The beauty of Fibonacci Numbers

Before Test #2 we learned the function to produce the Fibonacci Numbers in class.

Here is the sequence of Fibonacci Numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, …

And I believe everyone is familiar with how to get each number in the sequence. You just simply add the first two numbers to get the next:

0+1=1

1+1=2

1+2=3

2+3=5

3+5=8

5+8=11

...

And here is the function of Fibonacci we used in Dr. Racket to get numbers in the sequence:

; fib : number -> number

(define (fib n)
  (cond [(>= n 2) (+ (fib (- n 2)) (fib (- n 1)))]

            [else 1])) 

well, it is a unary function that produce a number from an inputed number.
n is the number of order in the sequence you want to get, and if n is greater or equal to 2 it also includes a repeated process to calculate fibonacci numbers in prior to n and then add the last two numbers.

I think the process of calculating Fibonacci Sequence is interesting. And this sequence is extremely famous for its beauty existing in nature, in science as well as in human bodies. I want to elaborate more on this findings:

Firstly when you divide one number in the sequence by the number before it, you obtain numbers very close to one another. In fact, this number is fixed after the 13^th number in the series. This number is known as the “golden ratio.”

GOLDEN RATIO = 1.618

233 / 144 = 1.618

377 / 233 = 1.618

610 / 377 = 1.618

987 / 610 = 1.618

1597 / 987 = 1.618

2584 / 1597 = 1.618

Perfect human bodies, faces and other beautiful things in the world always show a golden ratio of 1.618. The ratio is used by many artist in their artworks, and by designers in their products. For example the famous Mona Lisa.

Another beautiful application of Fibonacci Numbers is the "golden rectangular"

If you draw circles using radius of 1, 1, 2, 3, 5, 8, 13... and combine 1/4 of each circle together, you will get the following image: 

This beautiful shape of "golden rectangular" is found everywhere in the nature, such as snail, shell, flowers, etc. Here are some pictures showing the beauty of nature:

human work!

Well, it is a bit off topics from CSC104 but I find it's very interesting. There are more evidence of Fibonacci Sequence around us:) If you get some inspirations from this blog, then start to look around and enjoy the beauty of nature!