: The ratio is found in the dimensions of the Egyptian’s pyramids, yet there is no mathematical or historical proof that the Egyptians knew about Phi. Euclid: Most often, the finding of Phi is associated with the Greek mathematician, Euclid, who wrote about Phi in his series of books, Elements, around 300 B.C. Euclid is attributed with finding the golden ratio and many of its properties. WHO FOUND PHI? Fibonacci: Fibonacci is given credit for adding to the properties of Phi by establishing the Fibonacci Sequence, but it is uncertain if Fibonacci himself ever found the connection between his sequence and Phi. WHERE DID THE NAME PHI COME FROM? It was not until the 1900’s that the numerical value of 1.618 was given the name Phi. Until then it was only referred to as the golden ratio, divine proportion, golden mean, and golden section. American mathematician Mark Barr first used the Greek letter phi to designate the proportion Reasons for choosing Phi: Phi is the first letter of Phidias, who used the golden ratio in his sculptures, as well as the Greek equivalent to the letter “F,” the first letter of Fibonacci. Phi is also the 21st letter of the Greek alphabet, and 21 is one of the numbers in the Fibonacci series. WHERE WAS PHI FIRST SEEN? Phi was first seen in the design of the Great Pyramids. (2560 B.C.) It can also be seen used excessively in the design of the Parthenon. (447 B.C.) So, how is Phi derived? Jacques Philippe Marie Binet Developed a formula that finds any Fibonacci number without having to start from 1, 1, 2, 3, 5, 8, etc…. What old mathematicians found out about Phi x 1 1 1 1 1 ... Square both sides: x 1 1 1 1 1 1 ... 2 x 1 x 2 x x 1 0 2 Apply quadratic equation: x 1 5 2 Notice that phi differs by sign: 1.61803399... 0.61803399... What old mathematicians also found about x x 1 2 Can you find the pattern? x x 1 2 x 2x 1 3 x 3x 2 4 x 5x 3 5 x 8x 5 6 x 13x 8 7 x ???? n Binet’s Formula x fib(n) x fib(n 1) n Solve for fib(n). x A : fib(n) fib(n 1) n B : () fib(n)() fib(n 1) n Subtract B from A: n ()n fib(n) fib(n 1) [ fib(n)() fib(n 1)] n () n fib(n) fib(n)() fib(n 1) fib(n 1) n () n fib(n)[ ()] n ( ) n fib(n) ( ) n ( ) n fib(n) 5 n () n fib(n) 5 Finds any Fibonacci number, assuming at n=1, Fib(1)=1. What does Binet have to do with Phi? fib(n+1) If we look at Binet’s formula as it approaches infinity, it converges to phi. 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 fib(n) 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 fib(n+1)/fib(n) 1 2 1.5 1.666667 1.6 1.625 1.615385 1.619048 1.617647 1.618182 1.617978 1.618056 1.618026 1.618037 1.618033 1.618034 1.618034 1.618034 1.618034 Looking at convergence from a calculus perspective, what test should we use to test convergence??? The RATIO TEST! n 1 ( n 1 ) n 1 n 1 fib(n 1) ( ) 5 lim lim lim n n n n n n n ( ) fib(n) ( ) 5 n 1 lim n n Applications of Phi Phi in Nature There is no other number that recurs throughout life more so than does phi. When looking at nature, we see Phi, often times without realizing it. Phi in Nature The golden spiral is created by making adjacent squares of Fibonacci dimensions and is based on the pattern of squares that can be constructed with the golden rectangle. If you take one point, and then a second point one-quarter of a turn away from it, the second point is Phi times farther from the center than the first point. The spiral increases by a factor of Phi. Phi in Nature This shape can be found in many shells, especially in nautilus. Phi in Man The Phi proportion itself can be found in the very bones that form our body's skeleton. For example, the three bones of any finger are related to one another by 1.618. Also, the wrist joint cuts the length from fingertip to elbow at 0.618 Ratios equal to Phi Chin to Brow Tip of Head to Brow Navel to Bottom of Foot Tip of Head to Navel Phi in Design The appearance of phi in all we see and experience creates a sense of balance, harmony and beauty. Mankind uses this same proportion found in nature to achieve balance, harmony and beauty in its own creations of art, architecture, colors, design, composition, space and even music. Phi in Design Works Cited Freitag, Mark. "Phi: That Golden Number." Golden Ratio. 2006. 11 May 2006 <http://jwilson.coe.uga.edu/EMT669/Student.Folders/Frietag.Mark/Ho mepage/Goldenratio/goldenratio.html>. Obara, Samuel. "Golden Ratio in Art and Architecture." University of Georgia Dept. of Mathematics Education. 2003. 11 May 2006 <http://jwilson.coe.uga.edu/EMT668/EMAT6680.2000/Obara/Emat66 90/Golden%20Ratio/golden.html>. "Phi / Golden Proportion." Nature's Word | Musings on Sacred Geometry. 2006. 11 May 2006 <http://www.unitone.org/naturesword/sacred_geometry/phi/in_nature/ >. Place, Robert M. "Leonardo on the Tarot." The Alchemical Egg. 2000. 11 May 2006 <http://thealchemicalegg.com/leotaroN.html>. "The Arts - Design and Composition." Phi the Golden Number. 2006. 11 May 2006 <http://goldennumber.net/design.htm>.