Please explain phi and the golden ratio??

Question

I'm only in eighth grade so please explain it in an easy method to follow. Please attempt to answer the following questions:

1. When and Where is Phi used??
2. What is the difference between 1.618 and 0.618?? When is each used??
3. Please give some examples on how to use phi.

Any additional information is appreciated. Thank you.

Answer 1

The irrational number phi = (1 + sqrt 5) / 2 = 1.6180 (approx.) and is known as the 'golden ratio' or 'golden proportion'. It was known to the ancient Greeks, who considered the ratio 1 : phi to be the most aesthetically pleasing (nice to look at) ratio for the sides of a rectangle. I think they also built some buildings with rectangles in this proportion.

There is a famous construction which consists of a rectangle whose sides are in the ratio 1 : phi. If you divide this rectangle into a square and a smaller rectangle, that rectangle's sides are in the ratio (phi - 1) : 1 = 1 : phi. (i.e. It is the same shape as the original rectangle but smaller.) If *that* rectangle is divided into a square and rectangle, then once again, that rectangle's sides are in the ratio 1 : phi, and so on.

The number phi has several interesting mathematical properties, e.g. 1 / phi = phi - 1 = 0.6180 (approx.)

Also, you may have heard of the Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34 ... where each number in the sequence is the sum of the previous two numbers. It so happens that if F(n) is the nth Fibonacci number, the fraction F(n)/F(n-1) approaches phi. You are only in 8th grade, so you probably haven't studied sequences or convergence yet, but here is an example of what I mean:

1/1 = 1
2/1 = 2
3/2 = 1.5
5/3 = 1.67
8/5 = 1.6
13/8 = 1.625
21/13 = 1.615

You can see that the ratio is getting closer and closer to 1.6180.

Hope this helps.

Answer 2

Wikipedia is your friend:
http://en.wikipedia.org/wiki/Golden_ratio

The Fibonacci series: 1 1 2 3 5 8 13 21 ... is closely related to phi:
http://en.wikipedia.org/wiki/Fibonacci_number

Both phi and the Fibonacci series appear frequently in biology:
http://www.unitone.org/naturesword/sacred_geometry/phi/in_nature/