golden ratio and fibonacci sequence in nature

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.

Golden ratio and fibonacci sequence in nature