History of Math: Fibonacci Sequence

Fibonacci was an Italian mathematician who is sometimes considered to be the most talented western mathematician of the Middle Ages.  Fibonacci (Leonardo Pisano Bigollo) was also known as Leonardo of Pisa, Leonardo Pisano, Leonardo Bonacci, and Leonardo Fibonacci.  While Fibonacci is best known for spreading the modern (Hindu-Arabic) numeral system in Europe through his 1202 Liber Abaci (Book of Calculation).  Fibonacci, also in his Liber Abaci, introduced a problem about the growth of a population of rabbits.  The solution was a sequence of numbers (later named the Fibonacci numbers or Fibonacci sequence).

The Fibonacci sequence was known in the 6^th century, yet Fibonacci gets the sequence named after him because he is the one that introduced it to the west.  The sequence involves adding the last two terms in the sequence in order to achieve the new term, written as F(n+1)=F(n)+F(n-1).  However, although Fibonacci brought it to the “new world” he did change it a bit in his interpretation.  Some people of the time (and most today) started the sequence with 0, 1, 1, 2, but Fibonacci changed the sequence to start with 1, 1, 2, which in the “long run” does not change anything but the numbering of the terms.  Fibonacci did his calculation to the thirteenth place which is 233.

Using the Fibonacci sequence, the Golden Spiral can be made.  Using the Fibonacci numbers as guidelines, you can make squares that continually grow.  Then, connecting corner to corner with an arc, a spiral can be made.  The reason that this spiral has a special name, the Golden Spiral, is because in order to make this spiral, the dimensions are based off of the golden rectangle.  The golden spiral follows the golden ratio which is equal to

The golden ratio is also found below in this conch shell. 

However, how would you answer the problem, how do the Fibonacci sequence and Pascal's Triangle relate?  Give an example such as 89 to help with understanding?  After some research, you may find that the Fibonacci sequence is made up of the additions of Pascal's Triangle's angles.  Each line, always drawn parallel to the last, will make up the next number in the Fibonacci sequence.  So with Pascal's Triangle and the Fibonacci sequence in hand, errors can be checked for on both.  A good example is the number 89.  89 is the 11th number in the Fibonacci sequence.  This number is attained by adding 55 and 34 from the Fibonacci sequence.  Using Pascal's Triangle, 89 is attained by adding 1, 9, 28, 35, 15, and 1 (found below)

From this we can see that not only is the Fibonacci sequence important, it also have many practical applications.  The sequence is easily attainable for anyone that can do simple arithmetic.  It seems like such a simple sequence, but it has such large applications.