![]() ![]() Origin of FibonacciĪccording to Dunlap (2003), the Italian mathematician Leonardo Pisano was born into a merchant family in Pisa around 1175 A.D. Many artists have used the golden ratio throughout the history for their works as found it really beneficial and pleasing. Many of the greatest architectural structures in the world are designed in such a way that they have golden ratio in their proportions. the ratio between the diagonal and a side is divine ratio. Also in pentagon, when all the sides are equal, the diagonals when intersect divide each other in the golden ratio i.e. One of the best applications of the Golden ratio in the geometry can be, the proportion in the lines of the pentagram. Golden ratio fascinated the Greeks a lot, as it appears frequently in the field of geometry. (x+y)/x=x/y= golden ratio represented by Greek letter phi ( or ). For example: Let the whole length of the segment is x+y, where x is the larger part and y is the smaller part, then golden ratio can be defined as: He proposed that when a unit segment is divided into two lengths, then the ratio of the smaller part to the larger part of the whole segment is equal to the ratio of the larger part to the whole length of the segment. Euclid proposed the problem for the division of a line into golden section. Golden section and Golden ratio were first mentioned by Euclid in Elements in 300 B.C. Golden Ratio: Sum of all (fn)/(fn-1) produces golden ratio. ![]() ![]() from Fibonacci sequence f1,f2,f3,f4,f5.f2n-1, f2n, take numbers at even indices and sum them, their sum will be equal to number to f2n+1 – 1. Taking the Fibonacci numbers from even indices, their sum is f2n+1 - 1 i.e. from Fibonacci sequence f1,f2,f3,f4,f5.f2n-1, f2n, take numbers at odd indices and sum them, their sum will be equal to number to f2n. Taking the Fibonacci numbers from odd indices, their sum is f2n i.e. The Fibonacci sequence can be related to various items in the nature like: growing buds on the plants and trees, the growth of branches on trees, the spiral hexagon observed on the pineapple, the pinecone’s rows, the seed heads, the petals growing on various flowers such as sunflower, the starfish, shells observed on snails and nautilus, the spiral galaxies and the chambers of vegetables and fruits like apple, lemon, Chile etc. Thus the Fibonacci sequence and Fibonacci numbers can be defined recursively as:įn+2= fn + fn+1 where n>0 or n=0. After one year, let us calculate the pairs of Rabbit produced? Fibonacci Solution: Solution provided is the Sequence given by Fibonacci:Ġ,1,1,2,3,5,8,13,21,34,55,89,114 Thus, after completion of one year, pairs of rabbits produced will be 144. after reaching two months age, each pair produces another pair (one female and one male), and then mixed pairs are produced every month thereafter and no rabbit dies in between. Assuming all the months have same number of days and rabbit starts producing young rabbits exactly after two months they were born i.e. This sequence appears in numerous mathematical problems.įibonacci problem: Let us assume 2 rabbits (one male and one female) are born on 1st January. The Fibonacci sequence is the best example of any recursive sequence. By definition, the numbers in the Fibonacci sequence starts with either 0 and 1, or 1and1, and each subsequent number is the sum of the previous two numbers. The Fibonacci sequence has remarkable characteristics it can be applied to various fields like discrete mathematics, number theory and geometry. The Fibonacci number and sequence proposed by him gives a clear indication that mathematics can be connected to many things that may seem unrelated to it. One of these concepts is the “Fibonacci sequence” that was developed by Leonardo Pisano during the 13th century. In the realm of Mathematics, there are many concepts that can be applied to multiple fields of mathematics. ![]()
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