The document discusses the Fibonacci sequence and the golden ratio. It begins by introducing the Fibonacci sequence of numbers: 1, 1, 2, 3, 5, 8, etc., where each number is the sum of the previous two. It then explains that the ratios of successive numbers in the sequence converge on the golden ratio, approximately 1.618. The golden ratio is found throughout nature, such as in the proportions of the human body and in vegetation. The document also provides methods for constructing a golden rectangle using squares and diagonals, with a final ratio matching the golden ratio.

1. FIBONACCI SERIES
AND
THE GOLDEN RATIO
SYNDICATE # 04

2. WHAT IS THE GOLDEN RATIO?
LET US CLEAR WITH AN EXAMPLE,
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610
CONSIDER THE SERIES
TO GET THE NEXT NUMBER WE ADD THE PREVIOUS
TWO NUMBERS TOGETHER. SO NOW OUR SEQUENCE
BECOMES 1, 1, 2. THE NEXT NUMBER WILL BE 3.

3. Now, I know what you might be thinking:
"What does this have to do with the
Golden Ratio?
This sequence of numbers was first
“discovered” by a man named
Leonardo Fibonacci, and hence is
known as Fibonacci's sequence.

4. SERIES RATIO SERIES RATIO
2 2 55 1.617647059
3 1.5 89 1.618181818
5 1.666666666 144 1.617977528
8 1.6 233 1.618055555
13 1.625 377 1.618025751
21 1.615384615 610 1.618037135
34 1.619047619 987 1.618032787

5. CONCLUSION FROM TABLE
The Golden Ratio is what we call an
irrational number: it has an infinite number
of decimal places and it never repeats itself!
Generally, we round the
Golden Ratio to 1.618.

6. VALUE OF PHI
(AN IRRATIONAL NUMBER)
∅ = 1+ 5
2

7. NTH TERM OF FIBONACCI SERIES

8. IN TERMS OF PHI

9. GENERAL TERM OF FIBONACCI SERIES

10. SOME
SUPRISING EXAMPLES

11. The Proportions in the Body
The white line is the body's height.
The blue line, a golden section of the white line, defines
the distance from the head to the finger tips
The yellow line, a golden section of the blue line, defines
the distance from the head to the navel and the elbows.
The green line, a golden section of the yellow line,
defines the distance from the head to the pectorals and
inside top of the arms, the width of the shoulders, the
length of the forearm and the shin bone.
The magenta line, a golden section of the green line,
defines the distance from the head to the base of the
skull and the width of the abdomen. The sectioned
portions of the magenta line determine the position of
the nose and the hairline.
Although not shown, the golden section of the magenta
line (also the short section of the green line) defines the
width of the head and half the width of the chest and the
hips.

13. The ratio of your forearm to hand is Phi

14. Your hand shows Phi and the
Fibonacci Series
Your Index Finger

15. CONSTRUCTING
A
GOLDEN RECTANGLE

16. METHOD ONE
1.
WE'LL START BY MAKING A SQUARE, ANY SQUARE
LIKE THIS

17. 2.
NOW, LET'S DIVIDE THE SQUARE IN HALF
(BISECT IT). BE SURE TO USE YOUR
PROTRACTOR TO DIVIDE THE BASE AND TO
FORM ANOTHER 90 DEGREE ANGLE.

18. 3.
NOW, DRAW IN ONE OF THE DIAGONALS OF
ONE OF THE RECTANGLES

19. 4.
MEASURE THE LENGTH OF THE DIAGONAL
AND MAKE A NOTE OF IT.

20. 5.
NOW EXTEND THE BASE OF THE SQUARE
FROM THE MIDPOINT OF THE BASE BY A
DISTANCE EQUAL TO THE LENGTH OF THE
DIAGONAL

21. 6.
CONSTRUCT A NEW LINE PERPENDICULAR
TO THE BASE AT THE END OF OUR NEW LINE,
AND THEN CONNECT TO FORM A
RECTANGLE

22. 7.
MEASURE THE LENGTH AND THE WIDTH
OF YOUR RECTANGLE.
NOW, FIND THE RATIO OF THE LENGTH TO THE WIDTH.
ARE YOU SURPRISED BY THE RESULT? THE RECTANGLE
YOU HAVE MADE IS CALLED A GOLDEN RECTANGLE
BECAUSE IT IS "PERFECTLY" PROPORTIONAL.

23. EXAMPLES IN
NATURE

26. FIBONACCI SEQUENCE IN VEGETABLES