# Golden Section

52 53 aPPenDix ii fibonacci lucas forMul Definition of the Fibonacci Series F0 0, F1 1, then Fn2 Fn1 Fn Early Fibonacci numbers n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Fn 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 16 17 18 19 20 21 22 23 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657 24 25 26 27 28 29 46368, 75025, 121393, 196418, 317811, 514229 ... Binet formula for Fibonacci numbers Fn Fn Fn 5 Cassini formula for Fibonacci numbers Fn1Fn1 Fn2 1n Negative termed Fibonacci numbers Fn 1n1 Fn Factors of Fibonacci numbers Every nth Fibonacci number is a multiple of Fn, so Fn is a factor of every nth Fibonacci number. Thus F3 2 divides every 3rd Fibonacci number, meaning every third Fibonacci number is even F4 3 means every 4th Fibonacci number is divisible by 3, F5 5 divides every 5th Fibonacci number, and F6 8 divides every 6th Fibonacci number. Also if n is a factor of m, then Fn will be a factor of Fm. Summing Fibonacci numbers nFn Fn2 1, ie the sum of the first n Fibonacci numbers is one less than the n2nd Fibonacci number. Oddnumbered Fibonacci terms sum to the next evennumbered Fibonacci term while even termed Fibonacci numbers sum to one less than the next oddnumbered term. The squares of Fibonacci numbers nFn2 Fn Fn1 which means the sum of the squares of the first n Fibonacci numbers is equal to the product of the nth and the n1th Fibonacci numbers. Also Fn2 Fn Fn1 Fn1. The sum of the squares of two consecutive Fibonacci numbers Fn2 Fn12 F2n1. Definition of the Lucas Series L0 2, L1 1, then Ln2 Ln1 Ln Early Lucas numbers n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Ln 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 15 16 17 18 19 20 21 22 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39603, 24 25 26 27 28 29 64079, 103682, 167761, 271443, 439204, 710647, ... Binet formula for Lucas numbers Ln Fn Fn Cassini formula for Lucas numbers Ln2 Ln1Ln1 51n Negative termed Lucas numbers Ln 1n Ln The Cassini formulae As shown in the formula to the left, each Fibonacci number is the approximate geometric mean of its two adjacent numbers, alternately needing correction by 1 or 1, while above we see that each Lucas number is the approximated geometric mean of its two neighbors alternately corrected by 5 or 5. The two are further related by the expansion of Binet s formula as Fn Ln Fn52. Converting Fibonacci and Lucas numbers Ln Fn1 Fn1 , which is to say that the nth Lucas number is the sum of the n1th and n1th Fibonacci numbers. Related to this is the result Ln Fn2 Fn2. We also have Ln Fn 2 Fn1, and the fact that any four consecutive Fibonacci numbers sum to a Lucas number. Finally there is the simple, elegant equation F2n Fn Ln , and the fact that Fn Ln 2 Fn1. Hyperbolic Fibonacci and Lucas functions From Binet s formula we can derive the fascinating pair of equations L2n 2coshn ln F and its successor L2n1 2 sinhn ln F. In 2003, Alexey Stakhov published the following two remarkable identities sinFn cosFn sinFn1, and sinLn cosLn cosLn1. Trigonometrical functions involving phi Angle Sin Cos Tan 18o 36o 54o 72o Linking F, e and i Richard Feynman noticed the equation, based on Euler s, that eip F1 F. We also have the two results 2 sini ln F i, and 2 sinp2 i ln F 5. The golden string Intimately connected to F is the infinite binary rabbit sequence, which never contains 00 or 111 and arises in many ways 1011010110 1101011010 1101101011 0110101101 0110110101 1010110110 1011011010 1101011011 ... see www.mcs.surrey.ac.uk 2 1F 1 F aPPenDix i Phi equaTions Expressions for phi The simultaneously additive and multiplicative nature of the golden section is expressed in the simple quadratic equation a2 a 1 which has two solutions, one positive, one negative, F and F1 a1 1 5 and a2 1 5 2 2 thus F 5 1 1.61803398874989484882.. 2 and 5 1 0.61803398874989484882.. 2 This also gives the following important identities F 1 and F 1 F Taking the first of these and repeatedly substituting for F produces the simplest continued fraction F 1 1 1 while taking the second and repeatedly substituting for F produces the simplest nested radical F 1 1 F 1 1 1 F 1 1 1 1 1 1 ...... Approximate relationships between phi, pi and e The two formulae p E 65 F2 and p E 4F may both be derived from the Great Pyramid. Note too the approximate formulae e E F2 1 10 and the even more accurate e E 144 55 110 . 1 F 1 F 1 1 F 1 1 1 1 1 F 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ...... 1 1F 2 2 F 2 2 1F 2 1 F 2 1 F 2 1 1F 2 F 2 1F 1 F 2 1F 2 2 F 2 1 1F 2 2 F 1 1F