In this presentation, we see that how golden ratio and Fibonacci series are calculated or working? and What is the problem faced by Fibonacci in Fibonacci series.

1. Golden Ratio
and
Fibonacci Series
Made By:-- Naina Sharma
Class:-- X ‘A’
Roll Number:-- 06
School Name:-- Mother Divine Public School

2. Contents
› What is Golden Ratio?
› How is Golden Ratio Calculated?
› Golden Ratio in Design
› Where do we use Golden Ratio?
› What is Fibonacci Series?
› How does Fibonacci Series Work?
› What is the problem of Fibonacci Series?
› How is the golden ratio related to Fibonacci numbers?

3. What is Golden Ratio ?

4. What is Golden Ratio ?
› In mathematics, two quantities are in the golden ratio , if
their ratio is the same as the ratio of their sum to the
larger of the two quantities.
› We find the golden ratio when we divide a line into two
parts so that:
The longer part divided by the smaller part is equal to the
whole length divided by the longer part.
› The golden ratio is mathematical constant approximately
1.6180339887
› The golden ratio is denoted by phi.

5. Continued…
› It is also known as:
› Mean Ratio
› Divine Proportion
› Divine Section
› Golden Cut
› Golden Proportion

6. Continued…
› The Golden Ratio is what we call irrational number :it has
an infinite number of decimal places and never repeats
itself.
› The golden ratio is the relationship between two number
on the Fibonacci Series where plotting the relationships
on scale results in a spiral shape.

7. Continued...

8. Golden Ratio is Everywhere…
› The Golden Ratio is an ancient belief. It is known to be
around since Egyptian regime where it was used to build
the primeval pyramids. In recent times , the golden ratio
can be seen everywhere in nature , architecture , arts ,
design etc.

9. Continued…

10. How is Golden Ratio calculated?
› Two quantities A and B are said to be in golden ratio phi
› If a+b /a =a/b= phi
› One method to find the value of phi is to start with left
fraction a+b /a = 1+b /a = 1+1/phi
› Therefore, 1+1/phi = phi
› Multiplying by phi gives
› Phi+1 = phi^2

11. Continued...
› Which can be arranged to,
› Phi^2-phi-1=0
› Using the quadratic formula, two solutions are obtained:
› Phi=1+2.23/2=1.680
› And
› Phi=1-2.23/2=-1.680
› Because phi is the positive quantities phi is necessarily
positive:
› Phi=1+2.23/2=1.680

12. Golden Ratio in Design

13. Where do we use Golden Ratio?
A
• Architecture
• Painting
B
• Book Design
• Nature
C
• Aesthetics
• Perceptual Studies
› Golden Ratio is used in the
following:

14. Fibonacci Series

15. What is Fibonacci Series ?
› It is series of numbers that follow a unique integer sequence.
› These numbers generate mathematical patterns that can be
found in all aspects of life.
› The patterns can be seen in everything from the human body
to the physiology of plants and animals.
› The Fibonacci numbers are 0,1,1,2,3,5,8,13,etc.
› It is closely related to Lucas numbers and Golden Ratio.
› 1st in Indian mathematics.
› Invented by Leonardo Fibonacci.
› Discovered in an investigation of Reproduction of Rabbits.

16. How does the Fibonacci Series work ?
› The Fibonacci series is derived from the Fibonacci numbers.
The Fibonacci numbers are:
0,1,1,2,3,5,13,21,.34,55,89,144…..and so on.
› These numbers are obtained by adding the two previous
numbers in the sequence to obtain the next higher number.
› Example: 1+1=2,1+2=3,3+2=5,5+8=13,etc.
› Formula is: Fn=Fn-1+Fn-2.
› Every third number is even and the difference between each
number is 618 with the reciprocal of 1.618. These numbers are
known as the “golden ratio” or “golden mean”.

17. Fibonacci Series in Nature

18. Shallow Diagonals of PASCAL’S Triangle

19. What is the problem of Fibonacci Series?
FIBONACCI RABBITS

20. PROBLEM:----
Suppose a newly-born pair of rabbits (one male,
one female) are put in a field. Rabbits are able to
mate at the age of one month so that at the end of
its second month, a female can produce another
pair of rabbits.
Suppose that the rabbits never die and that the
female always produces one new pair (one male,
one female) every month from the second month
on. How many pairs will there be in one year ?

21. How is the golden ratio related to Fibonacci numbers?

22. Continued...
› It’s perhaps more accurate to phrase the question as “how are
Fibonacci numbers, also known as the Fibonacci sequence,
related to the golden ratio?”
› Either way, the answer is this:
› The ratio of each successive pair of numbers in the Fibonacci
Sequence converge on the golden ratio as you go higher in
the sequence.
› The Fibonacci sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,
etc., with each number being the sum of the previous two.

23. Continued...
› The ratios of successive pair of numbers is thus as follows:
› 1/0=8
› 1/1 = 1
› 2/1 = 2
› 3/2 = 1.5
› 5/3 = 1.66666666666666…
› 8/5 = 1.6
› 13/8 = 1.625

24. Continued…
› 21/13 = 1.61538461538462…
› 34/21 = 1.61904761904762…
› 55/34 = 1.61764705882353…
› 89/55 = 1.61818181818182…
› 144/89 = 1.61797752808989…
› 233/144 = 1.61805555555556…
› 377/233 = 1.61802575107296…
› 610/377 = 1.61803713527851…
› 987/610 = 1.61803278688525…

25. Continued...
› The golden ratio is an irrational number with an infinite
number of random digits that can be calculated as (the
square root of 5 + 1)/2. To fifteen places this is
1.61803398874989. So in the sequence above, you can
see that the ratio from each pair of numbers gets closer
and closer to the golden ratio.
› Go further into the Fibonacci sequence to the pair of
successive numbers of 14,930,352 / 9,227,465 and the
result is 1.61803398874989, accurate to 15 places.