**The Fibonacci Sequence**

By Blake Andert

**0+1**

**=1**

**1+1**

**=2**

**1+2**

**=3**

**2+3=5**

**3+5=8**

**5+8=13**

**...**

Green, Hank. "The Fibonacci Sequence: Nature's Code." Video blog post. SciShow. N.p., 17 Aug. 2012. Web. 24 Jan. 2014.

Bibliography

Hart, Vi. "Doodling in Math Class: Spirals, Fibonacci, and Being a Plant [2 of 3]." Video blog post. Vihart. N.p., 9 Jan. 2012. Web. 24 Jan. 2014.

Viatour, Luc. Da Vinci Vitruve Luc Viatour. Digital image. Wikipedia. N.p., 8 Sept. 2007. Web. 24 Jan. 2014.

Ida, Takashi. "“Vitruvian Man” by Leonardo Da Vinci and the Golden Ratio." “Vitruvian Man” by Leonardo Da Vinci and the Golden Ratio. Advanced Ceramics Research Center, 18 June 2012. Web. 24 Jan. 2014.

Parthenon Facade. Digital image. Padfield. N.p., n.d. Web. 24 Jan. 2014.

**Uses of Phi**

In Da Vinci's famous Vitruvian Man, the ratio of the square's side to the radius of the circle is golden.

Examples of the golden ratio in the human body include:

the ratio of the height of your face to the width of your face

the ratio of the length of your head to shoulders to the height of your head

the ratio of the length of your eye to the width of your eye

What happens when you divide adjacent Fibonacci Numbers?

1/1=1

2/1=2

3/2=1.5

5/3=1.666

8/5=1.6

13/8=1.625

...

377/233=1.618

610/377=1.618

1+(5 )

1/2

_______

2

Dcotezee. Phi Uc Lc. Digital image. Wikipedia. N.p., 10 Aug. 2007. Web. 24 Jan. 2014.

"The Golden Ratio in the Human Body." Modern Energy Research Library. N.p., n.d. Web. 25 Jan. 2014.

Architecture

"Architecture." Phi 1618 The Golden Number RSS. N.p., 5 Aug. 2013. Web. 23 Jan. 2014.

The pentagram naturally shows the Golden Ratio, as the ratio of its midsegments to the outer edges always equals phi. Pythagoras loved it so much that it became the symbol of the Pythagoreans.

Torra, Marc. Sacred Time. Digital image. Andean Spirituality. N.p., 2012. Web. 24 Jan. 2014.

Mona Lisa. Digital image. Wikipaintings. N.p., n.d. Web. 25 Jan. 2014.

Terrich, Matt. Mond Crucifixion. Digital image. Renaissance Art Project. N.p., 7 Oct. 2013. Web. 25 Jan. 2014.

Britton, Jill. Mond Crucifixion Pentacle. Digital image. Golden Section in Art and Architecture. N.p., 6 May 2012. Web. 25 Jan. 2014.

Approximately 1.6180339887

Kidder, David S., and Noah D. Oppenheim. "The Golden Ratio." The Intellectual Devotional: Revive Your Mind, Complete Your Education, and Roam Confidently with the Cultured Class. New York: Rodale, 2006. 326. Print.

Kidder, David S., and Noah D. Oppenheim. "Fibonacci." The Intellectual Devotional Biographies: Revive Your Mind, Complete Your Education, and Acquaint Yourself with the World's Greatest Personalities. New York: Rodale, 2010. 108. Print.

**Fibonacci**

Gardner, Martin. Fibonacci. Digital image. Wikipedia. N.p., 10 June 2010. Web. 29 Jan. 2014.

Leonardo Bonacci (aka Fibonacci) was born in Pisa in 1170. His father traveled a lot, and he grew up in what is now known as Béjaïa, Algeria. He was educated using Arabic numerals.

This series had already been known in India for 1300 years. Fibonacci was the person who introduced it into the Western world.

When he returned to Pisa, he was astounded that they were still using Roman numerals. He published Liber Abaci, which demonstrated the superiority of Arabic numbers to the traditional Roman numerals.

Fibonacci's achievements in math earned him great fame, and he was a friend of Emperor Fredrick II. He died around 1250.

**The Golden Ratio**

Two parts of an object are in golden proportion if the ratio of the longer part to the shorter part equals the ratio of the whole object to the longer part.

Stannered. Golden Ratio Line. Digital image. Wikipedia. N.p., 23 Mar. 2007. Web. 30 Jan. 2014.

a

b

_

=

a+b

a

___

=

1.618

Ahecht. Similar Golden Rectangles. Digital image. N.p., 26 June 2011. Web. 30 Jan. 2014.

Phi is unique because you can subtract one from it or divide one by it and get the same result.

The Parthenon

The Cathedral of Notre Dame

The Taj Mahal

The Great Pyramid of Giza

The CN Tower

Art

Meisner, Gary. "Golden Ratio in Art Composition and Design." Phi 1618 The Golden Number. N.p., 04 May 2014. Web. 06 Oct. 2014.

The Mona Lisa

Birth of Venus

Bathers at Asnières

**and the Golden Ratio**

Phi

If this is true, both ratios should be approximately 1.618, or phi.