The Golden Ratio

When you think of the golden ratio, what do you think of? I usually think of sunflowers, but why? That is because we see the golden ratio in nature very often, and it is present in the middle of a flower..
Screen Shot 2015-12-16 at 4.45.36 PM
That really pretty spiral that we see in the middle of a huge flower is known as the golden ratio, or a number that is approximately equal to 1.618. It is only approximately this number but the digits do keep on going without a repeating pattern, isn’t that weird?! This number does not just appear in nature, it also appears in geometry, art, and architecture. It is said that the golden ratio makes the most pleasing and beautiful shape to the human eye. Many buildings and artworks actually contain the golden ratio within them, like the Parthenon in GreeceScreen Shot 2015-12-16 at 4.48.09 PM,
but it is not known if it was intentionally designed to be this way or not.

How do we find this golden ratio? We divide a line into two parts so that the longer part divided by the smaller part is also equal to the whole length divided by the longer part. Kind of confusing so here’s a picture from mathisfun.com
Screen Shot 2015-12-16 at 4.50.48 PMThe funny looking symbol that you see at the end is the symbol that we use to represent the golden ratio, it’s easier than writing out all those decimals. But what’s interesting is that this golden ratio also has a connection with the Fibonacci sequence, there is a special relationship between the two. We know the sequence to be
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …. and so on and we find the next number in the sequence by adding the two previous terms. Now the golden ratio relates to the Fibonacci sequence when we take any two successive Fibonacci numbers, the ratio of those two numbers is very close to the value of the golden ratio. It seems as though the bigger the pair of numbers, the closer the approximation is to the ratio, that’s pretty cool!

Another interesting aspect of this ratio is that it is said to be the most irrational number. A reason why is because it can be defined in different terms, like this: Screen Shot 2015-12-16 at 4.55.25 PM and this can even be expanded into a more complicated fraction!
Screen Shot 2015-12-16 at 4.56.09 PMThis fraction appears to continue on forever and it is very complicated to understand but for these reasons alone, the golden ratio is thought of as the most irrational number.

Surprisingly, there is not much history out there on the golden ratio but it is said to have first been spotted in Euclid’s book of Elements where Euclid is giving applications such as the construction of a regular pentagon, an icosahedron, and a dodecahedron. I just think it’s really cool that it has made its presence in nature and in architecture! Check it out!

main-qimg-870d54588a81535792d8374dbaafb66b 1429695527113

 

http://www-history.mcs.st-andrews.ac.uk/HistTopics/Golden_ratio.html
https://www.mathsisfun.com/numbers/golden-ratio.html
https://www.mathsisfun.com/numbers/nature-golden-ratio-fibonacci.html

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