patterns in nature fibonacci sequence in nature

That’s a wonderful visual reason for the pattern we saw in the numbers earlier! If we generalize what we just did, we can use the notation that $F_n$ is the $n$th Fibonacci number, and we get: … and the area becomes a product of Fibonacci numbers. But the resulting shape is also a rectangle, so we can find its area by multiplying its width times its length the width is $8$, and the length is $5 + 8 = 13$… Well, we built it by adding a bunch of squares, and we didn’t overlap any of them or leave any gaps between them, so the total area is the sum of all of the little areas: that’s $1^2 + 1^2 + 2^2 + 3^2 + 5^2 + 8^2$. But we’ll stop here and ask ourselves what the area of this shape is. Now the length of the bottom edge is 2+3=5:Īnd we can do this because we’re working with Fibonacci numbers the squares fit together very conveniently. We could keep adding squares, spiraling outward for as long as we want. Now the length of the rightmost edge is 1+2=3, so we can add a square of side length 3 onto the end of it. Now the upper edge of the figure has length 1+1=2, so we can build a square of side length 2 on top of it: We have squared numbers, so let’s draw some squares. But look what happens when we factor them:Īnd we get more Fibonacci numbers – consecutive Fibonacci numbers, in fact. The resulting numbers don’t look all that special at first glance. What happens when we add longer strings? Three or four or twenty-five? So that’s adding two of the squares at a time. We get Fibonacci numbers! In fact, we get every other number in the sequence!

The Fibonacci sequence is all about adding consecutive terms, so let’s add consecutive squares and see what we get:

Okay, now let’s square the Fibonacci numbers and see what happens. Now does it look like a coincidence? In fact, it can be proven that this pattern goes on forever: the nth Fibonacci number divides evenly into every nth number after it! Cool, eh? How about the ones divisible by 3?Įvery fourth number, and 3 is the fourth Fibonacci number. Let me ask you this: Which of these numbers are divisible by 2?Įvery third number, right? And 2 is the third Fibonacci number. But let’s explore this sequence a little further.įirst, let’s talk about divisors. We already know that you get the next term in the sequence by adding the two terms before it.

Patterns in nature fibonacci sequence in nature