Phi, from Primordia to Parthenon

In 1225 AD, a man discovered a mathematical sequence found in a broad spectrum of different elements, from rabbits, to sunflowers, to conch shells, to architecture. The man: Leonardo Fibonacci. The sequence: the Fibonacci sequence.

Fibonacci, in response to a mathematics tournament in Pisa ordered by Fredrick II, developed his sequence by observing rabbit breeding. He found that one pair of rabbits, after one month, produced another pair of rabbits. By the second month, this pair of rabbits produced another pair for a total of two pairs. Thus, the sequence follows:

+ 0 MONTH, 1 PAIR

+1 MONTH, 1 PAIR

++2 MONTHS, 2 PAIRS

+++3 MONTHS, 3 PAIRS

+++++4 MONTHS, 5 PAIRS

Fibonacci took his observations and organized them into the numerical sequence 1, 1, 2, 3, 5, 8, 13, 21…where each new number is the sum of the previous two.  It is interesting to note that when the terms of the Fibonacci sequence are divided by the term after it, the resulting number approaches 1.6180399 …, or Ф.

3/2

1.5

5/3

1.66666666…

8/5

1.6

13/8

1.625

 

 

The limit of the Fibonacci sequence (or Phi Ф) as shown in the above graph is derived in the following way:

Two successive terms in a sequence,

b, c and b+c, can be arranged so that

 

Let Ф= limit of

Phi, commonly called the Golden Section, is found in many natural elements, including the primordia of flowers. The primordia of a flower grow so that the angle between the first and last is 137.5 degrees.  137.5 degrees, known as the Golden Angle, is derived from Phi:

360° - (Φ * 360°) = 137.5°mhtml:file://C:\Users\Abbey\Documents\MAT%20170\Dirk%20Bertels%20-%20PHI%20-%20the%20Golden%20Proportion.mht!http://www.dirkbertels.net/mathematics/Phi_files/image002.jpg

                                  (Example of a flower’s primordia taken from Dirk Bertels’s PHI (Φ) - the Golden Proportion)

Text Box: Primordial areaUtah Shakespeare 2007 and vacation2007 017.jpg 

 

Text Box: The measurements of the rectangles formed by the columns use the ratio of 1: Ф The Golden Proportion or the ratio 1: Ф is found in classical architecture, such as the Parthenon. http://www.geom.uiuc.edu/~demo5337/s97b/parthen.gif

(This image taken from http://www.geom.uiuc.edu/~demo5337/s97b/art.htm)

Fibonacci’s observation of rabbit breeding unearthed much more than a seemingly simple numerical sequence. In fact, Phi and the Golden Proportion and Angle are found any and everywhere, and can be seen in most every aspect of daily life.