Interesting Patterns Continued

C) Multiple each Fibonacci number with the next one in the sequence. Multiple each Fibonacci number with itself. Add the squared Fibonacci number. 

Pattern: When given a certain term, the multiple result of it is equal to the sum of the terms associated Fn2 added to all of the previous squared results (e.g. 2x3=6 and 0+1+1+4=6) 

D) First thirty nunbers of the Fibonacci numbers marked as multiples of 2, 3, 5 and 8. 

       

Pattern: The difference between one multiple and the next is equal to the term number (e.g. Multiple 2 = Term 3; all multiples of 2 will be three numbers from the previous multiple). 

Honeybees and their Family Tree

Female bees have one mother and one father, while male bees only have a mother bee. Bees' family tree is a Fibonacci sequence when accounting the amount of parents on both sides of the family.

Family Tree Creator: http://www.familyecho.com/

The Golden Ratio and Fibonacci Fun!

Fibonacci's sequence is seen throughout nature, but it can be clearly seen in the Golden Ratio. The Golden Ratio or phi (the greek symbol picture below) is roughly equal to 1.618034. Fibonacci's Numbers play into the ratio when one takes a Fibonacci number and divide it by the previous Fibonacci number. The ratio is not always perfect, but is averaged to the Golden Ratio. The graph below shows the trend between the Fibonacci number and the ratio. For the first few terms, the ratio is either 1 or 2, but as the sequence goes on it appears to be coming to a limit of 1.618034 (as seen below).

The Golden Ratio is used in many aspects of life. It is most apparent though in architecture. The perfect rectangle is a 1x1.6, or the Golden Ratio. A rectangle with these dimensions is so eye appealing that it is referred to as the Golden Rectangle.

Works Cited:

1) "The Life and Numbers of Fibonacci." Plus.maths.org. Plus Math Magazine, n.d. Web. 09 June 2014. <http://plus.maths.org/content/life-and-numbers-fibonacci>.

2) "Painting by Numbers." IEEE Spectrum 47.3 (2010): 20-21. Scientiareview. Scientiareview. Web. 10 June 2014. <http://www.scientiareview.org/pdfs/208.pdf>.

Interesting Patterns

A) For each Fibonacci Number- Relation to adding ALL the previous numbers:

Fn= Fibonacci Number, All previous numbers added together

F1= 1, 1

F2= 1, 1

F3= 2, 2

F4= 3, 4

F5= 5, 7

F6= 8, 12

F7= 13, 20

F8= 21, 33

F9= 34, 54

F10= 55, 88

Pattern: When you add all the previous numbers together it is -1 of the next Fibonacci number!

B) Adding alternative Fibonacci Numbers

F0 + F2 = 0 + 1 = 1
F1 + F3 = 1 + 2 = 3
F2 + F4 = 1 + 3 = 4
F3 + F5 = 2 + 5 = 7
F4 +F6 = 3 + 8 = 11
F5+F7= 5 + 13 = 18

Pattern: The sum of the alternating fibonacci numbers plus the following sum is equal to the third sum (e.g. F0+F2=x; F1+F3=y; F2+F4= x+y)!

First 26 Fibonaccis

First 26 Fibonacci Numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46358, 75025

Discovery and Definition of the Fibonacci Sequence

"A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair, which from the second month on becomes productive?"

- Fibonacci's Laber Abaci

Fibonacci first started forming his idea of the Fibonacci sequence when watching rabbits reproduce. He realized that at the end of the first month their would be 1 pair of rabbits, at the end of the second month their would be 2 pairs and then each month the amount of rabbits would keep growing. 

The rule for Fibonacci's Sequence:

 Fn= (Fn-1)+(Fn-2)

Definition of a Fibonacci Number:

A number that is the sum of the two numbers before it.

Works Cited:

1) " Fibonacci Sequence." Definition of Fibonacci Sequence. Math Is Fun, n.d. Web. 10 June 2014. <http://www.mathsisfun.com/definitions/fibonacci-sequence.html>.

2) "The Life and Numbers of Fibonacci." Plus.maths.org. Plus Magazine, n.d. Web. 09 June 2014. <http://plus.maths.org/content/life-and-numbers-fibonacci>.

Who Was Fibonacci?

Who Was Fibonacci? 

Leonardo Pisano Bigollo was born around 1175 AD in Pisa, Italy. Young Fibonacci lived traveled with his father, Guilielmo, to the city of Bugia on the Mediterranean, this is where Fibonacci would spend most of his childhood and his teenage years. His father worked as a customs officer on the docks of the city. Living in Bugia allowed Fibonacci to interact with many merchants and travelers. His conversations and experiences on the docks of Bugia were the basis for his interest in mathematics and the start of a real education. Fibonacci was exposed to Indian numerals, Greek literature and Latin, while living with his father. 

When Fibonnaci was about 25 years old, he moved back to Pisa and began his mathematical writings. Fibonacci's work was recognized by Emperor Frederick II and many other scholars. His writings consist of The Book of Calculations, Practica Geometriaem, and Flos. The most regarded work that Fibonacci has accomplished is the Fibonacci Numbers. 

Fibonacci died around 1250 AD in Pisa. He dedicated 50 years of his life to math before dying. 

Works Cited:
O'Neil, Christopher. "Fibonacci." Fibonacci. Rutgers University, n.d. Web. 09 June 2014.