What is 21?

     Triple-stumper from Tuesday, 2-1: In Science: “It’s the largest number in the Fibonacci sequence that’s also a day in a month.”

     I think I knew this at one point, but I couldn’t remember what the sequence was, and I wanted to!  It turns out to be quite simple: You simply start with 1, and add the two numbers just prior in the sequence to get the next number.  Okay, so 1.  1+0=1.  1+1=2.  2+1=3.  3+2=5.  5+3=8.  8+5=13.  And 13+8=21.  The next number in the sequence, 13+21=33, cannot be a day in a month.  So 21 is the correct question.

     According to An Incomplete Education by Judy Jones and William Wilson, Fibonacci came up with the sequence when he was trying to determine “how many pairs of rabbits can be produced from a single pair of rabbits, if every month every pair begets a new pair, which from the second month on is itself productive, assuming that none of the rabbits die…and each pair consists of a male and a female.”

     I remembered that there was some other significance to the sequence, and Jones and Wilson go on to explain that the sequence is found repeatedly in nature as well as art and architecture.  I did already remember my favorite example, the pineapple, as well as the “Golden Rectangle.”  Here we go:
     The pineapple:  This animation, from a website created by Jill Britton, shows it best:


       There are three different lines of “bumps” going different directions that pass through the same bump (in the center), and each line contains a number from the Fibonacci sequence (5, 8, and 13).  That ratio is consistent among pineapples.
     Now for the golden rectangle: First, the golden section is a line divided into two parts where the larger part is to the smaller part what the whole part is to the larger part.  That ratio is 1 : 1.618, the ratio between any two adjacent numbers in the Fibonacci sequence after three.  The golden rectangle has the same relationship with its length and width.  (That is, if you were to take the two parts from a golden section, each part would make the length and width.)  The facade of the Parthenon is a golden rectangle.  So is the human body, in multiple places.  An example: The distance from a person’s bellybutton to their feet is 1 unit, and their whole height is then the 1.618 in the ratio.
     (I hope this has not been too confusing!)