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The Fibonacci numbers and the golden ratio

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Introduction

Sofie Bronée 1v.         Fibonacci project.         11/04/08

The Fibonacci Numbers and the Golden Ratio

_____________________________________

image00.jpg

The Fibonacci Numbers        

The Fibonacci numbers are sequence of numbers. They are named after the Italian mathematician Leonardo of Pisa, known as Fibonacci. He published a book called “Liber Abaci”, and he was the first person to publish a book in Western Europe that used the Indian numerals 9, 8, 7, 6, 5, 4, 3, 2, 1. Fibonacci was perhaps the greatest mathematician of his time. But he is most famous for the numbers which has his name.

These are the first ten terms in the Fibonacci sequence.

U1

U2

U3

U4

U5

U6

U7

U8

U9

U10

1

1

2

3

5

8

13

21

34

55

To get the next term in the sequence the two previous terms must be added. This can be written as:  

Un = Un-1 + Un-2

A number in the sequence is the sum of its two predecessors.

The Golden Ratio

To find the Golden Ratio, a term has to be divided with the previous term

(i.e. . image01.png.. etc).

...read more.

Middle

0


image11.png

By investigating the Fibonacci numbers, I can make a conjecture, that the ratio of two consecutive terms gets closer, as they increase, to the Golden Ratio. By changing the start numbers I proved that it doesn’t make a difference which numbers we use.

To prove my conjecture I’ve changed the equation and solved it. Afterwards I found the discriminant (5), which leads me to find the roots. I plugged in the discriminant to the quadratic equation for finding the roots. The results for the positive root is, 1.618 and the negative root is, -0.618. This proves my conjecture. Look below to see my calculations and explanations.

I divided both sides of this equation Un = Un-2 + Un-1, by Un-1.

image13.png = 1 +  image14.png = Why we do it?

image15.png

image16.png= 1 + image17.png

Here, image16.png is one of the numbers in the column above, it gets closer and closer to 1.618, which I will call x. On the other side of the equation, image17.png is the reciprocal

of one of the numbers in the same column, so it gets closer to image02.png.

...read more.

Conclusion

The Greeks used the golden ratio in the design and construction of the Greek temple (Parthenon).

image06.jpg

image07.jpg

n

Un

Un+1/Un

1

1

1

2

1

2

3

2

1,5

4

3

1,666667

5

5

1,6

6

8

1,625

7

13

1,615385

To prove that the Greeks really used the golden ratio in the design, I’ve measured the sides. The first square is 1cm by 1cm; the next is 1cm by 2 cm and so on. Look in the table below to see the ratio between each side.

Appendix

1)

I changed to two “start numbers” to 1 and 7, and by making a table and a graph I proved that it doesn’t matter which numbers we use to get the Golden Ratio.

image08.png

To show that we would eventually get the same result, no matter what “start numbers” were, I plugged in -9 and 5. To prove that with even a negative number, we would reach the Golden Ratio.

image09.png

I also tried to plug in decimal numbers to see what happened. I plugged in 2,6 and -1,9.

image10.png

2)

The inverted Golden Ratio can be found by dividing Un-1 by Un-2. As you can see below, theinverted ratio is reached much faster, than the opposite.

image12.png

...read more.

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