Chris's Corner

Golden Ratio(Au? ratio?)

1, 1, 2, 3, 5, 8, 13…….

What is the next term of this sequence?

I will not answer this question, because the answer is OBVIOUS! 

What is the ratio between the two terms?

It is different all the time, right?

Question 1: As the sequence gets very very large, what is the ratio between two terms?

u=1+√(1+√(1+√(1+…)

Question 2: What is the answer of u?

Here is the real question: Is there a similarity between two questions?

The answer is there is the golden ratio.

The golden ratio is an irrational number which is the answer of the division that divides into two lines to make one square equal to the total area of ​​the other. This irrational number originated from Euclid, who invented the Euclidean Geometry, which is taught in high school. The sequence at the top is the Fibonacci sequence. This sequence’s ratio approaches the golden ratio. Even Though the golden ratio is less important than pi and e, it is used a lot in Geometry. For example, in the pentagon, the ratio between a side of the pentagon and the diagonal line is the golden ratio. Also, in a dodecahedron, the ratio of the radius of a circle passing through outer vertices and a circle passing through inner vertices is a golden ratio in the figure that appears when orthogonally projecting all vertices in a plane parallel to one face. 

Mathematically, the golden ratio is (1+√5)/2, which equals 1.6180339………

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