The Golden Ratio and Fibonacci Sequence in Nature!

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The Golden Ratio and the Fibonacci Sequence in Nature!

Come see how the Golden Ratio in nature!


The Golden Ratio and the Fibonacci Sequence is confusing stuff, but we will try to simplify it.


The Golden Ratio is a proportion, in which either ratio will equal 1.618.... the format for the ratio is . (The symbol is from the Greek symbol for phi.)

Basically, a length "a" divided by a length "b" equals the whole length "a+b" divided by the longer length "a"


This can also be expressed in many other ways such as
 and  .


In the second way to express the Golden Ratio, it is described in terms of itself. In the third one, it is described as a continued fraction, or a fraction that never ends.




Now here's the basics of a Fibonacci Sequence:


A Fibonacci Sequence is a pattern of numbers in which each number is the sum of the two numbers. Here's an example:
0,1,1,2,3,5...


The 5 is the sum of 2 and 3, 3 is the sum of 2 and 1, and 2 is the sum of 1 and 1.


This sequence can be written as a rule: . "Fn" represents the "Fth" term, "Fn-1" represents the previous term, and "Fn-2" represents the term before that.


The Fibonacci sequence was found by Leonardo Fibonacci (may be also known as Leonardo of Pisa) in in the 13th century, specifically 1202. Born in Pisa, Italy, he published 4 books, in which one of them introduced the Hindu-Arabic number system, containing number one thought nine, including zero. This number system replaced the Roman numeral system. More importantly, he found out about the Fibonacci sequence when he made conjectures based on his inductive reasoning of how fast rabbits could reproduce in an environment. He saw a pattern of how rabbits produced, leading to the discovery of the Fibonacci sequence.


The Fibonacci sequence and the Golden Ratio are very similar to each other. If you pick any two Fibonacci numbers, the ratio of those two numbers are very close to the Golden Ratio. It also works when you pick two random whole numbers to begin the sequence. Here is the Golden Spiral/ Fibonacci Spiral, which shows how the Fibonacci spiral can be related to the Golden Ratio.









The Golden Ratio and the Fibonacci Sequence could be used to find the prefect spiral for things in nature, like the seeds of a sunflower. Here is a great website to help understand that example, explaining how the Golden Ratio can affect a sunflower. There are many more examples on that website, but we will focus on this particular one.


http://www.mathsisfun.com/numbers/nature-golden-ratio-fibonacci.html


You can play around with how numbers affect the formation of a spiral in a sunflower. According to the interactive example on the website, the Golden Ratio can be represented perfectly as a spiral in a sunflower seed formation with the perfect "turn", but if you represent the  spiral as a rational number, then the spiral will revolve with a gap, or it wouldn't be a spiral at all. The Golden Ratio is the only number that can be represented with a spiral with no gaps, due to the fact that it is the most irrational number of all. Even "pi" and "e" can be approximated as fractions, but the ratio is very irrational due to how the Golden Ratio is expressed.


The Fibonacci sequence is also related to this example. Since Fibonacci numbers's ration are near the Golden Ratio, then you can get a good spiral to the flower.


I hope that was very informative and simplified! I hope that you guys read more of our posts!