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Unformatted text preview: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs. Leonardo Fibonacci In 1202, Fibonacci proposed a problem about the growth of rabbit populations. Inductive Definition or Recurrence Relation for the Fibonacci Numbers Stage 0, Initial Condition, or Base Case: Fib(0) = 0; Fib (1) = 1 Inductive Rule For n>1, Fib(n) = Fib(n1) + Fib(n2) n 1 2 3 4 5 6 7 Fib(n) 1 1 2 3 5 8 13 Sneezwort (Achilleaptarmica) Each time the plant starts a new shoot it takes two months before it is strong enough to support branching. Counting Petals 5 petals: buttercup, wild rose, larkspur, columbine (aquilegia) 8 petals: delphiniums 13 petals: ragwort, corn marigold, cineraria, some daisies 21 petals: aster, blackeyed susan, chicory 34 petals: plantain, pyrethrum 55, 89 petals: michaelmas daisies, the asteraceae family. Pineapple whorls Church and Turing were both interested in the number of whorls in each ring of the spiral. The ratio of consecutive ring lengths approaches the Golden Ratio. = = = = = = = AC AB AB BC AC BC AC BC AB BC BC BC 1 1 2 2 2 Definition of (Euclid) Ratio obtained when you divide a line segment into two unequal parts such that the ratio of the whole to the larger part is the same as the ratio of the larger to the smaller. A B C Pentagon Expanding Recursively = + = + + = + + + 1 1 1 1 1 1 1 1 1 1 1 1 Continued Fraction Representation = + + + + + + + + + + 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 .... Continued Fraction Representation 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 .... 1 5 2 = + + + + + + + + + + + 1,1,2,3,5,8,13,21,34,55,. 2/1 = 2 3/2 = 1.5 5/3 = 1.666 8/5 = 1.6 13/8 = 1.625 21/13 = 1.6153846 34/21 = 1.61904 = 1.6180339887498948482045 How to divide polynomials? 1 1 1 X ? 1 X 1 1(1 X) X + X(X X 2 ) X 2 + X 2(X 2 X 3 ) X 3 = 1 + X + X 2 + X 3 + X 4 + X 5 + X 6 + X 7 + 1 + X 1 + X 1 1 + X + X 2 2 + X + X 3 3 + + X + + X n n + .. = + .. = 1 1 1  X 1  X The Infinite Geometric Series (1X) ( 1 + X 1 + X 2 + X 3 + + X n + ) = 1 + X 1 + X 2 + X 3 + + X n + X n+1 + .  X 1  X 2  X 3   X n1 X n  X n+1  = 1 1 + X 1 + X 1 1 + X + X 2 2 + X + X 3 3 + + X + + X n n + .. = + .. = 1 1 1  X 1  X 1 + X 1 + X 1 1 + X + X 2 2 + X + X 3 3 + + X + + X n n + .. = + .. =...
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This document was uploaded on 01/17/2012.
 Fall '09

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