The Golden Ratio

In the previous section, we defined the logic of the Fibonacci sequence, used to produce a sequence of numbers, where each value in the sequence is equal to the sum of the previous 2. 

In this section, we will define the Golden Ratio, using the Fibonacci Sequence.

[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233]

As we continue to grow the Fibonacci sequence, adding sequential terms, each the sum of the prior two, we divide each new number by the prior value, and watch as a ratio emerges:

2/1 = 2

3/2 = 1.5

5/3 = 1.6666666....

8/5 = 1.6

13/8 = 1.625

21/13 = 1.615384615384615...

34/21 = 1.619047619047619...

55/34 = 1.617647058823529...

This process continues as the sequence grows larger, with each term being the sum of the previous two.

The Fibonacci sequence produces a pattern of numbers that occur everywhere in nature. As we ascend further and further along this sequence of numbers (as n approaches the infinite), we divide any point in the procession of the sequence, f(n), by the previous point, f(n-1), to discover the ratio of growth between two sequential numbers in our sequence and iterations of our logic.

As we progress further and further along the set of numbers produced by the Fibonacci sequence, we see a ratio emerge between sequential points in the procession. This Golden Ratio, given by Φ, the Greek letter Phi, is the factor of growth between f(n) and f(n+1). In other words:

      f(n) x Φ = f(n+1)     

This says that any number in the Fibonacci Sequence is roughly Φ times greater than the previous number in the sequence.

Where:
      Φ = Lim(n -> infinite)  f(n+1) / f(n)
      Φ = 1.6183399…

 As we can see above, the greater the value of n, the more closely Φ is approximated.

Both the Fibonacci sequence and the Golden Ratio occur everywhere in the universe, from the formation of galaxies to the growth of plants, and Φ occurs countless times in the human body. This sequence is the most basic way thoughts come to form more ideas.

These patterns have been observed, admired and replicated by many different generations and civilizations. The next section described the origins of the study of the Golden Ratio through its origins in geometry and discover how the Fibonacci Sequence produces the abundantly familiar shape of the Golden Sprial.

 

Thermodynamics, Number Theory and The Goilden Ratio
Creation, Evolution and the Golden Rule
Theory of Order
Why Fibonacci and Gibonacci sequences appear everywhere in nature,
and how simple combinatoric math can describe how a Universe with simple beginnings evolved into a complex form