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# Fibonacci Series

Student at Chittagong University of Engineering and Technology
27 Aug 2016
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### Fibonacci Series

1. Fibonacci Series GROUP 2: 1206006, 07, 08, 09, 10
2. What???
3. • Fibonacci series is an Integer sequence. Ex-1,1,2,3,5,8,13,21,34,55,89,144… first two numbers are - 1 and 1, or 0 and 1 (depending on source point) • closely related to Lucas numbers and Golden Ratio. • 1st in Indian mathematics. • Invented by Leonardo Fibonacci. • Discovered in an investigation of Reproduction of rabbits. Reproduction of rabbits formula Indian Mathematics
4. Need Help???
5. •Integer sequence is a sequence or an ordered list of integers. For example, a sequence 0, 1, 1, 2, 3, 5, 8, 13, … (the Fibonacci sequence) is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one. The sequence 0, 3, 8, 15, … is formed according to the formula n2 − 1 for the nth term. •Each Lucas number is defined to be the sum of its two immediate previous terms, thereby forming a Fibonacci integer sequence. The first two Lucas numbers are L0 = 2 and L1 = 1 as opposed to the first two Fibonacci numbers F0 = 0 and F1 = 1. Though closely related in definition, Lucas and Fibonacci numbers exhibit distinct properties. Ex- 2,1,3,4,7,11,18,29,47,76,123 •The ratio between two consecutive Lucas numbers converge to the golden ratio. Two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities Integer sequences Lucas Number Golden Ratio
6. Who???
7. • Leonardo Fibonacci (1170-1240),lived in Pisa, Italy. • He was a Catholic Christian in religion. • Most talented western mathematician of middle age. • Introduced the Hindu-Arabic numerical system. • Introduced Fibonacci Series in his book Liber Abaci (1202). • Introduced Digital Notation to Europe.
8. Fibonacci In Mathematics • Integer sequence. • Fibonacci spiral: an approximation of the golden spiral created by drawing circular arcs connecting the opposite corners of squares in the Fibonacci tiling. • A Recurrence relation (an equation that recursively defines a sequence.) • Fibonacci numbers occur in the sums of "shallow" diagonals in Pascal's triangle. • Fibonacci number is the length of the hypo-tenuse of a right triangle with integer sides. Ex: 1,2,3,5 etc. Pascal’s Triangle Fibonacci Series
9. Fibonacci In Nature • Appears in Biological settings • such as the fruit sprouts of a pineapple, the, an uncurling fern and the arrangement of a pine cone. Flowering Artichoke Arrangement of leaves on a stemFibonacci in Galaxies
10. Fibonacci In Art Birth of Venus Sandro Botticelli 67.9 in × 109.6 in • Intimately connected with golden ratio. • If we start the series with 0, we get golden ratio. If we start from 1, we get golden spiral. • Used in art. Such as – Birth Of Venus by botticelli, Parthenon etc. • The ratio 1:1.618 is very pleasing to eyes.
11. Fibonacci In Architecture • In constructing spiral stairs. • As golden ratio & Fibonacci spiral. • In constructing roofs.
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