2. The golden ratio
The golden ratio is a very special number. Represented by the Greek letter phi (`phi`), its value is:
`phi = (1 + sqrt(5))/2 = 1.61803339887…`
Two numbers are said to be in the golden ratio if the ratio between them is the same as the ratio between their sum and the larger of the two numbers. In other words, for two numbers a and b where a > b > 0:
`a/b = (a + b)/a = phi`
Mathematicians have been fascinated by the golden ratio since the days of Euclid (around the 3rd century BC). Since then, countless artists and architects have incorporated the golden ratio in their work via the golden rectangle, which many people believe is the most aesthetically pleasing shape.
Fibonacci numbers and the golden ratio
The golden ratio is closely related to our old friend the Fibonacci sequence.
First, let’s refresh our memory on what the Fibonacci sequence looks like:
`1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …`
Now, let’s see what happens when we divide each Fibonacci number by the previous number in the series:
`1-:1 = 1`
`2-:1 = 2`
`3-:2 = 1.5`
`5-:3 = 1.666`
`8-:5 = 1.6`
`13-:8 = 1.625`
`21-:13 = 1.61538462`
`34-:21 = 1.61904761`
`55-:34 = 1.61764706`
`89-:55 = 1.61818181`
`144-:89 = 1.61797753`
As we progress through the sequence, the ratio between successive Fibonacci numbers is getting closer and closer to a particular value. As you might have guessed, this value is approximately 1.6180334, otherwise known as `(1 + sqrt(5))/2`, or the golden ratio!
By now you’ve probably also guessed that the ‘golden’ in the Golden b refers to the golden ratio as well. In the next section we’ll discover where the golden ratio is hiding in the Golden b shape.
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