This document summarizes the Fibonacci sequence and demonstrates how to use it to draw a golden spiral. The Fibonacci sequence begins with 1, 1, and each subsequent number is the sum of the previous two. When dividing any number in the sequence by its predecessor, the ratio approaches 1.618. The document shows how to start with two squares of side length 1 cm, and continue adding new squares based on the Fibonacci sequence to form a rectangle. By drawing arcs connecting the corners, a golden spiral is revealed.

1. This presentation has been made for you by
FRANCISCO GUIJARRO BELDA
From Isaac Albéniz Secondary School
Leganés. Madrid. Spain.

2. THE FIBONACCI SERIES
Fibonacci was a mathematician who wondered himself about the fashion rabbits
bred. He studied an ideal situation and was able to determine a compelling series
of numbers out of this uncommon survey:
1, 1, 2, 3, 5, 8, 13, 21…
Can you figure out how the series works?
Any number from the series is the outcome of adding up the two previous
numbers. For instance, If you added up the two first, you would get 1+1=2. Again, if
you added up the second and the third ones you would get 1+2=3 and so on.
Besides, If you divided any component of the series by its precedent number, you
would get always the same ratio: 1,618 (with the exception of the first ones).

3. LET S APPLY THIS IDEA TO A SERIES OF
SQUARES
1) We are going to draw two squares that are equal. Their side has 1 cm length . We
put them together.
1
1

4. 2) Upon the largest side of the rectangle you got from the two original squares, we
draw a new one whose side will be, obviously 2 cm side.
1
2
1

5. 3) Again, we draw a new square on the bottom side of the rectangle brought about
by the square 2 and one of the original squares as you can see in the picture.
1
2
1
3

6. 4) You can keep working the same way as many times as you want.
1
2
1
5
3

7. 5) Next step will be to draw an arc with the compass. You will put the point of it on
the middle of the two 1 cm side squares and take a radius of the same length.
2
5
3

8. 6) Once you have drawn the semicircle,
You must join points
P and Q with the you will keep drawing another arc on
compass square number 2. Take center on point
B and radius the side of the square (2
Q cm)
2
P
B 5
3

9. 7) Keep going the same way with the other squares and you will have accomplished
the drawing of a beautiful golden spiral. I marked points C and D on the picture to
make you know where are the centers of the new arcs. I guess you can determine
what is the length of the correspondent radius.
D
C
5

10. 7) And here we go. It is needless to say that we could have developed the spiral
endlessly. Unfortunately we are hampered by our drawing sheet limits.