Fibonacci Sequence

Who is Leonardo of Pisa?

Famous for spreading the use of Arabic number system and knowledge of the Fibonacci sequence.

What is the Fibonacci Sequence?

Explore more at: http://www.magicmathworks.org/exhibits/numberpatterns/index.html

The Fibonacci Sequence

1 1

}+

2

The Fibonacci Sequence a recursive arithmetic sequence.

The Fibonacci SequenceThis 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

The Fibonacci Sequence

1 1 2 3

}+

5 8 13 21 34 55 89

This sequence can continue on forever…

The “Magic” of Fibonacci Numbers

Fibonacci Numbers in Nature

Understanding the Spiral

1. Draw a 1x1 square

2. Add another 1x1 square

3. Add 2x2 square

4. Add 3x3 square

5. Add 5x5 square

6. Add 8x8 square

7. And so on… For more on this visit: http://www.goldennumber.net/spirals/

Arcs drawn through each square connect to make a spiral

Seeing Spirals…

Taking a closer look at Nature

Visit this site to try it out for yourself: http://science.howstuffworks.com/life/evolution/fibonacci-nature1.htm

Count those spirals…

Fibonacci Numbers!!!

Let’s do it again…

Doodling in Math Class teaches us….

Make your own Fibonacci Spiral

http://chromatism.net/cfsg.htm

The Fibonacci Sequence leads us to the Golden Ratio

Exploring the Golden Ratio

This animation is located at: http://www.mathsisfun.com/numbers/nature-golden-ratio-fibonacci.html

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...

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.

The Golden Rectangle

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

Golden Rectangle and the Mona Lisa?

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

The Divine Proportion of the Human Body

Math is Beautiful

The Golden Ratio can be applied to the human face to determine beauty.

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-of-beauty.php

ResourcesBenjamin, 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/