The fibonacci sequence in nature series#

The fibonacci sequence in nature series#

The article I'm adapting this from is written a little better, so you can see what is going on, but they manipulate the infinite series in this way, which is not technically permitted. So we get that the partial sums converge to 1/89 and so x = 1/89, as we expected. S( N) = 1/89 - 89/100*.įinally, as N gets very large, those last three terms get very close to 0, so when we consider the infinite series, they do not contribute at all. Since fib(0) = fib(1) = 1, we can actually get rid of the second and third terms as well. So everything cancels out, except for the first and last few terms (this is a so-called telescoping sum).Īfter this nice cancellation, we're left with: S( N)-1/10*S( N)-1/100*S( N) = fib(0)/10 2 + fib(1)/10 3 - fib(0)/10 3 - fib( N)/10 N +3 - fib( N-1)/10 N +2 - fib( N)/10 N +4. But this is just 0, since fib(2) = fib(1) + fib(0). For example, we get fib(2)/10 4 - fib(1)/10 4 - fib(0)/10 4 from each term, respectively. Now look at the terms that have the same power of 10 in them. Notice that I've distributed the 1/10 and 1/100 into the sums by increasing the exponent of the 10s. However, we're going to use the fact that fib( n) = fib( n-1) + fib( n-2) to our advantage. + fib( N)/10 N +2.Ĭonsider S( N)-1/10*S( N)-1/100*S( N), which is something not immediately obvious that we should look at. This will allow us to treat it like a regular finite sum, as we're used to, and then use limits to make N arbitrarily large, i.e. we're going to stop N terms into the series. It's actually important to prove that this series converges, but I'm not going to do that. So, let's call the number that we're interested in x, i.e. Let fib( n) be the nth Fibonnaci number, i.e. In famous art pieces, 'The Last Supper' and 'The Mona Lisa', Leonardo Da Vinci used the Fibonacci sequence to create these masterpieces! Music composers such as Mozart and Bartok have used this same sequence in some of their works! Even in today's age of music, Maynard James Keenan from the band TOOL was inspired by the Fibonacci series and used it to create the rhythm and lyrics of their song 'Lateralus'.Okay, so I'm adapting this from here but modifying it to be a bit more rigorous / to my liking. Throughout history, the Fibonacci sequence has been applied to art in many forms. Early childhood is from 4-7, and middle childhood is 7-11.

The first two years of life are referred to as early infancy, and next is the toddler stage from two to four. Even as a human develop, we grow through 8 stages. The math and science behind our DNA is in the pattern of the Fibonacci sequence. That is the mysterious Fibonacci at work! From the number of petals that are on a flower, the way a pineapple or pine cone spirals, or the way a branch splits out into 3, it is all the same pattern! In relation to our own bodies as examples, think about the pattern of our DNA strands and how they spiral. Think of how the middle, or inside, of the flower, repeats this crazy cool pattern of the seeds.