Transcription of The Golden Ratio and the Fibonacci Sequence
1 THEGOLDENRATIO AND THEFIBONACCISEQUENCETodd Cochrane1 / 24 Everything is Golden Golden Ratio Golden Proportion Golden Relation Golden Rectangle Golden Spiral Golden Angle2 / 24 geometric Growth, (Exponential Growth):r=growth rate or common Ratio . with 6 and double at each step:r=26,12,24,48,96,192,384, ..Differences: ExampleStart with 2 and triple at each step:r=32,6,18,54,162,..Differences:Rule :The differences between consecutive terms of ageometric Sequence grow at the same rate as the / 24 Differences and ratios of consecutive Fibonacci numbers:1123581321345589Is the Fibonacci Sequence a geometric Sequence ?Lets examine the ratios for the Fibonacci value is the Ratio approaching?4 / 24 The Golden RatioThe Golden Ratio , = The Golden Ratio is (roughly speaking) the growth rate of theFibonacci Sequence asngets (325-265 ) inElementsgives first recorded definitionof.
2 Next try calculating nFn 5,and soFn n / 24 Formula for then-th Fibonacci NumberRule:Then-th Fibonacci NumberFnis the nearest whole number to n 5. the 6-th and 13-th Fibonacci 6 5=,soF6=n=13. 13 5=,soF13=In fact, the exact formula is,Fn=1 5 n 1 51 n,(+for oddn, for evenn)6 / 24 The 100-th Fibonacci NumberFindF100. 100 5= / 24 Some calculations with Everyone calculate the following (round to three places):1 = 2= / 24 The Amazing Number The Amazing Number = (we ll round to 3places):1 is, 1/ = +1that is, 2= + +1that is, 3=2 + +2that is, 4=3 + +3that is, 5=5 +3 n=Fn +Fn 19 / 24 Golden RelationGolden Relation 2= +1 The Golden Ratio is the unique positive real number satisfyingthe Golden / 24 Exact Value of What is the exact value of ? From the Golden Relation, 2 1=0 This is a quadratic equation (second degree):ax2+bx+c= Formula: = b+ b2 4ac2a=1+ ( 1)2 4( 1)2=1+ 52.
3 =1+ 52= / 24 Golden Ratio from other sequences , start with any two numbers and form arecursive Sequence by adding consecutive numbers. See whatthe ratios approach this we start with 1,3,4,7,11,18,29,47,76,123,.. :Starting with any two distinct positive numbers, andforming a Sequence using the Fibonacci rule, the ratios ofconsecutive terms will always approach the Golden Ratio !Recall the Fibonacci Rule:Fn+1=Fn+Fn 112 / 24 Why does the Ratio always converge to ?WHY?LetAnbe a Sequence satisfying the Fibonacci Rule:An+1=An+An 113 / 24 Golden Proportion Golden Proportion:Divide a line segment into two parts,such that the Ratio of the longer part to the shorter part equalsthe Ratio of the whole to the longer part. What is the Ratio ?14 / 24 Golden Rectangle Example. Golden Rectangle:Form a rectangle such thatwhen the rectangle is divided into a square and anotherrectangle, the smaller rectangle is similar (proportional) to theoriginal rectangle.
4 What is the Ratio of the length to the width?15 / 24 How to construct a Golden RectangleStart with a squareABCD. Mark the midpointJon a givenedgeAB. Draw an arc with compass point fixed atJandpassing through a vertexCon the opposite edge. Mark thepointGwhere the arc meets the lineAB. Note: Good approximations to the Golden Rectangle can beobtained using the Fibonacci / 24 Partitioning a Golden Rectangle into Squares17 / 24 Golden SpiralWhere is the eye of the spiral located?18 / 24 Pentagon and Ratio of the edge of the inscribed star to the edge of theregular pentagon is , the Golden Ratio of the longer part of an edge of the star to theshorter part is .19 / 24 Construction of a Regular Pentagon20 / 24 Golden AngleDivide a circle into two arcs, so that the Ratio of the longer arc tothe smaller arc is the Golden / 24 Continued Fraction Expansions = +.
5 +11/. + +17+. +17+11/. +17+ +17+115+. +17+115+11+..Every irrational number has a unique infinite continuedfraction / 24 Continued Fraction Expansion of Continued Fraction Expansion of . +.6180339=1+11/.6180339=1+ +11+.6180339=1+11+11/.6180339=1+11+ +11+11+ +11+11+11+..Another way to see this is: From the first line above we see that =1+1 ,Now substitute this expression for into theright-hand side and keep repeating: =1+11+1 ,..23 / 24 Convergents to the Continued Fraction expansion of 1,1+11=2,1+11+11=1+11+11+11=The convergents to the continued fraction expansion of are24 / 24
