How the golden ratio and the Fibonacci sequence are connected

Recall that the golden ratio is a special irrational number that is approximately equal to 1.618.  It appears frequently in geometry, art, and architecture. Also recall that the Fibonacci sequence is the series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 … Each number is found by adding up the two numbers before it.  For example 5 is found by adding 3 and 2. German Astronomer Johannes Kepler once wrote that “as 5 is to 8, so 8 is to 13, approximately, and as 8 is to 13, so 13 is to 21, approximately”.  What Kepler…