The document discusses the Fibonacci sequence and its relationship to the golden ratio. It begins by introducing Leonardo of Pisa, who helped spread the use of the modern number system and knowledge of the Fibonacci sequence. The sequence is defined as a pattern where each number is the sum of the two preceding ones, starting with 1, 1, 2, 3, 5, etc. This sequence appears throughout nature and can be seen in spirals of shells, pinecones, and sunflowers. The ratio of consecutive Fibonacci numbers approaches the golden ratio, about 1.618, an irrational number important in art and architecture considered aesthetically pleasing. The golden ratio can also be observed in proportions of the human body.

1. Fibonacci Sequence

2. Who is Leonardo of Pisa?
Famous for
spreading the use of
Arabic number
system and
knowledge of the
Fibonacci sequence.

4. The Fibonacci Sequence
}+
The Fibonacci Sequence a recursive arithmetic sequence.
1 1 2

5. The Fibonacci Sequence
This pattern continues and follows the rule:
xn = xn-1 + xn-2
}+
xn is term number "n"
xn-1 is the previous term (n-1)
xn-2 is the term before that (n-2)
1 1 2
3

6. }+
The Fibonacci Sequence
1 1 2 3 5 8 13 21 34
55
89
This sequence can continue on forever…

7. The “Magic” of Fibonacci Numbers

8. Fibonacci Numbers in Nature

10. Arcs drawn through each
square connect to make a spiral

11. Seeing Spirals…

13. Count those spirals…
Fibonacci Numbers!!!

14. Let’s do it again…

15. Doodling in Math Class teaches us….

16. Make your own Fibonacci Spiral
http://chromatism.net/cfsg.htm

17. The Fibonacci Sequence
leads us to the Golden Ratio

19. What is the Golden Ratio?
The ratio between successive Fibonacci Numbers
will eventually give you the Golden Ratio:
A
B
B/A
1
1
1
1
2
2
2
3
1.5
3
5
1.666666666...
5
8
1.6
8
13
1.625
13
21
1.615384615...
...
...
...
144
233
1.618055556...
233
377
1.618025751...

20. The importance of Phi Φ
If the longer segment divided by the smaller
segment is also equal to the whole length
divided by the longer part then you will have
the golden ratio.

21. The Golden Rectangle
Experiment with the golden ratio at
http://www.mathsisfun.com/numbers/golden-ratio.html

22. Golden Rectangle and the
Mona Lisa?

23. The Golden Ratio is considered to be
pleasing to the eye and can be found in art,
architecture and music.

24. The Divine Proportion of the
Human Body

25. Math is Beautiful
The Golden Ratio can be applied to the
human face to determine beauty.

26. Math of Beauty Interactive
According to the Golden Ratio, Jessica
Simpson is considered beautiful!
Visit this site to participate in a fun activity:
http://www.intmath.com/numbers/math-ofbeauty.php

27. Resources
Benjamin, A. (2013). The Magic of Fibonacci Numbers. Retrieved from
http://www.youtube.com/watch?v=SjSHVDfXHQ4
Fibonacci Sequence. (2013). Math is Fun. Retrieved from
http://www.mathsisfun.com/numbers/fibonacci-sequence.html
Hart, V. (2011). Doodling in Math: Spirals, Fibonacci, and Being a Plant. Retrieved from
http://www.youtube.com/watch?v=ahXIMUkSXX0
Inspiration Green (n.d.). The Fibonacci Sequence and Nature. Retrieved from
http://www.inspirationgreen.com/fibonacci-sequence-in-nature.html
Lamb, R. (24 June 2008). "How are Fibonacci numbers expressed in nature?“ Retrieved
from http://science.howstuffworks.com/life/evolution/fibonacci-nature.htm
Number Patterns. (n.d.). Retrieved from
http://www.magicmathworks.org/exhibits/numberpatterns/index.html
Spirals and the Golden Ratio. (August 25, 2012). Retrieved from
http://www.goldennumber.net/spirals/