The Golden Ratio

Also known as phi, the Golden Ratio can be derived by determining the proportion at which dividing the whole equals the proportion between the smaller and larger segment that together make up the whole.

Algebraically, this can be describes as:
x + y = 1
|x| = y / x
x (and y) must be positive since they make up a whole.

Isolate the variable:
y = 1 - x
y = x^2

Combine & Solve:
1 - x = x^2
...
0 = x^2 + x - 1
...
x = 0.618

This ratio is also discoverable by considering the ratio of adjacent numbers in a sequence that continues by adding the two prior numbers, starting with 0 and 1, the Fibonacci Sequence: oeis.org/A000045

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393,...

As the sequence continues, the ratio of adjacent numbers approach 0.618.

Why stop there?

Taking the ratio of all numbers in the Sequence produce more ratios that I call Fibonacci Multiples, which approach Lucas Numbers: oeis.org/A000032

..., 0.000733, 0.00119, 0.00192, 0.00311, 0.00503, 0.00813, 0.0132, 0.0213, 0.0344, 0.0557, 0.0901, 0.1459, 0.236, 0.382, 0.618, 1 ,1.618, 2.618, 4.236, 6.854, 11.09, 17.94, 29.03, 46.98, 76.01, 122.99, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39603, 64079, 103682, 167761,...

Given the similar relationship, is interesting but ultimately unsurprising to note how the sum of all ratios below the Golden Ratio of 0.618 approach 1 and the beginning significant digits begin to resemble whole numbers found in the Fibonacci Sequence.

Understanding that growth and decay is relative, we should then expect to see these ratios appear when we look for it...